Russell's Paradox - a simple explanation of a profound problem

[@nyc-exile]

My teacher told me that "all rules have exceptions" and I told her that that meant that there are rules that don't have exceptions. Because if "all rules have exceptions" is a rule then it must have an exception that contradicts it.

[@Goalsplus]

That's a good one.

[@harrymingelickr883]

Very True, However; it is a universal Constant that "He who has the Gold, Makes the Rules". Without Exception!!!

[@emaiden]

needed a minute to figure out how that worked haha

[@gehtdinichtsan309]

but if a rule has the exception of not having exceptions, it is still a rule with an exception; right?

[@OmniphonProductions]

​@@harrymingelickr883 _Except_ there are societies that have rules but don't have gold, _and_ there are societies that have rules but don't _value_ gold. 😉

[@louismartin4446]

I started reading Russel’s “the limits of the human mind” and I found out mine lasted one paragraph.

[@Oppenslimer]

Lmao

[@jhandle4196]

The more you know, the more you know what you don't

[@louismartin4446]

So true, I am a licensed engineer and have forgotten more about technical subjects than most people know- but education makes one a humble person- just knowing how much knowledge is out there of which 99.9% the average person does NOT know

[@jeanbriones1190]

@@louismartin4446 "of which 99.9% I don't know" you should have said. Why would the ignorance of other people make you humble?

[@CoolPigeon98]

​@@louismartin4446how did I know you were an engineer?

[@jonathanbenton2002]

Unlike many of your commenters, I don't have anything pithy to say about your presentation. I had never heard of Russell's Paradox or anyone else's Paradox. All I can do is tell you how much I appreciate how you described it. I did have to go back and review a couple of sections near the end, but I got it! You are passionate about sharing your knowledge with everyone who cares to learn. Even, and perhaps especially, people incarcerated in prisons. You are a gifted teacher, so thank you for sharing your knowledge with ALL of us.

[@directorstu]

Well said!!!

[@mcsuibhne005]

Boo urns. Have pithy on us!

[@squirrelbait2004]

Weirdly said!.... Now I'd like to learn why @janathonbeton2002 is in prison 🧐

[@jonathanbenton2002]

@@squirrelbait2004 I'm guessing it's because @sqrlbain2004 can't spell my name ...lol

[@theplinkerslodge6361]

@@squirrelbait2004 BC he belongs to that set.

[@alberthord9527]

For your consideration... 1. Math and language share more connections than those you have listed here. Among them, adding a negative connotation (such as the word "not") reverses the meaning of a statement in the same way making a number negative (such as multiplying by -1) reverses the value of that number. 2. Mathematicians used to commonly use physical models to prove and solve complex theories and equations, but I suspect those models were less useful (or at least more difficult to implement) when applied to less tangible problems (such as those involving negative values). 3. The paradox described in this video is not far removed from the paradox of taking the square root of a negative number. While the result cannot be expressed as any value along the number line, "i" is still useful in mathematics and physics. 4. When I have considered most common paradoxes, I have found that they cease to feel paradoxical once I obtain full understanding of the forces which create them. I, therefore, generally assume that a persistent paradox is evidence that there is more that Ido not fully understand. 5. Numbers are nouns: person/place/thing, specifically things. Yet they cannot be physically interacted with any more than you can measure out a cup of joy (a similar intangible thing). These intangible things are mental constructs which only "exist" conceptually. While they are not tangible, they are knowable. While they do not physically exist, they DO exist by virtue of having immutable characteristics. Changing the name we assign to these the conceptual constructs we know as numbers dose not alter their properties. As an example: 2 and 2 is 4. Dos y dos es cuatro. If the number, added to itself, does not equal 4, that number is not 2. Thus we see that language (which is mutable) is used to describe numerical values wich are immutable. Conclusion: Just as "i" is real but does not occupy a place on the number line, Just as the moon is a real place, but you cannot get there by driving north, south, east, or west, and Just asthe existence of "i" does not invalidate the number line, while the existence of the moon does not invalidate your atlas, The Russell paradox does not invalidate sets. Basic elementalism worked until Newton. Newton worked until Einstein. The presence of a paradox is an invitation to explore a wider realm of reality, tangible or otherwise.

[@SnafuReady]

And there are some rules that just aren’t true in life.. Easy example. #4 An army of 1 is not the same of an Army of 1 M. Facts

[@rachidvanheyningen]

Yep... and still captured by the limitations of human perception and the perspective of human limitation, as we all inherently are

[@SirKenchalot]

So to understand and grapple with i we invent a number-space instead of a number-line, adding an extra dimension in which geometry still functions as in other 2D spaces. So we need to find a new 'dimension' of set theory wherein this is no longer a paradox. I'm working on it...

[@KittchenSink]

For a 57 year old man who cannot even recite his times tables (my head just doesn't do maths), I'm stunned I actually followed that, I really did!! That speaks volumes about this guys ability to convey information. I applaud you Sir, especially for the ability to hold my attention for the entire video. I quite enjoyed that!! I've no idea what use it is to me personally, but it was fascinating!

[@GrantDayZA]

So are you really not good at maths or has it just been explained poorly to you in the past ...

[@KittchenSink]

@@GrantDayZA probably a bit of both swaying more toward poor teaching. Ive always been very good at art from being a kid. Don't get me wrong here I'm not saying love me love me I'm thick! I have a BA (Hons) degree in the social sciences, but honestly I've never been able to recite my times tables. If you fire one off I can tell you the answer eg. 7x8 or 9x6 etc. I just can't recite the whole tables they way your taught to more or less sing them if you get my meaning. I got by for a while but when they got to algebra and sticking letters in that was it, I just lost the plot and switched off and had a giggle instead. I enjoyed Pythagoras like working out areas was easy, and the simple letter stuff like 3 × X = 9, but when those equations got a bit serious my head just switched off!! In retrospect, I wish I had a maths head as now theres so much Id like to ask questions on but feel I cannot explain myself because I'll look stupid. I'm absolutely fascinated by Jeremy Strides math on Coral Castle, but I'm lost when he talks about prime sets etc. Anyways, I'm waffling now. But yeah, I just don't have a brain that handles maths, but.... We can't have everything can we!! Gotta work within your limitations so I'll keep on trying lol. Have a good day

[@profjeffreykaplan]

What a lovely comment for me to read! You've made my day. Thank you!

[@nichebundles7246]

I was thinking pretty much the same thing. I've often told people that I am "math stupid", and blunder through anything that involves math. Jeffery's presentation was both captivating and inarguable. At least I think it was - LOL. But you shouldn't take my word for it; I'm math stupid. BTW - I've never watched any of Jeffery's videos before today. He reminds me of a mix of both Sheldon and Leonard from the Big Bang Theory. "Sheldon and Leonard" is a set. Heh. See? I learned something. :D

[@AlejandroMoralesAraya]

@@nichebundles7246 I did took set theory in the university explained by a very good professor and I have to say that the way Jeffrey was able to explain all of this in only half hour, while keeping me focused (maybe helped by the use of LeBron and Garfield) was just flawless! PS. I was also looking at Sheldon's eyes at some point of the video

[@alexander0the0gray]

I really didn’t expect LeBron James to be so crucial to the fundamentals of set theory. What a legend.

[@nim127]

4-time NBA champion LeBron James*

[@alexander0the0gray]

@@nim127 Oh yes, thank you. Sorry

[@jackthomas3483]

How does this effect his legacy?

[@inconnu4961]

@@jackthomas3483 It seems like is legacy is a set!

[@skierpage]

​​@@jackthomas3483affect. But big props for not misusing "impact" like everyone else. The effect of LeBron losing in the playoffs was it affected his legacy compared with Michael Jordan.

[@joshwah4838]

I asked my girlfriend if we could have sets and she told me no because I didn't contain myself.

[@drakesmith471]

Lol, I love the humor you two have.

[@andycopeland7051]

Niiiiiiiice

[@richardwatson5437]

You're under arrest.

[@dashockpixle4140]

I sat here wondering how to respond to such brilliance for like 5 minutes

[@atzuras]

- We already have sets at home. - At home: { }

[@ujjwalchoudhary-nh5ii]

when i am in a LeBron glazing competition and my opponent is jeffrey kaplan

[@sanidhyaspandit515]

Facts 😂

[@ericdarkgoat4050]

Drink each time he mentions Lebron

[@ThatGuillermo]

I hope he doesn't win any more titles or Jeff's gonna have to re-shoot so much of this video

[@aljazslemc9569]

​@ericdarkgoat4050 doesn't matter what you are drinking, playing this drinking game is medically inadvisable 😂

[@HxTurtle]

I speak German and understand the letter Russell wrote to his colleague. the level of confidence he put into his writing that his recipient will just understand him amazes me.

[@sourcererseven3858]

yeah, I can barely read 90% of that handwriting 🤣 Interestingly, depicted are only the first and last page of the letter and the actual paradoxon is not described on these pages (he adds the formulaic representation of "abovementioned contradiction" in the post scriptum, but that's it). Pity, I would've liked to read the original wording and I'm far too lazy to hunt down the letter myself 🤣

[@mdhasnain_3764]

@@sourcererseven3858 what did he say in the letter

[@erikwillems3016]

@@sourcererseven3858 Yeah, because google is such a chore. But it's cool as he is so respectful and still shoots a hole in the theory with his paradox.

[@rauldoty]

It's important to keep in mind these men have dedicated their lives, decades, studying this. It's a form of language therefore shorthand is precise and expresses those years of knowledge.

[@timspiker]

It's just one of those random variables that got lucky. I wanted everyone in my class to side with me so we could together over rule the system. But I wasn't so lucky and I spend most days in detention.

[@conradolacerda]

The root issue is self-referencing, as noted by Douglas Hofstadter in his famous book "Gödel, Escher, Bach": any language that allows objects to make reference to themselves will contain a form of Russell's Paradox.

[@jameskelso839]

The one thing I would add to that, Conrad, is that as I said above, Language is not mathematics, and language can often be used to create a paradox, but, in truth, mathematics is an accounting not a linguistic description. Thus, the paradox is in our description not in fact.

[@conradolacerda]

@@jameskelso839 Indeed, in the case of set theory I think the paradox stems from trying to define a mathematical object ("set") out of nowhere by simply putting some words together, instead of having the concept arise bottom-up (which esentially is what the ZF scheme of axioms tries to do). But I also think that the issue is not limited to Mathematics; for instance, in Psychology, I notice a remarkable absence of a rigorous definition for "behavior" in the literature; I suspect that any attempt to define it via some sentence will be met with another instance of Russell's Paradox.

[@jameskelso839]

Well, I can't argue with that at all because behavior is an individual trait that no two persons have in common and there are those in my past who have convinced me that they have no idea what good or bad behavior is. I will absolutely say that I have studied mathematics a lot, and Cantors rules a widely used and there is literally no evidence of them ever failing in a calculation. No matter what any philosopher says about Russell the only thing that guy did was send a brilliant mathematician to therapy because he caused him to doubt his life work of furthering Mathematics. I have 4 terms of calculus behind me, and set theory works just the way Cantor set them out.@@conradolacerda

[@hckytwn3192]

@@jameskelso839 Don't think that's correct. Gödel and Tarski proved Russel's Paradox holds for math and all formal systems. Math can never be both consistent and complete, nor can it prove the truth of itself.

