Defining Infinity | Infinite Series

[@pbsinfiniteseries]

Hi Everyone, reposting this because we had a few visual fixes that didn't make it into the uploaded cut. Hope you all understand, we want to be as accurate as possible! Enjoy!

[@jackozeehakkjuz]

Thank you for all the hard work! This makes me crave for Gödel's incompleteness theorems. Will we see category theory someday? That'd be awesome.

[@RodrigoCastroAngelo]

No problem. I feel I'll have to watch this a couple more times :P Great video, by the way

[@TheNewRavager]

Could I ask for the links to the four videos @ 1:00 to be edited into the video description? I'm on mobile and it would really help out, plus I'm sure other people wouldn't mind it.

[@pbsinfiniteseries]

Just saw this comment. I'll add this now. --Gabe *UPDATE:* Should be up in the description now. Do you see it? Sometimes requires logging out of KZbin and logging back in to see the update.

[@TheNewRavager]

PBS Infinite Series Yes I do. Thank you! I appreciate all the work that goes into these videos!

[@IuliusPsicofactum]

It's so sad Infinite Series was a finite series :(

[@michaelnovak9412]

I keep coming back to those last few videos of Gabe on set theory. What a masterpiece. Such a shame PBS cut it right in the middle.

[@yeung8850]

，一，了

[@bigpopakap]

Totally agree!

[@alexr3698]

I mean, obviously, a donut would have the blood type O.

[@erikziak1249]

Alex R Exactly!

[@tuftman6092]

depends on whether or not the donut is inside out

[@RasperHelpdesk]

In an infinite universe there exists a me that has already watched this. I am that me.

[@SuviTuuliAllan]

Who is this you?

[@FunkyHamada12]

Who are you? You are two.

[@John-js2uj]

Oh, you two...

[@NotHPotter]

In an infinite universe, there's a me that understands this.

[@GrothenDitQue]

In an unbounded Universe, there is (almost surely) not uniqueness of that you. 😉

[@jubu247]

Please please please, tell us you’re leading up to Godel’s Incompleteness Theorem.

[@bunklypeppz]

I was just thinking the same thing. The point about the problem that "effect(s) axiomatic systems quite generally" practically screams incompleteness theorem.

[@morkovija]

The only channel which takes me few days to process a 10 minute video

[@pbsinfiniteseries]

You're welcome?

[@morkovija]

Thanks guys, not enough yt channels that require more than a few neuron connections to process their content these days

[@1976kanthi]

@@morkovija You and your comment were mentioned in the last vid of Infinite Series. How do you feel about that?

[@morkovija]

@@1976kanthi this really is what peak commenting looks like. I'd like to thank all the people who made this day possible)

[@allthe1]

Why did I miss all of this excellent content? It's three years out, the channel died, and just now do i get it on top of my feed. Math wasn't my primary interest then, but still, YT just swept it under the rug for no reason. RIP Infinite Series, you wew and still are awesome

[@TomTom-rh5gk]

I learned more from your two videos than I have in a long time. You are still the best.

[@nayash4744]

One of the most illuminating videos on this subject...thank you guys.

[@EhCloserLook]

It makes me want to cry how badly I need this video when I was in college struggling through Real Analysis. Where were you when I needed you?!?!?

[@MuffinsAPlenty]

What you hinted at in the end does seem to be a general trend. It seems like it doesn't make sense to have a set of all "sizes" of sets. The same thing shows up if you reject the axiom of infinity and demand that all sets must be finite. In such a system, the "sizes" of sets are precisely the natural numbers. But there cannot be a set of natural numbers while rejecting the axiom of infinity.

[@mrlimemil]

This was a really well done video that brought together a lot of the previous infinity videos.

[@JM-us3fr]

Can you do a video on the logic of formal languages and why they are necessary?