[@colinmunro2632]

If physics one day becomes complete, each event and each entity in the universe will have a one to one correspondence with a mathematical entity. But the description of the universe is contained within the universe, so it is self-referential, and hence results in the paradox which allows for events we cannot explain.

[@TheJuice92]

I have no interest in math and somehow i just watched this whole 28 minute video on something ill never use. Youre a great content creator, bravo.

[@PureGlide]

It is important to remember: if guns don't kill people, people kill people, then toast doesn't toast toast, toast toasts toast.

[@DN19756]

20 minutes into the video before I realized you were writing backward. Impressive, and thank you for teaching me something new.

[@garybrisebois2667]

Interesting observation! Or he might have been recording using the front view camera, which would reverse the image...?

[@HowardMorland]

@@garybrisebois2667 I think he is right handed and writing forward. The video image is flipped horizontally.

[@samsam7648]

@@HowardMorland definitely left handed. Think about it a bit longer

[@jcrackcorn]

@@samsam7648you are definitely wrong, think about it a bit longer

[@pragmatickaos852]

It's amazing how some people choose the most unlikely explanation instead of the most simplest. "YoU weRe WriTiNg bAcKwArDs WoW!" *facepalm* No, he's writing normally and the video is flipped horizontally.

[@Astronomadical]

Never thought I could have such an enjoyable time watching a 30 min video on advanced mathematical theory. I chuckled and even laughed multiple times. Well done sir

[@Skyscraper21]

same here

[@lindascanlan6317]

I as well and I'm math phobic.

[@davidmaxwaterman]

Wow, it was that long? I didn't even notice.

[@mertkaaner1735]

Advanced?

[@qwertyflags]

I couldn't help but laughing every time he said "LeBron James, 4 time..."

[@KaiserAegis]

As soon as you got to explaining to the paradox, I knew exactly what the issue was because it's conceptually identical to several other paradoxes I've studied, including the Liar's Paradox and the Grandfather Paradox. I've noticed that this sort of problem tends to arise in almost any kind of abstract, self-referential system, if you dig deep enough.

[@choamlockstep]

I solved the paradox to my satisfaction by simply pretending we lived in a quantum universe which has settled down into three dimensions (three mutually perpendicular straight lines whose point of intersection is a single point whose most basic characteristic is extension, which is the lie we define by giving it points connected in a straight line and which is a set filled with all measurable elements or points; it isn't stable and wobbles around a bit, and the "barber" all the while exists in an indeterminate number of point states until it is measured and comes into being when it actuates the 3D point at which the three lines of points become one point from an infinite set of points...the geometric approach...

[@choamlockstep]

thank you!

[@DejayClayton]

​@@choamlockstep I have the same concept, except that I consider datums of existence to be measured in dimensional probabilities, instead of explicitly naming them as quantum. Those two ideas may be the same or not; I haven't thought about it enough. However, I do agree that datums "settle down" into specific dimensions only when required to maintain probabilistic relationships with other datums. Thus, measuring particles within a 4D domain can cause those particles to seemingly pop in and out of existence, when in fact their presence within a particular point in the 3D domain only occurs when absolutely necessary. Perhaps this relates to quantum phenomena because observation seems to be a key factor in "forcing" the probabilistic relationships to adhere to the constraints imposed by observation?

[@muzza_r]

Well done you

[@mainbka]

yep, it's like an infinite logic loop without a break condition.

[@GarrettBryan-q7l]

Being told I don't have to remember certain things is surprisingly comforting. What a wonderful video. I truly enjoyed your presentation of Russell’s paradox..

[@BiffTech05]

Wait, am I supposed to remember that?

[@Hokie200proof]

This is the kind of shit where Wittgenstein was like "yea, turns out Philosophy, my entire life's work, is just a language game and, in the end and very much like a game, it can be fun, frustrating, challenging, but ultimately meaningless." ... and everyone booed him and immediately went back to playing their own games, trying to prove him wrong by proving him right

[@michaelc7955]

There is a way that seems right to a man , but they can be the ways of death. Proverbs 14 : 12 - 16 . A simple man believes anything, but a prudent man gives thought to his steps. Proverbs 14 : 15 .

[@kberken]

I was a math major a million years ago. I wish you were one of my math profs! You are a great teacher!

[@TheButlerNZ]

Looking at that 1st sentence, I guess the bar for being a math major is set fairly low... q8D

[@notelonmusk690]

I love the bridge between the linguistics and mathematics. I too believe that math is a branch of logic and that there are many parallels between language and math. Great Video!

[@ishmaelforester9825]

I would say there is very little in aim and intention. The correspondence is obvious, but a mathematical language sounds like a mathematical romance, dull and ugly. Stay in your own field, and do well by your own harvest and herd.

[@jaxsazerac4904]

Well you kind of need language to explain math to someone. That includes computers.

[@PaulBurnsed]

9:59 10:00 10:00 10:01 true

[@kensurrency2564]

language and mathematics are symbols. symbols represent things or ideas. we use symbols to communicate ideas to each other. so they are both imperfect symbols. we’re not perfect beings, having not discovered the perfect language with which to communicate. i believe bertrand found a glitch in our matrix. but: it’s ok. the matrix still works …

[@bluevalentine2009]

When I was about 7 years old. I came across a calculus text book and opened it. Of course I did not understand it, but after looking through it, I said to my dad- dad that looks like someone thought up their own math and wrote it down. I'm 46 so I noticed this argument 39 years ago. Imaginary numbers. It's all you need to know to realize math starts as an idea before it becomes math (differential equations is another example). The only math is 2+2 =4. All other "math" was an idea before it was math. They can certainly be mutually exclusive- math and language- but they are also undoubtedly connected. Astronomy is where we see the biggest "man-made math" in my opinion. Certain terms, the best example being parsecs, is a made up term using made up measurements. However, lastly, all math at one point was "invented" one day in the past. One day in the past someone said a circle is 360 degrees and we all said -okey doke.

[@jimstack2863]

At my age (77), I am not going to wade through 18,643 comments to check if someone else has made the same comment as I am making here! I apologise in advance, however, if that is, in fact, the case. When I first came across Russell's Paradox, more than 50 years ago, I explained it to myself as follows: if A is a set, then A is not the same thing as {A}, the set containing A. A set, in short, cannot be a member of itself, and the Paradox arises because the erroneous assumption is being made that a set can be a member of itself - your Rule 11. On the few occasions in the last 50 years when I have thought about this again, I have come to the same conclusion. I concur with the other comments about the quality of your presentation. Well done!

[@sebasforest963]

Upon watching this (very good) video, I immediately thought this paradox might be caused by a double negation effect... but your idea seems to solve it in a satisfactory matter. A is not {A} and thus {A} is not {{A}}.

[@IanGorman-v5d]

As more of a physicist than a mathematician I have always held that there are no exceptions to a rule. If an exception is encountered then the 'rule' is not a rule and the 'rule' requires modification such that the exception no longer exists under the modified rule. Set rule 11 is at fault as you have identified. Think of it spatially - set A has a boundary as it 'contains', and the set that 'contains' set A has a second boundary around Set A and is spatially different from Set A - therefore a set cannot contain itself.

[@MrRoguetech]

Yes, rule 11 is prima facia absurd. If our set is {A}, and "itself", then itself is {A,{A}}, but then becomes {A,{A,{A}}}, etc. Any set that contains itself is automatically an infinite set, even if the first set is null. It makes no sense that you could "create" a set that includes what is being created. The set doesn't exist until you create it, so you can't very well include it within itself (except something like "a set of all things that don't yet exist", in which case... No sets exist that contain itself.)

[@TiwsDay]

@@IanGorman-v5d But not all sets are physical and as such concerned with boundaries. The set of all sets contains itself surely? Simply ask: Is the set of all sets a set?

[@MrRoguetech]

​​@dereksmith5934 No set is physical. Hence the question. If sets were physical, then we could just measure them to see if any contain themself, or attempt to manufacture one. (And until then, the only correct answer would be "We don't know.") A set that contains all sets does not contain itself, because it only contains all the sets that existed up until you created that set. You could then say "a set that contains all sets that ever will exist", but... If you make a rule that says it can't contain itself, then it can't contain itself. Sets are sort of an Aristotelian concept - they exist in the aether, and are a perfect example of themselves. So the question is what is more perfect - a logically contradictory set that is infinitely regressive (ie "paradoxical"), or one that can not contain itself? Totally up to you, unless of course you want to complain about your own answer like Jeffrey. You want your set of all sets to contain your set of all sets as well as all sets... doesn't make sense to me, but whatever. It's your set, you do what you want with it. edit: But we could extend the question to all things, not just sets of things. Can the boundary of a thing be within the thing? Something like an apple, the answer doesn't matter. But with things like time, space, and universes, it starts to matter. Do the questions "What happened before the beginning of time," and "What is beyond space," make sense? If not, boundaries are not inside of the thing (and the answers are "the boundary is the boundary").

[@warpetrel]

What's more mind boggling is the effortless backwards writing

[@jasonbrown678]

If only there was no world in which every single video recording application didn’t include a flip option,

[@theredpandaman1]

@@jasonbrown678oh thank god, I thought he was actually left handed for a second

[@DonBrandt]

Nice Try, he records backwards here.

[@fisheye42]

This “facing me” writing was distracting. My brain kept flipping it back to backwards, like a double-negative, so I could barely read it. (I can read and write backwards, but that’s just a single flip). -3M TA3

[@adebowalemichelgranger-mcc4228]

It's a photographic trick.!

[@jessicaoverthinks]

Honestly, there's a lot beyond my understanding. So it was weirdly reassuring to hear about the genius guy whose brain just straight-up blue screened because of this paradox.

[@keatodiet]

How could it be reassuring? Because if set theory was “beyond his understanding” then something tells me this dude is not gonna be hospitalized over reading a letter he doesn’t understand.

[@dirk143]

Fun fact: basic set theory was part of Mathematics education in elemetary school in germany of the 1970s (and not a small one) . My estimate is, this forever reduced Germanys BIP by 2-3%.

[@jockbw]

Your honest comment is genuinely reassuring, because weirdly you solved the paradox. If there are not things beyond understanding, then the concept of understanding itself becomes nonsensical. The narrative construct of reality is a instance of mathematical induction, moving from the known to the unknown. Reality is chaos and the unknown, determinism is an emergent property of the process of understanding. So if you take away hope and possibility (which resides in the unknown) you take the life out of reason and the reason out of life. Fcvk I just blue-screened myself 😂

[@jacobwiren8142]

Your head was designed with paradox-absorbing crumple zones.

[@jockbw]

@@jacobwiren8142 it’s like the “designed-to-be-dropped” cartilage system never disappears. It merely adapts to be hit you take.

[@brynbstn]

It's the same as the liar paradox, "this sentence is false". Whenever you allow self-reference in a logical system (where true/false are the expected outcomes), you enable the paradox. "Sentences can refer to themselves", "Sets can contain themselves", "Predicates can refer to themselves" - - these are all equivalent, and all problematic in a T/F logical system. The solution, as Russell and others proposed, is to not allow self-reference (or self-containment), which makes sense because self-reference creates endless loops for the T/F evaluation, as you aptly demonstrated. The solution is to show how self-reference, though feasible semantically, isn't logically valid in subject/object relationships - to get into that is beyond the scope of this comment. Another solution is to allow self-reference but to specifically handle endless loops as "undefined", i.e. have 3 possible outcomes: T, F, Undefined. Great video but I disagree if you have hit on anything new using Predicates.

[@thenonsequitur]

He wasn't claiming he hit on something new by showing predication is structurally equivalent to set theory. He was claiming that predication is how we naturally think. And that the paradox arises from the way we naturally use predication in language (and maybe in thought), and can't be solved way by saying "self-reference isn't allowed". Because in language, self-reference is allowed. And the rules of language are observations, not rules we can change.

[@brynbstn]

@@thenonsequitur Yes, Jeffrey's proposal is that the problem is real because it's noticeable in real language - the use of predicates, which is a new twist on the problem. However the problem with predicates is no more real than the problem with "this sentence is false", which has been around for a long time, and which he didn't bring up in the video, oddly... The video is initially about logic. When you bring up logical problems in language, you need to address the relationship between logic and language... what's valid/acceptable in one system is not necessarily so for the other. "This sentence is false" is a misconstruction (in logic) or a syntactic curiosity (in linguistics).

[@stolensentience]

@@brynbstn yeah this video, while interesting, felt anti climactic for me. He seemed to leave out some important bits to make it as mind blowing as it seems to be for him

[@Supe063]

@darren collings ... and like a paradox, you can tie up more than one boat

[@brynbstn]

Here's an example to clarify that the relationship between language and logic is not fully correspondent: "Four plus nine is one". This a completely valid statement, syntically. Is it correct? "No" most would say because 4+9=13. However it depends on the mathematical domain you are using. You assumed the standard number line. What if we were talking about the numbers on a clock? The point here is that, the logical domain behind a statement should not be assumed. It has to be defined. The syntax tells you nothing of the logical system under question. A valid syntax does not necessitate logical validity.

[@PeterDivine]

On the predicate paradox: The main issue you seem to be grappling with on this is functionally comparable to the old, simpler paradox: "This sentence is false." If it's false, it's true; if it's true, it's false. So which could it be? The most descriptively accurate answer I can think of is that it is neither, because it has no constant referential point upon which to base its definition. What can the sentence even proffer within it as "false"? What truth is it trying to debunk? None, because no such truth was extrapolated. Its only point of reference is itself, but it _ipso facto_ eliminates that point by labeling it false, thus leaving it a useless self-contradictory abstraction, vacuous of point, logic, sense or reason. And keep in mind that for definitions literary or otherwise, _constant_ referential points are not to be underestimated in their essentiality. Without them, the means to describe them become variable and generalized to the point of uselessness. Consider, for example, the set that contains all sets, [X]. Okay- does that set include itself, [X] + [X+1]? Does it include that set, as well, [X] + [X + 1] + [X+2]? You'd have to keep on reiterating the addition of the set within itself ad infinitum, but doing so leaves you with an infinitely escalating value - and if your set contains an infinite value, can you really say you have a definition for it, considering the whole point of these sets was as a means to define whole numbers and now you have to find a single whole number for a sigma function? This doesn't mean that math is broken, it only means that generalized categorizations give naive (heh) interpretations of mathematics that don't hold up without much greater scrutiny. If Zeno can be wrong about his ideas on motion being an illusion and Euclid can be wrong on his ideas of geometry, so can some professors be wrong about their ideas on sets. Nobody ever said this math stuff was easy, unless they did, in which case they can file under [set x: x contains all people who are shameless liars.]

[@hespa8801]

just wow.!!!!, you just broke the iteration of this amazing professor I would love to see you do it in a video as good as this one.

[@anthonydanielgann1]

it's just a play on words, sort of like values@@hespa8801

[@J-YouTube324]

Nice description. Sometimes you can string together words that look cool (this sentence is false) but in reality are just silly words that end up being meaningless or incoherent, logically useless. Words twisted back on themselves.

[@alsam4678]

Maybe sets should be accounted for through the passage of time. It seems like their also may only have meaning for our minds which is subject to time.

[@birb1947]

I'm learning basic programming (C#) so your comment reminded me of something adjacent. If you define some object such as a string to be some... well... string of letters, that is fine, it can even be a pre-existing string (analogous to sets containing other sets), but you cannot fundamentally define the string object to be itself, as 'itself' doesn't exist yet, it can be defined as null, but cannot be defined as itself or any variation on itself. I find this quite interesting, as this paradox appears to subtly arise even before the introduction of the "this sentence is false" style paradox. It seems that it is ok to say "this sentence is true" because the action of declaring it doesn't invalidate its 'inital state'; the sentence agrees with itself. conversely, the paradox "this sentence is false" invalidates its initial state as it doesn't agree with its own definition. But the problem with both of these sentences is that they are evaluations on a sentence that is still being constructed. It is fine to define a statement that is altered by a different statement, like defining "bool A = true" followed by "A = false" to change its state, but saying that "bool A = A" or "bool A = !A", analogous to saying "this statement is true" and "this statement is false" respectively, is impossible. Writing this now, I am just realising that you can extrapolate this to compound sentences. "this sentence is a statement, and that statement is false" is allowed (afaik) because the statement has been defined in the former half of the compound sentence, and has been then made false by the latter, which is fundamentally disconnected from the former. You can also say "this sentence is a statement, and it is false", but this opens two possible interpretations. Is the "this sentence" false, or is the statement false? I put this down to the vague nature of the sentence itself, but I'm not sure. If I had to guess, this intepretation suggests that a set that contains all sets that contain themselves (I'll define as V) must not contain itself, as the set constructor logic for V cannot have V as an input as it hasn't been completely defined according to its conditions. This (I assume) would hold for any set constructor that, upon full compilation, satisfies its own conditions.

[@stevewithnell911]

I have two engineering degrees and didn't come across sets until my 5 year old daughter needed help with her homework...and last week my wife asked if I understood conjugated verbs which I didn't (she volunteers in infant school, where the topic arose). If I had have known about those two things, my career path and performance would not have changed. It is satisfying to learn about sets and conjugated verbs, and I feel happier in myself now. Make of that, what you will!

[@dukeon]

This problem of self reference, infinite recursion, strange loops, or whatever one chooses to call it comes up again and again. Gödel’s incompleteness theorem is essentially another form of it, Hofstadter has made a career writing about it, and classical philosophers knew all about it and expressed it in many ways that we might boil down to the Liar’s Paradox or the most efficient form, “this statement is false”. They’re all logically-topologically equivalent. Good presentation for lay people, I like your channel and have subscribed. Going to check out your other videos. Cheers.

[@D3sdinova]

Yes, but no, but yes, but no, but yes, but no adinfinitum.

[@BoxdHound]

It strikes me as analogous to dividing by zero.

[@shirgall]

@@BoxdHound It has a certain undefined quality about it.

[@janjezek9913]

You can also create a problem of undecidability. You can have statement "this statement is true". It can be proved to be true and it can be proved to be false. If you put the statement into a set with some true statements and then you say "all the statements in the set are true" then you have another undecidable statement.

[@FreedomToRoam86]

My answer to the learner asking these questions would be to go dig a hole. Then another hole. And then another. And keep doing that until he figures out it is stupid to dig holes just to dig holes, and stops.

[@Numberonecanbeater]

You took 30 min to explain something that my prof couldn’t explain in 2 hrs, lifesaver 🙏

[@toddallan7086]

Literally be explained in 5 mins its not that mind blowing this bored me hard

[@tognah6918]

​​@@toddallan7086 okay todd allan

[@WalterLiddy]

This despite the fact he actually over-explains at almost every point along the way.

[@twilight7457]

No, he didn't.

[@paromita_ghosh]

@@tognah6918 lmao

[@jonathanbakalarz7786]

As a mathematically challenged person, this video made my head hurt and I had to rewatch several parts, several times. But, I understood much, perhaps even most, of what you said, and must agree with the other comments before mine, that you are a gifted teacher. Thank you for sharing this.

[@edwardlulofs444]

My brain ONLY works mathematically. So, of course, I have trouble relating to most of the world. So, in that sense, you are lucky to be able to relate to so many people.

[@Pepespizzeria1]

The fact you want to learn even though you're not mathematically minded is awesome in of itself

[@edwardlulofs444]

@@Pepespizzeria1 Anyone can learn from a great teacher. Even a genius can fail with a poor teacher. Compared to other nation I think that the USA ranks 47th below the country with the best educational system. Finland has been rated number 1 for many years. USA spends $800 billion a year on defense to have the best military on the planet. I guess the military is what America values the most. It's NOT you, it's currently the teachers.

[@edwardlulofs444]

@@Pepespizzeria1: Good attitude. I bet that it's not Jonathan Bakalarz's fault that he feels that way.

[@armisis]

1. Contradictions as Proof of Totality Contradictions, rather than being seen as problematic, might actually be indicators of a more expansive, infinite reality. This viewpoint aligns with certain philosophical and metaphysical traditions that view totality (or infinity) as something that transcends the binary nature of formal logic. In this view: Contradictions aren't errors or failures of a system but rather signals that the system is confronting a more profound, complex reality. In an infinite system or totality, all possibilities are included-even those that might seem mutually exclusive or contradictory in a finite, linear framework. From this perspective, contradictions reflect a greater truth about reality's non-dualistic nature, where opposites like "true" and "false" or "finite" and "infinite" might coexist, not as errors but as part of a more complete picture of existence. 2. Paradox as Expansion, Not Limitation Instead of seeing paradox as a limitation of understanding, you argue that paradox is an expansion of understanding. It points to places where our current frameworks, rules, and systems break down, not because they are wrong, but because they are incomplete for dealing with the full scope of reality or totality. This is especially relevant when we attempt to deal with infinity, the ultimate concept of totality, which often leads us into paradoxical thinking. For example: Russell’s Paradox may not just be a limitation of naive set theory; it could be interpreted as a signal that our traditional logical frameworks aren't broad enough to handle the concept of totality. Quantum mechanics, where particles can exist in superpositions of states (seemingly contradictory by classical logic), could be seen as a physical example of reality’s non-binary nature. What appears paradoxical in classical thinking opens up new fields of exploration and deeper understanding in quantum theory. In this way, contradictions serve as a call to exploration-they tell us that the system we're using isn't wide enough to capture the true scope of reality and that we must expand our thinking. 3. Non-Duality and Infinite Reality Philosophically, the idea that infinity includes contradictions resonates with non-dualistic traditions, where reality is seen as a unity beyond the opposites of logic and contradiction. In non-dualistic metaphysics: Infinity is not simply "endless" but also "all-encompassing" in a way that includes both being and non-being, truth and falsehood, and even paradox itself. Paradoxes, therefore, are part of the infinite nature of reality because infinity doesn’t abide by the strict divisions of human logic. It is beyond "either/or" thinking and embraces "both/and" realities. From this viewpoint, contradictions or paradoxes aren't failures to be resolved-they are natural aspects of totality. To truly understand infinity, one must be open to the possibility that it transcends logical consistency and that paradox is an inherent part of the way the infinite unfolds. 4. Contradictions in Mathematics and Logic Even within formal systems, contradictions can sometimes push boundaries forward, expanding the system in new directions: Gödel’s Incompleteness Theorems, for example, show that in any sufficiently complex system, there will always be statements that are true but unprovable within the system. This seems paradoxical, but it reveals something deeper: the limitations of formal systems to ever fully contain or describe themselves. In non-Euclidean geometry, for centuries, Euclidean geometry was thought to be the only "consistent" way to understand space. But when mathematicians explored the consequences of altering Euclid’s parallel postulate, they encountered seeming contradictions with classical geometry. This contradiction wasn't a dead-end-it led to the creation of entirely new geometries that expanded our understanding of space and led to general relativity, which describes the curvature of spacetime itself. Thus, contradictions don't necessarily signal failure. They can lead to the creation of new frameworks, expanding our understanding of what is possible. They serve as guides to exploring uncharted territory in knowledge. 5. The Infinite and the Incompleteness of Systems If contradictions and paradoxes are seen as inherent to infinite systems, they also suggest that no single system can capture the totality of infinity. This idea is central to Gödel’s work, as mentioned earlier, and speaks to a larger truth about infinity: Infinity cannot be fully contained within any finite framework, whether that’s logic, mathematics, or physical theory. Any system that tries to describe totality will necessarily have limitations, and paradoxes are the points where these limitations become apparent. Rather than being limits on understanding, paradoxes may be evidence that the system is engaging with something larger than itself-something that can’t be fully comprehended by that system alone. 6. Contradiction as a Gateway to Deeper Understanding In summary, you're proposing that contradictions are not flaws but evidence of totality, of a system’s engagement with infinity. They point us toward the infinite, all-encompassing nature of reality, where binary logic breaks down and a more expansive form of understanding is required. This is not a limitation but an invitation to deeper exploration. In this view: Contradictions don’t limit our understanding-they expand it, by pushing us to think beyond conventional frameworks and explore new possibilities. Infinity includes paradox because it transcends the limitations of dualistic thinking. To engage with infinity is to engage with paradox, not to resolve it but to embrace it as part of a larger whole. Thus, contradictions and paradoxes are proof of the infinite-not because they need to be resolved, but because they reflect the expansive, all-encompassing nature of totality. They are not limits of understanding but opportunities for transcendence and expansion of thought.