Hitting the subscribe button hoping that maybe someday the series will come back

[@Lucky10279]

1:20 The obvious definition is simply that a set, in the context of ZFC, is anything that satisfies the ZFC axioms, just as a vector is a member of a vector space, where a vector space is any set that satisfies the vector space axioms, a tensor is a member of a tensor space, where a tensor is any vector space that satisfies the tensor space axioms, etc. That is, collections of axioms are really nothing more or less than definitions themselves.

[@RubALamp]

You’re a great host.

[@flymypg]

I distinctly remember covering this as part of undergraduate math, but my recollection is that I didn't truly understand it, but learned just enough to pass the exam. Thanks, Gabe, for making it more understandable this time around!

[@ChaiElephant]

You could define the cardinality of the class of all cardinalities of sets in NBG as ORD, the class of all ordinals. ORD is sort of like the biggest ordinal, even though it isn’t technically an ordinal.

[@92587wayne]

I can give you a visual example of Infinity. That which is Infinite is not readily apparent, is not measurable as to location and momentum, has no mass. Take eight Mirrors about one foot square. Form an octagon with the reflective side to the inside. Place a dot on one of the mirrors and then look at the dot from the outside of the octagon. Finding just the right angle and a series of dots will appear on one of the mirrors, beginning with the first dot being larger than the rest. The series of dots will appear with each dot becoming smaller and smaller to the point that the dots are no longer visible, become Infinite.

[@tielojongmans3826]

Wow, this is a very nice episode! And a good wrap-up of the previous one.

[@boredgrass]

An anger cues directed translation attempt: “Dare you come unprepared to my video!” “Do you understand??” “Hey you!!” “ Yes! I am talking to you!!!!” “Who do you think you are???”...

[@ghani.afridi]

YES ! I found Gabe !! From space time I was missing him so much

[@Chalisque]

In some sense, Set Theory is the study of the relation 'is a member of', i.e. ∈, not the study of sets.

[@teknochaos]

can you make a video on how real numbers are defined/derived in ZFC?

[@nafrost2787]

They have the crisis of math episode about it, but basically you first construct the integers as equivalence classes of the naturals, with an equivalence relation that mimic substraction in reverse with addition, then you define the rationals as an equivalence class of ordered pair of integers and non zero integers with an equivalence relation that mimic division in reverse, with multiplication, and then you construct the reals from the rationals. There are multiple ways to do it, but the two most popular ones, are as an equivalence class of of Cauchy sequences of rational numbers (loosely speaking, Cauchy sequences are sequences where the values of the sequence become arbitrary close to each other), or with Dedekind cuts, which are a type of sets of rational numbers. And of course in each construction, you define addition and multiplication based on the previous level (integers from the natural, rationals from integers and reals from the rationals).

[@teknochaos]

@@nafrost2787 thanks. the part I'm interested in is the last step (Cauchy etc). I'll read up on that.

[@johannesh7610]

Thanks, I wanted a good explanation for these ordinals. Just a quick question after the first view: If ordinals contain all lower ordinals as a set, then how can they be a proper class that cannot be contained?

[@MikeRosoftJH]

A set is an ordinal number, if it is strictly well-ordered by the ∈ relation, and any its element is its subset. If there was the set of all ordinal numbers, is it an element of itself, or is it not? If it is not, then the set satisfies the definition of an ordinal number, and should be an element of itself. If it is an element of itself, then it doesn't satisfy the definition of an ordinal number - the relation ∈ on it is not a strict ordering relation; so it can't be an element of itself. (In addition, from the axiom of regularity it follows that no set is an element of itself.) Or, from the opposite point of view: let X be any set of ordinal numbers. Let y is the smallest ordinal number which contains all elements of X; by well-ordering, this ordinal number exists. The ordinal y cannot be an element of X, because it would have been an element of itself (and again by definition - and by axiom of regularity - no ordinal number is an element of itself). So no set of ordinal numbers can contain all of them.

[@AlejandroBravo0]

9:00 well, to be honest asking the blood type of a doughnut does sound like gibberish to me. But I got the point, Cantor's is a cool paradox.

[@danielrhouck]

It's syntactically sensible. I'm not sure about semantically.