[@XionEternum]

This reminds me of the "failure paradox" as well. In a nutshell; if one sets out with the goal to fail, then they can only succeed. Because if they fail then they succeeded at failing which invalidates the failure, but if they fail at failing then they succeeded at failing which is still a success.

[@MCart1215]

@@vanceoz4080the goal is just to fail - I don’t think the paradox works when you assign it to something

[@kayembejuanlwaba1013]

❤

[@linsqopiring6816]

Not true. For example if I want to fail on a math test I can just not answer any of the questions and with certainty I will fail the math test.

[@MCart1215]

@@linsqopiring6816 it doesn’t work when you apply it to another action.

[@linsqopiring6816]

But Louie Depalma from the show Taxi did have a similar theorem which proved to be valid. Bonus points to anyone who knows what I"m talking about.

[@bevo65]

You have a gift. I actually understood this whole thing. I wish I'd had you teaching all of my philosophy and math classes in college.

[@herbertlong3981]

I wish he would endeavor to explain why Zenos Paradox isnt really a paradox, because it surely is not. That would make an interesting video

[@Wasko2]

If you understood it, you would realize that you can't really understand it ;)

[@thestatusjoe9949]

Great video. I think it perfectly illustrates the fundamental flaws inherent to viewing language as a logical system. Even Wittgenstein, once considered the greatest champion of linguistic logic, decided later in his life to abandon that path. Language is not and never will be logical, because the purpose of language is not description but communication. All language is at its core more concerned with forming connections and being useful than with being accurate to reality

[@rainey82]

I was sifting through the comments because he never restated is original claim, “math is a human construct”? You put words to my thoughts. Math is objective, and any apparent paradox speaks to the limitations of our tools, namely language and human thought.

[@acvarthered]

I must disagree. The purpose of language is not communication. The purpose of language is deception. There is no reason for language to be as complex as it is if it is designed to communicate ideas. It only needs the complexity we see if its goal is to deceive.

[@acvarthered]

@@rainey82 NO! Math is not objective! Math is completely subject to human minds. If human minds (or some other complex mind) do not exist then math does not exist. Being dependent on a complex mind by definition makes it subjective. You clearly do not understand the definitions of objective and subjective.

[@rainey82]

@@acvarthered Fair enough. Math is a language. Our understanding of the universe, described through math, can be complete or incomplete. The principles of the objective universe are objective.

[@samosapikora9169]

@@acvarthered So did you just communicate or deceive?

[@Aditya-f8t5z]

Russell's Paradox is nice.....because it draws u in with simple words....and then appears to reveal an amazing paradox....that can be avoided in my opinion..... As in the problem is not in the paradox...it's in shaping the argument....the manner in which sets are designed....they can be avoided by using some more foundational statements.... For a set to exist, the ability to define a set must exist....as in the concept a of a null set is quite intriguing....a set that has nothing....however, it is a set....so we already have implicitly assumed that there is "something" that we call a set which is different from nothing - that is the first idea - and then we put "somethings" into this original differentiator....As in, before the empty set is created...there is something even more foundational....nothing....THEN comes the idea of a set, or rather a null set.....then comes the idea of non-empty sets..... But a nice paradox nonetheless............🥰

[@mrosskne]

Work on your rigor.

[@Aditya-f8t5z]

@@mrosskne No.......🥰🥰🥰🥰🥰🥰🥰 I work on what I want.....🥰🥰🥰🥰🥰🥰

[@Aditya-f8t5z]

@@mrosskne Roses, white roses at my funeral.....I want white roses at my funeral.....🥰🥰🥰🥰🥰🥰🥰🥰🥰

[@Aditya-f8t5z]

@@mrosskne There was time when I thought Bertrand Russell worded the Paradox in the way he did to bait the German mind.....I don't know why I thought that way....interesting thought nonetheless.....😊 for me that is......😊

[@mrosskne]

Your funeral.

[@Goosebone]

Thank you for a perfectly clear explanation. I am in my 50's now and value it so much when I can find a concise and intelligent explanation for something I should have learned a long, long time ago but just didn’t. It's like scratching a 30 year itch. Subscribed. Appreciated.

[@AJ-yw7hf]

However, this video's explanation was *Not* concise, and it's *Not* something you should've learned a long, long time ago. :)

[@timhuson4800]

Agreed on the quality of the presentation. Though I've studied this problem, Kaplan's presentation brought some aspects into focus. But the real questions raised have not been discussed yet, such as the relationship between set theory and ordinary language, in general and on the point of this paradox. Kaplan says "just saying things about things." But that's exactly what happened in the video itself, it was all "saying things about things." So what is the difference? Needs to be a sequel, I guess there is.

[@copasetic87]

Your passion for teaching is exemplary. If half the teachers around the world had this type of energy and devotion, students would stay riveted. This is what we need. Also, if I name my cat "Is a cat," Then "Is a cat" is a cat!

[@maltflesh]

thats a set of sets

[@Patrik6920]

..well.. the really fun part it isent at all a paradox... its a problem all/most programmers has solved multiple times... the destinction between data-(sets) and meta-data-(sets)... its where Le Browns confusion comes from... ex {x: x data sets} X can be say shoe sizes, and the data for X is 30, 31, 32 etc... the metadata X is a description for the data X nothing more.. hence no paradox... say we make a dataset of cat colours, the colours are gray, white ... the meta data is cat colours, and the data is gray, white etc the meta data is obviously never a part of any set, but the set can point to itself and it creates no paradox Cat colours: gray, white, {Cat Colours} aka as self reference wich is used in objective programming we can also use it to create collections of sets thats contains references to themself or other sets that contains referenses to it (aka Cycles in programming) without creating any paradoxes... ex Variable:INT A, B, C Dataset:OBJECT-ARRAY AA:OBJECT, BB:INT, CC:FLOAT Data A:1, B:2, C:3 Data AA:AA, BB, CC ( This dataset contains a cycle 'AA' ) Data BB:A, B, C Data CC: 1.0, 1.5, 2.3

[@henriquepereira2811]

No, because in that case "is a cat" is not a predicate, it's a noun.

[@SlimboyFatcat]

You're brilliant! The cat named "is a cat" is a member of the set of all cats. Therefore, "is a cat" is an element contained within the set {x : x is a cat} ...Kudos, @copasetic87

[@DangerDthatsMi]

we actually need all the teachers to be like that. And they should be. And many many many actually are already. lmao "is a cat" is not a cat!

[@JP-hj2gs]

You’re a phenomenal instructor/teacher! I took basic college algebra just because it was a requirement for my Bachelors… I have always hated math. The way you explained this was EXCELLENT and I totally understood it.

[@AddilynneLastname]

‼️‼️‼️i LOVE how this inadvertently PROVES mathematical factionalism when that was the EXACT OPPOSITE of its intent, *CHEFS KISS*, A M A Z I N G, he invented a construct because he couldn't cope with the fact that numbers are constructed and not a natural thing, then proved the invalidity of fictional mathematical constructions, thus proving further that numbers aren't natural and proving himself wrong while lacking all self awareness and thinking he's proving himself right, THIS IS HILARIOUS, and all because he couldn't cope with the truth, LOL AF AF, as a linguist DO NOT drag language into this, WE KNOW language is constructed, its you mathematicians that won't admit math is constructed, we DO NOT have the same narcissistic outlook, we DO NOT lack self awareness in the way yall do, language is not infallible and neither is math, there is no paradox, the paradox only exists when you assume infallibility that is not there, wordplay is lowest form of poetry, and all binaries are false, sets, predicates, etc, are not required to be true or false, indeterminance is the answer to these edge cases, youre assuming your mathematical outlook is natural or true and in doing so you create the issue, once you've released that and let go of that incorrect mindset the actual answer becomes clear, it is indeterminable, B E A U T I F U L, when those who seek infallibility realize our very tools for determining truth themselves are not infallible and therefore no infallibility about anything can be inferenced or logically constructed, welcome to to thinking people's club, it's all complicated, all the time, welcome

[@werner134897]

Btw, I like the way you explained it very much. Very clear arguments. I wish all my fundmaental and logic math classes were given this clearly.

[@koboDresden]

I haven't read much philosophy and even that was a long time ago, but I remember Plato going on about the skill philosophers need to practice - abstraction, maybe? Well, he didn't use that word, it was something about learning to see the idea behind the things. The point is that it is a skill to be practiced. So maybe professors don't always explain the best way possible, but you gained the skill anyway - which is useful, since it is rare that math comes the most consumable way possible. My logic prof was also quite dry btw. Well, I rambled, sry.

[@scottmitchell2757]

This paradox is one of many paradoxs in a set of known paradoxes. That make up the set of all paradoxes.

[@Niffunn]

Wich in itself is a paradox lmao

[@scottmitchell2757]

@@Niffunn unless you can include it in the set of paradoxes. Which then becomes it's own paradox of a set of paradoxes of that paradox..

[@ronjones-6977]

My Paradox has the flight numbers 5.0/4.0/-4.0/0.0 and is unstable at higher speeds, but can still be stable when thrown with enough hyzer.

[@nishanttrivedi332]

You’re a brilliant teacher; it’s not an easy feat to deliver such an entertaining intro to set theory in several minutes! Looking forward to checking out your other vids. Thanks!

[@chandreshmehta2399]

Our shanti Mantra पूर्णमिदम् पूर्णमद:..... This is explained in only four lines. Google it.

[@PropPounderFPV]

How is he good? He is literally terrible in every way. He has no clue what he’s talking about and litterally talks way too much. He talks for the sake of talking in this video.

[@floggerrush5]

I remember learning this for the first time and it blowing my mind and all I could could think of is those Infinity mirrors. Where yeah the reflections that just seem to go on forever and all I can think of now is that once you create the set that contains all sets that do not contain themselves. You've created a new set that then goes into that set, which creates a new set of all sets that do not contain themselves and it becomes an infinitely recursive set ever expanding the number of new sets. I never saw it as a problem so much as a demonstration of infinity but my math teacher never liked that explanation.

[@marvin.marciano]

But couldn't this infinitely recursive set contain another infinitely recursive set equal to itself? Your point depends on premise that infinity cannot include itself, but it can.

[@coldshot1723]

As someone who has always sucked at math, I'm actually shocked that I pretty much understood everything you said.

[@Elisabeth19031978]

Me too, because he uses images like a basketballblayer and cats.

[@JimmyTheGreek2000]

So you do not suck in math ... you suck in logic !

[@miguelvaliente1475]

Then you are set.

[@newmankidman5763]

@coldshot1723, you did not suck at math. It was your teachers that sucked at teaching you. In high school, Albert Einstein had the same problem as you did, but fortunately for the World, he realised that it was his teacher that sucked at teaching, and not he at learning

[@dr.doolittle4763]

@@newmankidman5763 Absolutely, I fell into the same dilemma, having a shitty High school match teacher

[@kingfisher9553]

I have no interest in mathematics and no advanced training in mathematics, but i can follow the concepts --and more to the point - I love listening to characters who love what they do, and Jeff, you are a fascinating character. And that is a compliment.

[@Saujas]

The world of math appreciate it more than ppl who don't want to know what they are doing

[@ahgflyguy]

My subscription has been earned. My calculus I professor depicted this paradox without reference to Russell, using the words “autological” and “heterological”. He summed it up by saying that just because the answer to a yes or no question isn’t “yes” we can’t necessarily assume the answer is “no”.

[@dinomar7818]

Indeed, the answer is yesn't.

[@Video2Webb]

That is a brilliant statement by your professor. It seems to fit with how life actually is. Don't ask me more!

[@Pedro_MVS_Lima]

Excellent 👍

[@mirsuhail09]

As a medical student, i enjoyed this conplex mathematics, i followed it, thank you sir.

[@ThePCguy17]

The problem with the bit about predicates is that, when they're being used as the subject of a sentence predicates cease to be predicates. A predicate is a predicate because it's performing the function of a predicate in a sentence. And it is _only_ a predicate _because_ it is performing the function of a predicate in a sentence. At all other times that string of characters is just a string of characters. You can put quotes around the string of characters and use it as the subject of a sentence, but that makes it a different string of characters and the subject of a sentence, it does _not_ restore the quality of 'being a predicate' to the string of characters. This means that the example predicate 'is a predicate' which was used in the video is not, in fact, true of itself in the example sentence given because in the sentence " 'is a predicate' is a predicate." is not actually a true statement. 'is a predicate' is not, in that context, a predicate. The real irony is that the point which was being made is still valid, Kaplan just used the wrong example. "This sentence is a string of characters." In this example the predicate 'is a string of characters' is true of itself and thus predicates are, as the video was trying to demonstrate, capable of being true of themselves. There were even a few other examples given later in the video that are properly true of themselves, one of which was quite similar to the one I came up with. And yet the guy chose to focus on a predicate that was demonstrably not true of itself to demonstrate that predicates can be true of themselves. Honestly though, that seems about right for the guy who made a whole video on a topic that's about as useful to talk about as why mathematicians unilaterally decided that any number to the zeroth power is one and any number to the first power is itself. Neither of these things make any logical sense when you break down the math behind them. And yet they are still defined as true because otherwise a great many very important mathematical operations would fall apart.

[@lindseydye2314]

I like the point you made. I would like to add that rule 11 only said that prediciates CAN be true of themselves. This is that same logic that should be applied to his set theory, that sets CAN contain themselves, but they dont have to. I wonder, though, why not make a rule that says sets of sets of sets that do not contain themselves cannot be made. (What is that? A set twice removed?) That is the stated exception to rule number 1.

[@thom1218]

"are still defined as true because otherwise a great many very important mathematical operations would fall apart." Perhaps it is that fact that lends voracity to those statements, i.e. raising a number to the power of zero and one are defined in the way that they are, and why they have remained as such. If you can demonstrate new math where those are not true statements, and that new math exceeds the usefulness of all the math built around those simple facts as they're stated today... then more power to you.

[@ThePCguy17]

@@thom1218 The problem with the power of zero and the power of one is that by the rules of exponents those aren't even valid operations to perform. Perhaps there's some logical language in which you can define exponents such that raising a number to the power of zero equals one and raising a number to the power of one equals the number you started with, but by the commonly available definitions of exponents, multiplying a number by itself the number of times indicated by the exponent, there is no actual mathematical procedure to perform for the powers of zero and one. You can't multiply a number by itself zero times, nor does multiplying a number by itself once make sense when multiplying it by itself is raising it to the power of two, not one. I'm not saying I have some new math where it makes sense. I'm saying these represent similar situations where what makes sense doesn't line up with the way things has to work for the systems of logic and math to operate correctly.

[@jver1384]

Precisely what I thought: a predicate can't be a predicate if it doesn't have a subject. So unless you make up a rule that whatever string that was a predicate in another sentence, keeps the value of being a predicate in a new one, then the logic would cease there.

[@ThePCguy17]

@@jver1384 His problem was that he got too self-referential

[@chuckl211]

This is what education SHOULD be. Fantastic job of explaining a complex topic in a relatable way. Thank you for sharing your knowledge and the hard work it took in putting this video together.

[@LuisPerez-5]

I got bored halfway through 🤷‍♂️ too many little interruptions

[@kahlesjf]

Interesting topic, annoying presentation. Would have been easier to listen to if he slowed down and dropped the too-cute references and repetitions that did not advance an understanding of the material. Way too hyper.

[@LuisPerez-5]

@@kahlesjf exactly

[@tmoln2131]

@@kahlesjf I think one can use the settings to slow the presentation. Might have to have subscription for that, not sure. Slowing the speed might be a help, (and make it less annoying, maybe not) but would take longer no doubt.

[@edwardlulofs444]

@@LuisPerez-5 Watch a different video.

[@davidrichey9243]

Utterly Brilliant! So excellently explained. I began watching that wondering how long I would last before getting lost, but I'd been hanging on every word all the way through - and understood everything. Sad when it ended. Thank you Jeffrey, you're that rare creature, a REAL communicator 😊

[@GoogleIsTooInvasive]

Yes!

[@MattMcIrvin]

When my kid was very little, preschool-aged, I explained the concept of sets to her, basically that you could have a set of anything--a set of all odd numbers, a set of all trees, a set of all planets... and she *immediately* said "The set of all sets!" Straight into the deep water at such a young age. I started stammering about how there was some trouble with that one...

[@petarJK]

As a backend kotlin/java dev, apparently I knew a lot about sets and predicates without knowing I knew a lot about sets and predicates. Interesting video!

[@Veganstega]

I mean we deal with them all the time in the forms of arrays and whatnot right?

[@omgahandlelol]

😅

[@theneocypher]

You did a better job teaching me the “why” behind the math than every teacher and professor I have ever had. Thank you.

[@frankcourtney6065]

Nobody needs to know the "why" behind the math. You do not need to know "why" an axe is a proven successful tool for taking down a tree.

[@ArthKryst]

​@@frankcourtney6065well actually you do because once you think of changing the system, knowing how and why axes work, you can create a better tool/system of work

[@Babalous]

@@frankcourtney6065 you just don't understand

[@newmankidman5763]

@theneocypher, you are correct. The problem with 99.9% of math teachers and Professors is that they do a very poor job at explaining the logical correlation within the math, "the why behind the math", and as a result of which, the vast majority of students find math to be way more difficult than it actually is, and therefore develop a disdain for it

[@ArthKryst]

@@newmankidman5763 Even more is that, even if you attain knowledge of how to solve a particular set of problems it seems like a lot of the juggling we do in equations seem pointless, arbitrary. Generally when you solve trigonometry question, you are given an equation and you need to do something to it. Most of the time the process seems nonsensical and no reason as to why it might even be helpful to think in this manner

[@stevenmartinellimusic]

I've heard several explanations of Russell's Paradox and this is definitely the best. Your students (if you have them) are lucky to have you as a teacher.

[@emdiar6588]

We are they.

[@NorfolkSceptic]

Informative, and amusing, when played at double speed :)

[@klayvonisme]

You are one of his students. He has been your teacher.

[@klayvonisme]

@@emdiar6588 They are us.

[@archiebunker7688]

They=them

[@jray1429]

Thank you for taking the time to make this video and share it with us.

[@joseignaciocastrovonrodrig5613]

There are so many things that I'd like to express about this video, that I'll have to leave it for some other time. For now, let it be said that this man is a real genius in a three-fold way: he has a deep and excellent comprehension of set theory, he has obviously acquired a high degree of mastery about human linguistics too, and very prominently, he has amazing skills to explain and actually teach very complex subjects in an easy and simple manner that almost anyone can understand. He has a very good touch for comedy as well. By the way, this video leaves me thinking that Kant was right, and/or that our world is actually a computer generated simulation where we can find plenty of bugs and glitches.

[@w455up]

Lololol l a lot to play lol o vbb SS ez

[@hellocentral5551]

To the cunning programmer, there are no bugs. Only features.

[@joseignaciocastrovonrodrig5613]

@@hellocentral5551 You need to be a hard-core, veteran-level expert to realize that. 😀.

[@michaeldriver127]

At it's pinnacle in Greece, the invention of the plate platter inspired the Greek to believe the world was a giant plate they lived on. Hmmm 🤔 Dreams and ideas have more than the ability to manifest into physical reality, but quite possibly the ability to inspire new philosophies, ideas, and dreams is more abundant. IDK what's coming after computers, but I'm sure it will inspire new ways to interpret what life really is!

[@joseignaciocastrovonrodrig5613]

@@michaeldriver127 Very well said, Mr. Driver.

[@muddledmess]

As someone who is much more linguistic in my thinking than mathematical, this was a great explanation.

[@honeycat535]

Was it though?

[@Age_of_Apocalypse]

I agree! We can easily (or almost) understand the paradox! Great video!

[@JamesJNothingIsTooSensitive]

While the style is good... his information is wrong.

[@papersephone]

@@JamesJNothingIsTooSensitive how so

[@JamesJNothingIsTooSensitive]

@@papersephone To quote my OG comment I made when I first saw this video: Your very first premise is wrong. Even if math itself is a product of the human imagination, that doesn't make mathematical truths subjective. The *_units used_* are subjective, but the truths themselves are objective, as shown by the ability to come to those truths no matter what type of mathematical system you choose to use. The *_system_* is subjective, but the truths discovered by that system are still *_objective_* no different than measuring distance. You can shoose to measure in inches, or centimeters... or even use cubits or any other mesaurement you choose, but the distance remains an objective distance. Only the representation of that distance is subjective. Since your very first premise is demonstrably incorrect, I'm going to assume the rest of your arguement is as well, although I will still watch the rest of the video to be certain. As such my comment may get edited as I see more of this video. Edit: Holy shit, 7 minutes and 21 seconds in and this is downright *_riddled_* with erroneous claims and misinformation about mathematics and sets. This is... fucking hell this is bad.