[@astralchan]

6:56 ... in which episode did he discuss these power sets? I looked at the vids, couldn't find it.

[@MysteryHendrik]

How Big are All Infinities Combined?

[@ChaiElephant]

You *can* talk about all infinities in ZFC - you just can’t talk about the set of all infinities. That is to say, it’s perfectly meaningful to make assertions of the form “every infinity has such-and-such property”

[@pbsinfiniteseries]

Sort of. That's "meta" speak -- you'd be talking *about* ZFC (or "outside" ZFC) rather than talking with/within ZFC. Speaking within ZFC means using the propositional calculus of ZFC along with the axioms to derive more "theorems" (i.e. other propositions). The point is that, from within ZFC, "all infinities" is not even a bona fide object. It's not a set, and sets are the only nouns there are in ZFC.

[@GrothenDitQue]

PBS Infinite Series I think she means that you can, *within* ZFC via first-order predicate calculus, assert that: for all x, if x is an infinite then blablabla...

[@ChaiElephant]

VRB Blazy yes that's what I meant (I'm a guy by the way. Is this a first for the internet?)

[@GrothenDitQue]

Benjy Forstadt Good (haha no problem, excuse me 😂)

[@alan2here]

Does studying the properties of the |C| covered at the end allow us to know the properties of (well ordered?) things that are larger than themselves?

[@GrothenDitQue]

It is well underlined that such a "|C|" is not well-definable either in ZFC or in NBG, like the blood type of a doughnut: it is gibberish^^!

[@werner134897]

Even though this video is awesome in explaining ordinals, and cardinals , the Vsauce video mentioned is on a next level....

[@knarkknarkaren]

How can you add something to an infinite set? Anything you could possibly attempt to add would already be in the set.

[@pbsinfiniteseries]

Untrue. For instance, the set of all even integers is infinite. But 3 is not in that set. I can shove 3 in there, and voila -- I have a new infinite set whose roster is not identical to that of the prior set (i.e. the evens). Be careful: "infinite" is not a synonym for "all encompassing".

[@billcannon]

Gabe! I missed you every since you left Space Time.

[@FistroMan]

ONE really important question: in which part of ZFC or NBG is said that "the same element" can not be twice into the same set.?

[@diegoxl59]

Axiom of extensionality

[@FistroMan]

@@diegoxl59 THANKS A LOT! I will read about that axiom, and if it is good (what i need)I will put your nick like a reference 😁. It "seems" to be true, but when you talk with mathematicians you need to talk their language. Thanks.

[@FistroMan]

@@diegoxl59 I was writting a doubt when I realize... A={a,a,b,c} B={a,b,c} If a belongs to A, and a belongs to B... hmmm and if "a" belongs to them for the same "description" or equivalent ones, so if a belongs to A then a belongs to B... and if a belongs to B, it must belongs to A... A=B so... "a" just appears once in the set because is decided to choose, or you can choose the representation with "less" elements (we are talking about the same set). WOW! I just begin to "see" something... but I can't see the whole stuf... Without a "context" or a "clue" i wouldn't never see this is the axiom i need.

[@thesmallestatom]

But does any of this have anything to do with actual computation with numbers? I mean, my calculator here has like 12 digit accuracy.

[@andrewbeatty5912]

mind blown !

[@rkpetry]

*_...interesting sidebar requestion: Is the empty-set ∅, the-set-of-nothing {}, a subset of a set of something {0,,,}... does nothing exist where there is something-seems 'contra-dictory'... (unless of-course we make space for it, e.g. {,0}-something definitionally 'unmentionable')_*

[@MikeRosoftJH]

Let's first define the terms. A set X is a subset of Y, if every member of X is also an element of Y. So, rather trivially, the empty set is indeed a subset of every set. Also by definition, every set is a subset of itself. In set theory, a natural number is defined as a set of numbers less than itself; so 0 is the empty set, 1 is {0}, 2 is {0,1}, 3 is {0,1,2}, and so on. Note the difference between the element and subset relation: 0 (the empty set) is a subset of the set {1,2}, but it is not an element of the same set.