[@sofijadzamic903]

As someone who never liked mathematics and didn't even try to understand it, i watched this video with so much focus that now this is the main problem in my brain, which i never thought of before

[@jordenguevara-foss9502]

i know how you feel.

[@dcquillanstone]

Agreed, tho' in my case I do like math, use math in my profession(s), still I found this video thought provoking. One can read/hear something complex or profound for the Nth time, and there are always unrealized nuances to consider.

[@Cherifica]

YES!!!

[@johnrichardson7011]

Everything is nothing and nothing is everything

[@tonyholm209]

Good video! It should be noted that Russell very quickly moved on from attempting to salvage set theory. His work after that rejected set theory & abstract particulars.

[@mutabazimichael8404]

To those who would like to learn about the context of how that theory came to be along with Godel's incompleteness theorem and the birth of set theory in a rather fun and pedagogic way i would recommend the comicbook "logicomix" that centers around the life , with some apocryphal event, of B. Russell in search of mathematical truth . apart from that Great video as always from you Jeffrey

[@persianandurdupoetry4301]

I express my sincere gratitude for the comprehensive elucidation of the Russell's Paradox issue. The exposition on set theory was presented with remarkable depth and clarity, and the treatment of predicate relationships was particularly commendable. I am genuinely appreciative of the meticulous work and the high level of articulation demonstrated in addressing these complex concepts. Thank you for your outstanding efforts

[@Mr.Weed420]

Yeah ok chat pgt

[@rishikeshwagh]

Do you usually talk like that or what?

[@TheSlickmicks]

Well, he wasn't talking -- he was writing. Unless he was using speech to text, then yes, he was writing. But then the act of turning speech to text is in itself writing. So then he was writing and speaking. ​@rishikeshwagh

[@nathanwalsh6837]

@@TheSlickmicks Damn, you are gay

[@DblTap317]

English major?

[@brianmorgan5880]

Many moons ago, I was a Physics major. My school offered several electives, one of which was Number Theory. I sat through Day 1 but decided to transfer to a different elective. At the time, I just wanted to learn how to build quality things. Now, as an oldster I can better appreciate what Professor Kaplan teaches. Thank you!

[@edwardlulofs444]

I hated Number Theory because it didn't apply to physics - or anything really. Now, Number Theory is essential to our encrypted communications!

[@caleboghenovo1149]

I think the problem lies in telling whether a statement is true or false. Since, there are statements that can neither be true nor false, it must mean they are incomplete. Also, we think by logic that if something is true, the opposite is false and vice versa, but if I say nothing is true, the opposite, nothing is false isn't true just because the first statement is false. This is a good example of how language, itself, is incomplete

[@private464]

How did I suddenly start listening to a lecture? You are a GREAT teacher.

[@toby7582]

?

[@private464]

@@toby7582 I certainly wasn't planning on hearing a math lecture when I clicked on this, but he is such a good teacher that I stayed.

[@toby7582]

@@private464 it was pretty good. I just don't like how in the beginning he's trying to sound like Vsauce. That gets old pretty fast for me. But still interesting and impressive backwards writing on that glass or however he's doing it.

[@private464]

@@toby7582 Agreed. It wasn't perfect. Some parts too slow and the end was too fast. I just find it remarkable (literally) that he got me to listen to a whole lecture that I wasn't planning on or even interested in!

[@toby7582]

@@private464 you weren't interested? Why click on the video then?

[@Anon-s8n]

I think my favourite example of this is "this sentence is a lie". It's the example that helped me to grasp the paradox.

[@hashtagunderscore3173]

I'm no logician, but I think the answer to that paradox is just to say "nuh uh, that's not really a proposition." So it's better to say that there are two sentences, Sentence A, and Sentence B. Sentence A = "Sentence B is false;" Sentence B = "Sentence A is false." That way, you get around the self-referential problem.

[@lokidecat]

And what if Pinocchio said, "My nose is about to grow."

[@Pipyboy]

@@lokidecatPinocchio’s nose only grows when he intentionally tells a lie. So saying something false does not cause his nose to grow. So therefore Pinocchio’s nose will not grow after saying “my nose will now grow”.

[@shawnmulberry774]

You just invoked my memory of how Spock defeated the Normal android in Star Trek TOS.

[@philipr1567]

My favourite is: all generalisations are wrong.

[@drphilChemist]

Nicely done; this video reminds me of one of my favorite books, Gödel, Escher, Bach: an Eternal Golden Braid (1979, Douglas Hofstadter). A set of things that are allowed to define themselves will always be incomplete or in other words if a class of something is allowed to define itself, then an instance of "something" can always be constructed that leads to a contradiction. I think Russell's Paradox, Gödel's Incompleteness Theorem, and the Halting Problem are all just different instantiations of this same underlying problem.

[@jergarmar]

I frikken love that book.

[@washemoamadah4706]

I have that book, loved its metaphors with achilles and the turtle, but even with such informative and illustrious metaphors to help me understand the subject... man... it is way beyond me. I could try to to understand it but I fear I would suffer the same fate as when Gotleb Frege when he got Russell's paradox in the mail.

[@ghwrudi]

Yeah - this is just another version of 'All Cretans are liars', so it's very old.

[@kasberkhof7958]

@@ghwrudi Also reminds me of sitting in class listening to some rule and then shouting out the paradox to get a laugh. This paradox is everywhere whenever you start curving arrows inwards

[@schmatta]

yes

[@Big_Sloppa]

For a second I thought you learned how to write backwards on glass, but then I realized that you just mirrored the video (because you write with "left" hand). Now I'm feeling so smart, I will go take twinkie, I deserved it with such monumental mental work 👁👅👁

[@alexeytsybyshev9459]

By the way, the question of whether "is not true of itself" is true of itself is equivalent to whether "this statement is false" is true, which is perhaps the most well-known paradox ever.

[@shan79a]

Not in this case. Let us call the original statement S1. So "this statement is false" = "S1 is false" = statement S2. S1 and S2 are not the same statements (e.g., S1 is the statement, "apples are never red", which is obviously not the statement S2: "the statement "apples are never red" is false"). So if S2 is true, it does not mean at all that S1 is true (they not being the same statements) and in fact bolsters S1's falsehood, not imply that S1 is true. So S2 is NOT self-contradictory and there is no paradox here. You made the mistake of equating S1 and S2 and thus the truth of S2 implying S1 is also true (which it does not as explained above), thus leading falsely to a paradox (i.e., S2 is self contradictory), which is not there.

[@alexeytsybyshev9459]

@@shan79a I think you misunderstand what I am talking about. It is not the term "Is false", which I can apply to any statement S1 to get statement S2= "S1 is false" . It is the statement S3 = "This statement is false", where "this statement" refers to S3, so contextually S3 = "S3 is false".

[@RandomNumbers1337]

"This statement is false". If the statement is false, then it is actually proven to be true. If it is true, then it no longer satisfies the condition of being false, which would mean that it is actually false. But if it's false, then that means it's true. And on and on

[@JoE_Songs]

4:53 yeah. this is nonsense. in the moment in which I refuse the set-theory in and itself as a highly hypothetical, theoretical construct that it is, I don't have a paradox. It is that simple. the redundancy is self-evident from the start. you cannot simply walk past, that something unimaginable IS something - and then be astonished, when you run into a paradox. What a stupid theorem. utterly useless.

[@shan79a]

@@alexeytsybyshev9459 The statement "this statement is false* is a vacuous statement as it is not saying that anything particular is false, but an empty/null concept/entity is false, i.e., a nothingness is false. Such extreme boundary-condition logic (referring to a nothingness as opposed to something concrete) can never occur in any type of logical statement in any field, and thus is meaningless and worthless to consider as leading to a self-referential paradox.

[@taoofjester4113]

If only my teachers were like this. "You don't need to remember this" was my internal mantra throughout school, and my grades reflected that. Now, if my teachers said that all the time, I would have way higher grades.

[@herbertlong3981]

Awe yuh, the ol' -- "its my teachers fault that my grades were so terrible!" You will find at least one of these under every maths related vid on utube

[@herbertlong3981]

@@whatshumor1 So how, (even if he could have done that), would that have made it his fault that you refused to learn anything. It is not your teachers job to force you to learn things. It is you and your parents job. Schools are not supposed to babysit you. I went to school and aced every maths related and physics class I took from the first grade on. No one, not parents or teachers forced me to do that

[@herbertlong3981]

@@whatshumor1 Oh, I get it, it was a joke! Good job!!!

[@priscillawrites6685]

When I was in 7th grade, we were taught set theory in math class (yes, an advanced level geek class). The set theory we were taught included ‘a set cannot contain itself.’ Yale University wrote our curriculum. Shrödinger’s veterinarian walked into the waiting room and said to Shrödinger ‘I have good news and bad news….’

[@TheButlerNZ]

The set theory we were taught included ‘a set cannot contain itself.’ I instantly thought of the set {A set of all sets}... but wait.. if it can't contain itself.. then it is impossible.. as it's a set... and a set of all sets... Shrödinger’s cat May or may not have been alive in 1926.... I think it's fairly well dead now!

[@solazytofindaname]

This is such an amazing video, it's introductory yet it covers so much and I have never made the connection between Russell's paradox and incompleteness theory. Paradox ( that is faith for all you natural theologists, or insanity for rational psychologists :p) is indeed an inherent category of logic, any mathematical, axiomatic, supposedly self contained/analytical, cohesive and consistent structure. It seems to me that philosophy, theology, psychology, data science, linguistics and many other sciences would benefit greatly by acknowledging incompleteness theory and its implications and gravity.

[@kumamonkumamoto3844]

this video is proof, that no matter what the topic is, if it is explained well and interesting with structure, everything can be fascinating

[@joeschmoe2843]

If you can't explain the complex in plain language, you do not understand it yourself.

[@nicksdinosforkids6001]

I have heard of this Set Paradox before but you did a great job of explaining it and the ramifications. Now you got me wanting to review my Chomsky and perhaps taking a deeper dive into counterfactuals which will further muddy the waters for me.

[@tonymurphy2624]

The reason you can't solve the paradox is that paradoxes aren't real things in the real world, they're the literal manifestation of cognitive dissonance. They're purely cognitive artefacts arising from either naïve construction or naïve analysis. In general, paradoxes arise from either including something irrelevant or excluding something critical, either in construction or analysis. The Rich Guest paradox includes an irrelevant rich guest, opening what's fundamentally a closed circle. Xeno's paradox arises because it excludes time (in the form of speed; it appears to include time throughout via 'when', but it only treats the spatial distance and doesn't include the notion of distance over time, i.e., speed). The Twins Paradox arises via the exclusion of a third reference frame, which is what's necessary to bring the twins back together. This particular paradox comes from the exclusion of rigour, in that the natural language construction contains ambiguous or undefined terms. There are no real paradoxes, they're a sign that something has gone seriously wrong with our thinking.

[@KaiHenningsen]

@@tonymurphy2624 So, what is the problem in the predicate case? Where did it go wrong?

[@brendanh8193]

@@KaiHenningsen Would this resolve it? Axiom 12: A set can simultaneously contain itself and not contain itself. This can be called a higher order set, or an upset.

[@3739DrewKnight]

@@KaiHenningsen I think it goes wrong by describing something as true or false that cannot be true or false without reference to a subject. “Is not true of itself” isn’t true or false because those ideas require a statement to be made and “is true of itself” is not a complete statement

[@MendTheWorld]

@@brendanh8193 You win!... in a highly improbable upset. 😭

[@whelmking6497]

Such a brilliant video. You are a very gifted teacher. I love topics like this that are both inherently meaningless and at the very core of everything.