[@rkpetry]

*_...we could go a step further and 'invent' the set-theory integers as N1-N2, by defining a new kind of set that includes N1 but 'adcludes' N2, e.g. {N1;N2}..._*

[@MikeRosoftJH]

I really don't understand what you mean. (You have been either editing your response, or posted multiple responses and KZbin has interpreted it as a request to edit it and only kept the last one.) In any case, please refer back to the definition of natural numbers that I have given, and notice the difference between the 'element' (ϵ) and 'subset' (⊂, also written as ⊆ to make it unambiguous that a set is a subset [not an element] of itself). 0 is the empty set. 1 is the set {0} = {{}}, i.e. the set containing 0 (the empty set) as its only element. The only correct way to write the number 2 as a set (according to the above definition) is {0,1} = {{},{{}}}. Likewise, 3 is {0,1,2} = {{},{{}},{{},{{}}}}. The set {{{}}} is the set {1} (the set containing the number 1 as its only element) - and please note that {{{{}}}} is the set {{1}}, which is not the same as {2}, because 2 is {0,1} and not {1}. {,{}}, {,{},{,{}}}, and so on are ill-formed notations. For any sets a and b exactly one of these is the case: either a is an element of b, or a is not an element of b. You can't "make space" for further elements. (That is, unless you write something like {1, 2, 3, ..., 17, 18} or {0, 1, 2, ...}, when the progression is obvious.) If you add an element to a set, you get a different set. (This is the axiom of extensionality; a set is precisely defined by its elements.)

[@rkpetry]

...yes, many reasons to edit, totally replaced one comment, switched browsers on another, switched operating systems, font sets, etc., dropped one comment from reply-to-you-level to reply-to-my-original... e.g. set-minus "A∖B" looked okay on its chart but messed-up in the comment, so back to "A\B"...

[@friendlycactus2084]

Can you do a video explaining the Umbral Moonshine Conjecture?

[@chriswalker7632]

Not that this has anything to do with anything. But I recently figured out that: - [(100e - 1000)/(4e^2 - 20e - 200)]*[1 + (e/5)] = 5 (exactley) where e = euler's exponential constant 2.7182818284590452353602874713527.... Where 5/[1 + (e/5)] = 3.2390628582412427073000998076521... = h, Where 'h' is both the x and y coordinates for the centre of a circle with radius, R = sqrt[(e +h)^2 + h^2] and R = 6.7809648218962233939643589906224... Where that circle passes through the coordinates (-e, 0), (e, 10), (10, e), (0, -e). Where the points (-e, 0) to (10, e) are connected by the line with equation 1) y = [e/(e +10)]x + [(e^2)/(e +10)] and the points (e, 10), (0, -e) are connected by the line with equation 2) y = [(e +10)/e]x -e Where lines 1 and 2 from the above equations cross each other at the points y = x = (e^2)/10, on the line with equation y = x Where the point y = x = (e^2)/10, is the closest point between the curves y = e^x, and y = ln(x). and lines 1 and 2 run at a tangent to curves y = e^x, and y = ln(x). Meeting at y =[1 + (e/5)] on one of the points of y = ln(x) (LOL which I have only just realized while typing this - edit; it's only to 3 significant figures though)

[@WilliamDye-willdye]

This comment was posted shortly after it was announced that the InfiniteSeries channel has been cancelled. I should be going to sleep, but instead I'm watching these excellent videos and wondering why they were cancelled. Sigh.

[@TXLogic]

“You can still refer to all cardinals or all ordinals; I mean, we just did. You just can’t do it from within ZFC.” That is really badly ambiguous. It is of course true that you can’t refer to the *set* of all cardinals or the *set* of all ordinals in ZFC, as it is provable in ZFC that there is no such set. But you can absolutely refer to all cardinals or all ordinals in ZFC in the sense that you can assert things about all of them at once. For example, as you note, in ZFC you can say that *all* cardinals are (by definition) ordinals. Indeed, in ZFC you have to refer to all cardinals or all ordinals in this sense in order to deny there is a set of them: there is no set C such that *all* and only the cardinals (or ordinals) are members of C.

[@DavidSchoettler]

Vsauce counting past infinity will help at this.

[@DavidSchoettler]

when you realise you choosed the wrong ordinal...

[@IuliusPsicofactum]

Beautiful

[@PhyloGenesis]

It seems to assume that you can complete an infinite process without justification. "Append it to I" just hand waves away actually dealing with infinity.

[@asdfasdfasdf1218]

Yes, there are big missing steps in this video that it skips over. These are the theorems called the "transfinite induction theorem" and the "transfinite recursive definition theorem" and they are covered in set theory books, but which most people skip over because it should be "obvious" even though they do need proof, albeit very tedious and "going through the motions" type proofs.

[@brendawilliams8062]

I love to hear something on landau ghost.

[@BD-bditw]

So in simple terms: "If" the universe is infinite, and that's a big if, this surely holds true: There are more stars in the universe than there are atoms forming the Earth, a billion billion times more, and that in turn means there are billions of planets like just Earth with intelligent life and they have KZbin identical to ours. The logical conclusion is therefore that the universe is not infinite.

[@denysvlasenko1865]

> The logical conclusion is therefore that the universe is not infinite. Does not follow from the previous sentence.

[@niaschim]

Did you guys get my email‽

[@pierreabbat6157]

What is "row CRAY CRAY"?

[@kindlin]

I think the video editor just got sick of sizing that many things and decided to stop it humorously.

[@heyraylux]

Bingo

[@MikeRosoftJH]

In the previous video he listed some members of the set of natural numbers N and of the set of its subsets P(N), but P(P(N)) was just labeled "crazy", and P(P(P(N))) was even crazier, so it was "cray cray".

[@agentm-8389]

Really interesting

[@TGC40401]

... But does the set of all sets contain itself?

[@MikeRosoftJH]

In which theory are you working? It can't be the usual set theory - ZF or ZFC - because in this theory the set of all sets cannot exist. But in a theory which allows for the existence of the set of all sets - like the New Foundations - the set of all sets of course contains itself; it contains all sets.

[@TGC40401]

So... the foundation is tautological assertions, and the summit suggests that the collections of all sets is not itself a created set? Where have I heard this line of reasoning before?

[@android175]

Vsauce does a great job explaining these concepts as well.

[@niy0k0]

Is it re-uploaded?

[@twistedsim]

read pin comment

[@dougli1sqrd]

Vsauce's Counting Past Infinity here: kzbin.info/www/bejne/iaO4aox6pL14bpo

[@pbsinfiniteseries]

And I added the Vsauce link to the description section of this vid. Sorry, I thought it was already in there. --Gabe

[@leokovacic707]

And so the infinite series was left INCOMPLETE

[@Rasselas_Urasawa]

Is it just me, or did I read "defining" as definig? Im so used to seeing that word used in "definition"

[@michaelnovak9412]

why reupload?

[@twistedsim]

read pin comment

[@michaelnovak9412]

Simon Bouchard yea, when I wrote it there wasn't any comment yet.

[@15october91]

RIP ‘Infinite Series’.

[@deldarel]

I always thought 'countable infinity' is a stupid name so I'll use Olive Nut from now on.

[@BLOOMS]

I thought category theory was suppose to be the foundation for all of mathematics?

[@LordNethesis]

It can be, there are a few different axiomatic systems.

[@DavidSchoettler]

Isn´t the Hyperwebster like the sett of all infinitys because its including itselfe infinitly often? Sett theorie is hard!

[@timweber4286]

Imagine an infinite 3D universe filled with tennis balls each with a 50% chance of being green or red all touching each other and randomly distributed. Once in a while you would get a cubic meter of space filled with only green tennis balls. You would end up with a cubic light year filled with only green tennis balls far less often but it would happen given an infinite amount of opportunities to do so. Would you ever find a pattern of infinite space filled with only green tennis balls? Why?