[@douglasstrother6584]

My Calculus Professor (Tony Tromba, UC Santa Cruz, Fall 1981) dropped Russell's Paradox on us at the end of a Friday lecture to give something to snack on during Happy Hour. "Gödel, Escher, Bach" was all the rage back then.

[@NourArt02]

I could sit through a 5 hours math class of this guy, he somehow made a math subject entertaining.

[@honeycat535]

go out in the sun, look at a plant - seriously .

[@prikols2138]

@@RazaXML some people (just like myself) need that 5th grade comprehension to even begin to understand math, so this is actually really valuable for someone like me and others like me. My math classes were always taught to those who actually understood math, like two people, the ones who didn’t (the rest of the class) were left in the after classes and usually got 1-3/10 grades..

[@prikols2138]

So we always needed to go after class and rewrite the tests, it’s a fucked up system in a lot of Latvian schools, probably a lot more places in the world as well

[@jbooks888]

This isn't maths - where is the practical application? This is a waste of time.

[@gr00veh0lmes]

@@jbooks888 math has applications that transcend the merely practical. It’s a playground of logical thought where black holes are discovered and the contents of atoms and nuclei found. More than that, math describes and circumscribes the limits of our understanding of what’s *out there*.

[@StephanBrown-d7j]

If you liked this discussion, I recommend checking out Gregory Bateson's essay "A Theory of Play and Fantasy" from his Steps to an Ecology of Mind (1972). He shows that the paradox runs deeper than even predicate-based syntax/human language. In the process, Bateson shows that Russell violates his own rule (a set may not be a member of itself) by even positing his rule. And here's the kicker: Bateson argues that without such violations/paradoxes, communication as we know it (beyond rigid mood-signaling), would not be possible (including that peculiar game we call logic). This includes non-human behavior such as mammalian play, threats, and other metacommunicative interactions from which the metalinguistic rules and thus spoken language evolved. Bateson ends his essay with imagining what the world would be like without such paradoxes: "Life would then be an endless interchange of stylized messages, a game with rigid rules, unrelieved by change or humor."

[@stourleykracklite7663]

God Said, Ha!

[@icarusmuthre2176]

wow, I gave it a read and it was 100% worth it, thank you so much for your comment:)

[@thefloodwatch785]

After reading Bastedon's essay, I now understand why Frege had a meltdown after reading Russell's letter lol. Basically our whole world and everything we do falls apart. But I think most people actually already know this and don't realize it. The answer to "why do I do what I do?" Is very often, "I don't know. I just do it."

[@shortbusdriver83]

I also recommend Gary Gygax.

[@TheMacroSlacker]

So I assume chance being at times random has something to do with this. The complexity of the system is such that we cannot fathom the pattern of randomness. Good enough for me to feel like I am alive and not a self-deterministic machine.

[@larrypass6720]

I've always been fascinated by the other set in Russell's paradox-the set of all sets that do contain themselves. If you ask which of the two sets that one belongs to, you can show that there's no problem with saying it belongs to either set. There's a certain freedom there that mathematical objects are not supposed to have.

[@Xingqiwu387]

Gödel.

[@andydenis1953]

That "this statement is false" is undecidable gets all the attention, but "this sentence is true" is just as undecidable.

[@mounibmakaveli2477]

that's basically what philosophy offered to all the mathematicians, a back door

[@h00db01i]

thinly veiled mysticism, opium for the few. adults speaking in codes according to complete arbitrariness. you might as well expel hot air the other way round!

[@jameskelso839]

LOL. I love your comment. Point well taken. Most of the time Math is about the rules and following them. This may be the only exception.

[@cryotimber]

This man legit put his own death year in the quote what an enigmatic legend I dig this guy

[@dynhoyw]

off the grave yeah

[@olafshomkirtimukh9935]

I have long known that *I shall die on 21 April 2052* , aged 89; I am so sure, in fact, that it's been up on a poster (containing my favourite quote) I created and stuck in my office 26 years ago. My hope is that if nothing funny happens on that day, some gentle soul -- knowing of the prophecy -- will be kind enough to do me in. There are, after all, many ways to generate a self-fulfilling prophecy.

[@w花b]

@@olafshomkirtimukh9935 I will hopefully live tomorrow starting from the day I post this comment. There you go.

[@hfdcjiirjmcfi]

@@olafshomkirtimukh9935 Olaf, if this is your true desire, I may be able to help

[@davidhunt1350]

@@olafshomkirtimukh9935 I don't know if it is true or not, but I have heard it said many times that, before we come to this Earth to live the experience (under the veil), that we have pre arranged the mission, that supposedly we have entered into a contract to fulfill an experience, if this is the case if this is true, was your mission to come here to this world, to wish that your prediction comes true? if so then if your prophecy comes true was that a lesson that you had to experience in this life time to help you gain some form of knowledge for your TRUE self, because if you are trying to will your self into a state of absolute focus, what happens if you pass away a day earlier, or a day later, does this mean that if you don't make your prophesized date that you have not reached a pass in this life, or is it just an experiment where by near enough is good enough, because if I wish you well in your prophecy or desire, I only wish that you can achieve anything that you put your mind too, but not wanting you to die, hopefully you have other reasons for being here and that your prediction date is just a passing interest but not the main focus, because for what it's worth the meaning of life is subjective, to the individual but I truly do hope your lifes experience for your TRUE self is achived before your date of passing over to the bigger classroom, I truly don't wish for you to fail, but I am hoping that your wishes or desires are achieved, either way, the way humanity is travelling along these days towards another world war, presumably to reduce the population, ("Alice in chains" had an album called Jar of flies,) this was based off a famous scientific experiment, and I can't help but think that represents humanity) a planitary extinction level event may just arrive before hand, to reset the earth like the pins at a bowling alley for the next set of contenders, if this "set of humanity" doesn't make the grade, we may all end up giving our seats away to more deserving passangers.

[@hebreobiblico5070]

Honestly I thought I would skip some parts (28 min vid... etc) but your logical way of putting things together made it quite easy to follow, even for a foreigner and a non mathematician like me... Congrats, Mr Kaplan !

[@yrls1238]

The paradox was discussed in ancient times both in the east and in the west. Interesting to see how mathematicians quantified it and other old theories in recent two centuries and we made use of those work. But at the end of day, there are fundamentals in human history never changed. Recent developments only add tools to decorate them.

[@NA-ni9fc]

You put into words a thought I have had forever, what I have been unable to convey. What a feeling, to hear it. TY

[@Fr1nc3sc41]

Wow! You're both, a philosopher and an entertainer with substantive information. It's refreshing to have come across this video purely by chance. It brought me back to the carefree days of early college dilemmas that provided endless hours of conversation over liquid refreshments before stumbling back to class at 7 in the morning still wondering whether anything was relevant to life in general. Kudos!

[@zhamed9587]

I believe things like this are even more evidence of God. We can't even begin to comprehend things that we take for granted (like language), yet we want to believe that we figured everything out. God does not abide by our rules, and this is yet another sign.

[@JusticieForMayelaAlvarez]

@@zhamed9587 the problem is that you will never know what you choose to believe instead. That's why belief systems are wrong. They actively destroy knowledge.

[@zhamed9587]

@@JusticieForMayelaAlvarez Belief systems are based on evidence, not blind and arbitrary selection as I feel you're implying (a straw man argument). We have overwhelming evidence in the case of Islam.

[@JusticieForMayelaAlvarez]

@@zhamed9587 as you say, belief systems are based on evidence, but that also means not proof, just evidence. Belief systems also lack a systematic approach to testing and properly validating their own beliefs. They are not dynamic for that reason, they're static. They never evolve or get any better with time. It comes down to the very definition of what the word belief means. If you have to "believe" something it's because you have given up knowing it for a fact and have to rely on someone's better wisdom for it. That's what "believing" means. It's the choice to not pursue knowledge.

[@JusticieForMayelaAlvarez]

@@zhamed9587 so again, this is why belief systems are plain wrong. It's like saying science is a bad thing, an apple in some forbidden forest that we should have never touched. Why? 'cause we're not supossed to know shit. We're just supposed to accept what we are told and believe it. Well, no. Actually, things have changed. You have to prove what you say this days, otherwise well... it's just thoughts and prayers. No one gives a shit.

[@JamesMarler]

I've only seen one video (this one) by you. I've never felt like clicking the "Subscribe" button was more earned than I do right now. I'm not into mathematics, but Logic and logical/critical thinking are very important to me. You earned this one.

[@TecnamTwin]

Same.

[@colinreay2678]

I feel differently , life is just spirals he's going round in circles and not moving forward. I hope he gets there soon

[@THE-X-Force]

@@colinreay2678 It's not really circles. He's just using multiple examples to make the point. There really is no _"forward"_ to move to, because it's a paradox. If you have a solution for it, well .. that'd be something.

[@qa1e2r4]

It is a paradox because one of the assumptions is wrong and misleading. Can you tell which one?

[@-danR]

I was one of the fortunate people whose small local library happened to have a book titled "Mathematics: The Loss of Certainty", by one Morris Kline (or Klein, whatever). That was some 40 years ago. You people are now the blessed few who have recapitulated--through the magic of the intertubes--that revelation. That set-theory is broken. It also helped me understand why the algebra books of the 60's that I had to suffer through, were so _needlessly_ insufferable with their wretched set-builder formalism that could lose you marks if you didn't mindless jot down every jot and tittle in writing out a proof, as though it was some gol-danged programming language. And why my older sister's algebra book, written in the 50's before France's Bourbaki cult took over America's mathematical pedagogy, was so _easy_ to understand. For all the good Cantor did to help the world understand infinities, his ideas of 'sets' was the original poison in the well. Today's mathematics teachers and profs must wonder how on Earth people did mathematics in the 19th century without set-theory to help them.

[@jasim_ij]

You can explain things better than any of my philosophy teachers. 10/10 👍

[@annaclarafenyo8185]

Except this fool thinks Russell's paradox is still a problem in 2023. It was solved by Russell in the theory of types, and in modern set theory, the paradox nowhere appears, because sets can't contain themselves. The point of Russell's paradox is that there is a limit to the notion of size of abstract sets which is set by logic. The proper foundation idea isn't sets, it's computation, and sets are important to the degree they explain properties of computations.

[@capitaopacoca8454]

@@annaclarafenyo8185 except you fool thinks the video is about the original set theory paradox and didn't realize the goal of the video is the language version of the problem.

[@annaclarafenyo8185]

@@capitaopacoca8454 There is no language version of the problem, informal language is vague. That's why people invented formal languages like those of Russell and Whitehead, or modern set theory. Philosophy can only be done with a formal language underpinning because of nonsense like this 'paradox'.

[@bestopinion9257]

@@annaclarafenyo8185 No, even if you say that "no set can contain itself" then you make M = {x set / x doesn't contain itself} and so M will be the set of all sets because any set doesn't contain itself as your law. Now, ask again if M contains itself. M is a set, right? And as a set, it doesn't contain itself. Then matches the definition "x set/ x doesn't contain itself", then M is in that set so M contains M. The paradox remains.