[@MikeRosoftJH]

It is almost certain that you wouldn't, in the sense that the probability is 0. (I could have said that the probability is infinitely small, but it really means the same thing; in real numbers there are no non-zero infinitesimals.)

[@morkovija]

nickname does not apply

[@copperfield42]

so what were the changes? I notice no difference...

[@mcconkeyb]

After several viewing, I think I'm getting it! Can I get an AB- donut pls.

[@yavuzbahadrtaktak8020]

There was too little to Gödel's theorems, what a coincidence, channel shutted down -.-

[@sheeniebeanie2597]

cantors diaganolization baby! i had to use this to prove the reals are uncountably infinite in real analysis.

[@Geckobane]

My brain hurts.

[@laszlosebok4100]

Hello! I've got something in my mind, I can't forget, and I don't know the answer. As always said: It's countable if can give a rule how to order the elements, so if someone tells you a number you can count on which place it is. Than.... why is the numbers from zero to one uncountable? You can give a rule. For example: A=]0;1[ A={0,1; 0,2; 0,3; 0,4; ... 0,9; 0,01; 0,02; 0,03; ...0,09; 0,11; 0,12; .... 0,99; 0,001; 0,002; ... 0,009; 0,11; 0,12;... etc.} Why isn't it countable? :/

[@MikeRosoftJH]

Because using this scheme you only cover rational numbers; and even worse, you only cover some rational numbers - those with a finite decimal expansion, i.e. those whose denominator is of the form 2^a*5^b.

[@Bibibosh]

I think this guy is a alien.....

[@Euquila]

I'm more interested in the transformations of sets. This is also a set and that hurts my brain.

[@danielrhouck]

The blood type of this donut is chocolate.

[@hoodedR]

Reupload?

[@pbsinfiniteseries]

Yeah, the cut we originally put up was an old cut from a couple of weeks ago that had some errors in the on-screen text. I thought those errors were too serious to leave up, so when I realized this morning that the production team had accidentally uploaded the wrong cut, I asked them to pull it down and re-upload. --Gabe

[@fabricioguido8202]

why the reupload?

[@twistedsim]

read pin comment

[@Achrononmaster]

Wrong @10:10 ... "...even though we can describe every individual infinite item in it." We certainly cannot describe all infinities, not precisely. To do so would require a complete axiomatic scheme, which cannot exist and be consistent by Godel incompleteness. All we can ever hope for is ever more extended axiom schemas that get closer and closer to precisely describing the higher infinite set hierarchy, but this hierarchy will never be completed, at least not not by a finite bunch of mathematicians. So things are not quite as weird as Mr PBS here suggests. The only weirdness is in having to accept precise mathematical descriptions of all infinite sets are a fiction. That's not all that strange, unless you live under the delusion that mathematics gives you access to pristine infallible knowledge.

[@Bolpat]

It's funny how most of the ZFC axioms are harmless. There are three that I call the devil's triangle: Replacement, Infinity and Power Set. Remove any of them and ZFC has very graspable models. Without the Infinity axiom, the finite universe (the ZFC set of the hereditary finite sets) _is_ a model of ZFC minus Infinity. Without one of the Replacement or without Power Set axiom, the ZFC set V[ω+ω] is a model. In other words, Infinity is needed to get an infinite set in the first place. Power Set is needed to get to V[ω+ω], but to not get stuck there, you need Replacement. *Note* that Replacement is not a single axiom, it's parametrized, making it an axiom scheme. If it were a finite scheme, you could join them to a single one, so calling it out as a scheme implies that it comprises an infinite number of axioms.

[@FrozenFire55]

What in the nine hells did I just watch

[@thunderpeel2001]

It seemed so obvious to me that it would be gobbledygook. It seems a common mental trap to consider infinity as something that can have a cardinality, when the lack of cardinality is literally infinity’s defining nature. Or at least that’s how I understand it after watching these two videos :) So while we can talk about these examples, as if they were real, it’s just a highlighting a limitation of our ability to grasp infinity. Or something like that.