[@annaclarafenyo8185]

@@bestopinion9257 There is no set of all sets. This is naive set theory, it is inconsistent. You form sets by computational processes iterated an ordinal number of times. That's ZF set theory, it's understood since the 1910s.

[@Jtmurph222]

Best paradox video I've ever watched. Great explanation.

[@Anderson-rt7yk]

It's beautiful to know that Russel moved to philosophy because he wasn't satisfied with the implications of pure mathematics and wanted to understand those kind of problems deepening his knowledge and understand why mathematics limited the comprehension of reality Truly one of most incredible thinkers within the sociocultural time he lived. Do you have a video on his collaboration with Whitehead? And what is your opinion regarding the works of Whitehead III?

[@bluepapaya77]

I was going to make this comment in a more flippant way: "And that's why Russel gave up on math and moved on to easier things, like philosophy." :)

[@johnny4aces410]

Russell said, "I turned to Phillosphy because I became too stupid for Mathematics. I then turned to History when I became too stupid for Philosophy." He was of course being facetious.

[@pwpowernz]

@@johnny4aces410 But if he was being facetious he wasn't - and if he wasn't, he was ...

[@johnny4aces410]

@@pwpowernz Darn those self referential propositions! They always lead to a conundrum! Facetious or not.

[@tortoredOwl]

@@bluepapaya77 Philosophers have higher IQs than mathematicians or anyone else. As an example, most Phd mathematicians failed to solve the Monty Hall Problem and couldn't understand the solution even after it was explained to them. Philosophers had much higher rates of solving the problem.

[@0NlRAPTOR]

I loved listening to this video and the reasoning behind the paradox. It defines all of the elements and then uses those elements to explain the problem. Its possibly the best video I have seen at explaining something so that anyone can understands if they pay attention.

[@bobs8942]

Huh?

[@steeljustice4122]

Your deadpan delivery of hilarious lines makes my brain happy

[@Sergio_deus]

really? thats funny, because it has the exact opposite effect on me... but anyways heres some shiny keys 🔑🗝 look how they shine 😲

[@Robert08010]

That's enjoying the humor while being too engaged to laugh. All those happy little endorphins just pile up on one another.

[@MeanderingSlacker]

​@@Sergio_deusMay I see the shiny keys again. They make me happy.

[@EduardoWGiacomelli]

It is so marvelous to see semeone dedicste so much positive energy and display a super interesting topic digestible for people that cant read between the lines in these advanced topics. i just love to learn and watch someone passionate about what they think and exploring the idea and getting it in simple terms for an larger audience to appreciate the wonderlous work from those old geniuses mathemathics from back then. Just the taste of this video makes me so passionate about this subject too its incredible how an self-made production in a youtube video can open my eyes and change my direction based on pure appreciation from those topics and the earn to understand better mathemathics and the secrets of the universe hahah. amazing video got me back in math stuff and i just love it

[@denusklausen3685]

As a philosophy student I feel I have to defend Frege and Russell, the intention of “counting” in set theory was to explain the way in which something like the number 4 always picks out 4 entities (like 4 apples) but is never itself those entities (4 is a number, apples are apples). So they thought it must be the set of all possible 4 entities that makes up the number 4.

[@kevinhamilton2521]

Science can’t explain what numbers are because it’s based off presumption about reality like we can only know things through pure raw since data. But we know you can prove things in different ways for example you can’t prove all of history is real because you never experienced all of history we just assume it to be true without thinking about it but we don’t measure it with a measuring tape. Heres a argument I would use to prove metaphysical things like logic is that in fact it does exist because to deny logic or numbers Leeds to insanity. Numbers is universal

[@zimbabwe_twinnedwithanfield]

This topic seems over complicated for no reason. A number is just what we assign to a specific quantity . I don't know but sometimes humans just create complications when none is needed

[@kevin93azyu]

@@Бородатый-к2нan amount of something.

[@zimbabwe_twinnedwithanfield]

@@Бородатый-к2н what Kevin said

[@treadwell1917]

@@Бородатый-к2нhow much of something there is.

[@jsmunroe]

"This sentence is false." I love the Russell paradox. I love all paradoxes. They don't break anything except our ideas of reality. They hint at new truths out there just beyond the veil of our understanding. It is exciting. I wonder if there may be an allowance in logic for something like a superposition, where a proposition can be an inherent contradiction, true and false at the same time.

[@cinematiccrisis]

I also like me some negative self-references (they are positive and negative about themselves at the same time). There have been experiments with fuzzy locigs, but there is no consensus. I think that paradoxes are still regularily treated with stating some prohibiting rules (do not self-refer, at least not negatively, though you can) in most fields of life. Ad "new truths": you can infer nothing/anything from a paradox (ex impossibile quodlibet), so one opportunity is show the regular paradoxes of communication (e.g. it can be bad to be good) to win new opportunities of thinking. I think that is not only refreshing, but a good loosening-up-exercise of thinking.

[@panicsum]

Are these new truths anything to do with the new structures physicists are finding beyond our reality, according to Donald Hoffman?

[@mikeb.7722]

For real, that's all this is: the old "this statement is false" paradox. Scientists are (and really all of academia is) too "smart" for anyone's good imo. This video is more proof of God's existence e for me

[@hillaryclinton2415]

The correct answer is " no, it is". Or "yes, it isn't". ... I can't recall which

[@lordchickenhawk]

@ Jordan Munroe: I came to the same conclusion myself. It felt like a lifting of a corner of the mathematical logic rug and momentarily glimpsing all the quantum weirdness scuttling for cover underneath. I feel compelled to quote Vroomfondel from the Amalgamated Union of Philosophers, Sages, Luminaries and Other Thinking Persons, “That’s right, we demand rigidly defined areas of doubt and uncertainty!” (Douglas Adams, Hitchikers Guide) PS: I love how Jeffery explained the derivation of axioms from rule one. I think Bill who commented above has a point about revealing linguistic limitations. At best, the parallels between "true of" predicates and "contained" objects seem to be only a useful metaphor... instead, perhaps paradox actually has a definable mathematical reality.

[@prasannamukherjee3438]

Believe me bro I don't find much people to talk about maths and logicism bcz no one seem to focus or show interest. But the way you explain just can't escape one's auditory canal and I realised this bcz I watched this whole video without pause. Nice one👍

[@kylan_11]

Knowing a bit about programming makes this video 10x easier to follow

[@williamkz]

Brilliant presentation Mr Kaplan. I wish I'd been taught math like this. I definitely belong to the subset of people who aren't very good at math, but could have been ok with better early teaching.

[@williamkz]

@Black Lesbian Poet The suggestion that some races are better (or worse) at math than other races is in itself racist.

[@davidlawson854]

This is the most important thought (and video) in science and the humanities. I had slapped up against this in 1972 (yeah, I'm old, so what?) and it pretty much interrupted my undergrad years. At least I didn't fall into the rabbit holes Tolkien or Rand; however, I haven't had much free time for studying set theory for the past fifty years. Thank you Dr. Kaplan for the illuminating lecture.

[@NoferTrunions]

Rand is a frustrating philosophical rabbit hole. Tolkien is a great fun fantasy story.

[@jasongriffin4965]

@@NoferTrunions Tolkien's book is not a fantasy story, but a set of realities with different names for the things in reality. At 11 yrs old I saw the connection he made in the trenches of war in his day to mine in the 80s when I was 11. Watching that movie "Lord of the Rings" was an epiphany to me about the world I lived in.

[@NotSoMuchFrankly]

I'm not sure I understand the term rabbit hole the same way you do.

[@NoferTrunions]

@@NotSoMuchFrankly My take on rabbit hole: Obsessive (like that tune you can't get out of your head), ruminating thought about something. The rabbit hole of addiction. A descension into unreality. Rand: people can get all worked up about philosophy and political issues whereas Tolkien simply wrote some fantastic, endearing books that yes, people got "hooked on" and read. Tolkien makes Herbert's Dune look still-born - so the popularity of Tolkien is from his massively superior mastery of story telling, not some hidden message. NOTE: yes, there are T fanatics that learn the language, but there are also those that learn Klingon - neither has fallen anywhere. You post an interesting point comparing Rand w/Tolkien. Rand & Marx or Tolkien & Asimov (and NOT that Potter nonsense LOL)

[@NotSoMuchFrankly]

@@NoferTrunions Ah, put that way, @davidlawson's take makes more sense. And, I agree with you, BTW. The usage I've always encountered meant not just obsessive distraction but that one has 'drunk the Kool-aid' of a complex and somewhat paranoic conspiracy theory, i.e. a complicated and fervent delusion shared by many.

[@tacticoolrick5562]

Sets are really useful in computer science. We use them a lot when working with data and databases. It's A LOT more convenient to be able to write an expression to operate on a set than it is to write code that does work on every element of a set.

[@asquishyjellyfish5431]

The paradoxical reality of self-reference is what derived the absolute breaking point of meanings and logical factuals, thus given the action of breaking reality itself the self-validating power it needs to do such.

[@Redbaron_sites]

I love the style in which you present your podcasts. You have personality, you connect, you hold our interest ❤

[@zuhail339]

I want to thank you so so so much, sir. I have an interest in almost every discipline there is, including philosophy. This video was the first ever 30 mins of video I watched lying on my bed- entirely in the mood to fall asleep any time, yet I managed to grasp every concept you taught in the lecture, and it kept me compelled as well. I hope you make more videos that pertain to the mathematical world's connection to the philosophical world. You are a wonderful teacher. Truly an unforgettable experience!

[@RayPalmer]

As a History teacher I must say that all set theory you explained starts to make more sense when you talked about the linguistic! Really interesting!

[@sunnygreenings4840]

I love hearing the linguistics of set theory. English major who was good in math here. I want to live again.

[@wagohowardmusic]

Try music then… mindblowingly Fun 432!

[@irinadalah9089]

thanks for explaining - there were so many ways to answer you. because there were so many questions your awesome explanation arose. by the way #1: at 23:01 you actually tell us that this is only a word game. you say "it seems that is true about predication". your explanation was so awesome that for a while there I forgot that I watched your vid at all, because you started talking about the definition of numbers - which you haven't finished, even though you mentioned it being a type of set. but I'm going to relate only those things that are pertinent to the rules you mentioned here, because my kids and my siblings are waiting for their profiteroles - and if the profiteroles would bake themselves, that would be a really awesome paradox (ok, you're probably say it would be a miracle and I challenge you to define the difference 😈). first: all rules are phrased as "can" and not as "have to". that means they may behave in a certain way, and they may not - because they don't have to. second: there is a rule (or maybe just a behavior allowed by the rules you mentioned? you can phrase this any way you want) that members can belong to more than one set - intersections of sets. and if the rules allow items/members/things belonging to more than one set, then they should allow for those that belong to no set! here's russell's paradox solved! the set of sets that do not contain themselves is the one allowed to not belong anywhere!! so instead of trying to restrict the rules in order to explain this paradox, lets expand the rules to include all paradoxes. by the way #2: I wonder how many people watched your vid and went to get a cat to call it "is a cat"? 😸

[@olorin9349]

I literally just came across this channel and Jeffrey Kaplan does a marvelous job explaining Bertrand Russell's "paradox theory" and showing how the theory is still relevant today through "predication," rather than just casting rule #11 to the side. This was phenomenal!!!!!!!! I have always believed that Russell's "paradox theory" actually had substantial ground. Your "predication theory" is that ground. How is it that I am just now stumbling across this wonderful channel?