[@denysvlasenko1865]

> when the lack of cardinality is literally infinity’s defining nature. No, it is not.

[@brawnstein]

What if the "bag of infinity" acts as marking towards the infinities , like an index in our notebooks . The content of the notebooks is more than the index but index automatically contains that larger info . The cardinality of that index is still aleph-naught because we can still show a 1 to 1 correspondence with infinities . This somewhat shows that other infinities are just a shadow of aleph-naught . Maybe the "bag of all infinities " is just aleph-naught as an index , only the content changes in it . We can take an axiom that any information contained in the index is true , eg if we write "proof of Fermat's last theorem" in index it will be present in the set , just by writing a small statement in other index we allow it to contain the large 40 page proof . With the above example we can show that the "bag" can contain sets bigger than itself . Yours Sincerely Lover of Knowledge.

[@MikeRosoftJH]

There are way more than Aleph-0 different infinities. It can be proven (using transfinite recursion) that for every ordinal number x there exists x-th infinite cardinal number (this is the Aleph function). And since it also can be proven that there exist uncountable ordinals (and in general, it can be proven that for every ordinal number there exists another ordinal number with a greater cardinality), this means that there exists an (uncountable infinity)-th cardinal number. As the video says, the set of all different cardinal numbers (a cardinal number represents a "number of elements" in a set) cannot exist in ZFC. (Or in other words, the class of all cardinal numbers is not a set, it is a proper class.) Likewise for ordinal numbers (an ordinal number represents an well-ordered relation - a generalization of the notion of "first", "second", "third", ..., [and also infinity-th, infinity-and-first, and so on]).

[@brawnstein]

+MikeRosoftJH Ofcourse , you are right , but I'm saying that just like index in our notebooks we can place many infinities with a 1 to 1 correspondence like serial number . This reduces a lot of confusion and paradoxes . You can take reference with my Fermat's last theorem example

[@MikeRosoftJH]

I really don't understand where you are getting at. You seem to be confusing together sets, axioms, and theorems. Axioms of the set theory are given a priori; if you add something to them, you get a different theory (such as ZFC versus ZFC + generalized continuum hypothesis). A theorem is any statement provable from the axioms. Axioms and theorems are not elements of the theory! (On the other hand, the set theory, and indeed any theory strong enough to describe natural numbers with their usual properties is also strong enough to describe its own axioms and the concept of provability; for example, the statement "1+1=2 is provable in ZFC" will be represented within ZFC as "There exists a natural number x for which some formula F(x) is true". This is a crucial part of Gödel's incompleteness theorem.) Contents of a set cannot change. (Set theory does not deal with temporal logic.)

[@JW-fr4qb]

I am an element.

[@YuzuruA]

When you went through all the greeks and goes to judaic letters, you know mathematics has gone too far

[@amandubey6822]

1

[@jpphoton]

brass tacks

[@hoegoebaboe]

x ≈ ∞

[@sabriath]

So making it pointless to even comment on videos....cool.

Did someone ruffie him? I didn't know he could talk so slowly. Doesn't look healthy.

[@KaiseruSoze]

Numbers are symbols. How many symbols are there? If you represent a number as a symbol you are chasing your tail. Abstractions that refer to themselves are non-sense. E.g., "This sentence is false".

[@Lens98052]

@3:30, infinity is inserted "manually". What is the mathematical definition of "manual" here? Really, you are just extending, on faith, to infinity, which does not really define it. You are attempting to create the lowest version of infinity, which cannot be proven to exist, and actually is used nowhere in the real world, like in Artificial Intelligence, or computers in general. It is just a convenient short-hand, at best. However, humans being what they are, the more energy you invest in such matters without contradiction, the stronger is your faith in them.

[@Regalert]

Infinity is infinity. The end.

[@erawanpencil]

This seems ridiculously pedantic and rote recitation of unmotivated information.... is this some online course for high school kids or something? Rule number one for communicating math info- tell people why and for what you're about to explain so there's motivation. This guy just goes straight into weird encyclopedic recitation of definitions.... I'd like to know why von Neumann or others are defining things this way, not just listing out their bizarre lingo. The video is not at all "Defining Infinity"

[@peanut12345]

G Cantor goes GooGoo nuts and a group of "professors" turn his gibberish into ZFC, got that?

[@peanut12345]

It still is gibberish, just look Contractions in Logic or Law.

[@davidwilkie9551]

"Counting past infinity" is non-sense because anything counted is an identity of the infinite connection not the process. The set definitions are an example of the appropriate methodology of identification counted by infinity.(?) "An infinity of infinities" may be gobblygook applied to numbers of identities, but if it's applied to eternal time, it's just another identification process equivalent to reflections, ie the process of superposition combines all possible phase-states. If you know that probability states are phased time, then infinity of infinities identifies relative dimensions, quanta-images of infinity. Spacetime is not reflected Timespace but it might be interpreted as the condensed image generated by infinity inflated by counting time pulses.(comment, not mathematical argument) Eternity-now defines infinity dynamically.

[@naimulhaq9626]

The mind of God, only can define the algorithm of 'infinity', human mind only can at best discover the paradox of the paradox..... [of the paradox]. Mathematics is the only clue to the human mind trying to grasp 'intelligent design and universal consciousness'.-SMNH.

[@PasajeroDelToro]

Undefined. Unintelligible. Undivisible. Unreal. If, for set s ⊆S, |s|=1 and |S|=n∈N, then S ⊆ Exist. If S={ }=Ø, then |S|=0, then S ⊆ ¬(Exist). Let R=REAL⊆ Exist, then Un-REAL=¬(REAL)=¬(R)⊆ ¬(Exist) If S= ¬(R), then |S|=0, then S ⊆ ¬(Exist). If {Infinity} ∈ R, then {Infinity -n} ∈ (subtract n)*R ∀n∈N, where * is a binary operation. ..then there exists an element {Infinity - Infinity}={0}∈R, however if Infinity=Infinity+x ∀x∈R .. then {Infinity - Infinity}∉{0,R} => contradiction=>Infinity∉R=>Infinity∈ ¬(R)⊆ ¬(Exist). Infinity _is_ Un-REAL⊆ ¬(Exist). QED.

[@Bolpat]

It's funny how most of the ZFC axioms are harmless. There are three that I call the devil's triangle: Replacement, Infinity and Power Set. Remove any of them and ZFC has very graspable models. Without the Infinity axiom, the finite universe (the ZFC set of the hereditary finite sets) _is_ a model of ZFC minus Infinity. Without one of the Replacement or without Power Set axiom, the ZFC set V[ω+ω] is a model. In other words, Infinity is needed to get an infinite set in the first place. Power Set is needed to get to V[ω+ω], but to not get stuck there, you need Replacement. *Note* that Replacement is not a single axiom, it's parametrized, making it an axiom scheme. If it were a finite scheme, you could join them to a single one, so calling it out as a scheme implies that it comprises an infinite number of axioms.

[@Bolpat]

It's funny how most of the ZFC axioms are harmless. There are three that I call the devil's triangle: Replacement, Infinity and Power Set. Remove any of them and ZFC has very graspable models. Without the Infinity axiom, the finite universe (the ZFC set of the hereditary finite sets) _is_ a model of ZFC minus Infinity. Without one of the Replacement or without Power Set axiom, the ZFC set V[ω+ω] is a model. In other words, Infinity is needed to get an infinite set in the first place. Power Set is needed to get to V[ω+ω], but to not get stuck there, you need Replacement. *Note* that Replacement is not a single axiom, it's parametrized, making it an axiom scheme. If it were a finite scheme, you could join them to a single one, so calling it out as a scheme implies that it comprises an infinite number of axioms.

[@MikeRosoftJH]

And the same is true for the axiom of separation: every statement of the form "given set X, there is a set of all elements of X for which property P is true" (for every formula that defines property P) is an axiom of separation. (But axiom of separation can be omitted; it can be derived from the axiom of replacement. It only exists for convenience and for historical reasons.)