Every Infinity Paradox Explained

[@ThoughtThrill365]

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[@hardworkingcriminal4873]

Note to self never stay at Halbert's hotel. I'm not tryna keep switching rooms everytime a new schmuck shows up

[@seanpierce9386]

No worries, just ask your neighbour to move instead. And they can ask their neighbour, and they can ask theirs…

[@name-nam]

​@@seanpierce9386 double it and give it to the next person

[@invenblocker]

You tell me about it. I was just minding my own business in room 654, and all of a sudden the receptionist told me to pack up and climb the stairs all the way up to some new room with a 197 digit number. Not even sure what the point is, no one else moved into my old room, it was just left vacant.

[@nissantzvitovey]

To be honest, they could solve their issues by simply making sure that they always have infinitely many rooms available at any given time. Have everybody move once, and you're done. And if they have a problem with it, there are plenty of ways to make the proportion of empty rooms arbitrarily small the higher their numbers.

[@neji2401]

An infinite amount of people can refuse to do that, and you'd still have an infinite amount of people who accept!

[@Pocketnemo]

The lamp is gonna be off by the end of the 2 minutes, because if it's not, Dad's gonna yell at me for wasting electricity.

[@PrzemyslawSliwinski]

The paradox of the Gabriel's horn is that you need an infinite amount of dye to paint it inside with a brush, but only a finite amount of dye to fill it (and paint it anyway).

[@piercexlr878]

Arguably you need 0 paint to paint it since its infinitely thin

[@PrzemyslawSliwinski]

@@piercexlr878 On my horn the paint is r/3 thin.

[@piercexlr878]

@@PrzemyslawSliwinski I feel like that has to converge, but I'm gonna guess it doesn't

[@piercexlr878]

@PrzemyslawSliwinski Oh right you can convert that into an area which gives a finite amount of paint

[@PrzemyslawSliwinski]

@@piercexlr878 My r tends to zero as x tends to infinity.

[@klikkolee]

My long-time statistics hot take is that the "expected value" operation is given undue significance. Its name is very suggestive of it being the inherently correct way to judge risk and reward, but the fact of the matter is that it is fairly uncommon for it to actually yield a value which you can, in any meaningful way, expect. The St Petersburg paradox shows this well. There is less than a 7% chance of getting $32 or more from the game. A person simply cannot, in the ordinary sense of the word, expect more than $32, regardless of the value of the operation which has been named "expected value". Pascal's mugging is another "paradox" that just comes down to the expectation operator being treated as gospel. A mugger says if you give them your money, they will give you x amount back later. No matter how improbable you believe it to be that they will uphold the deal, they can give an x for which the "expected value" of taking the deal is positive, giving muggers a surefire strategy against people who treat the "expected value" operation as the ultimate arbiter. "Expected value" is only genuinely accurate to its name when the risk is taken infinitely many times. Humans aren't immortal. The risks also often have an ante of some kind, and if the risk failed enough times, you can't afford the ante anymore and can't try again. Even with large corporations, which can absorb many more and much larger failed risks than people can, risk assessment goes beyond expected value -- they qualify or quantify the level of risk itself and compare that to a risk tolerance

[@Daniel-ng8fi]

I was a professional poker player for a long time, so EV is a concept I'm vary familiar with. It can be incredibly beneficial to think in terms of EV in countless real life scenarios. However, real life is just as you said, not infinite. So you also have to factor in utility, otherwise you get into the absurdities you mentioned and the concept loses the value and importance it has.

[@partyxday]

Expected value is very useful in science. Ignorant comment.

[@PC_Simo]

This, very much, makes sense 👍🏻.

[@piercexlr878]

Expected value is great for getting an idea of how things shpuld trend. But you have to temper it with analysis into short term risks. The combination works really well for evaluating an overall idea by seperating short and long term benefits

[@n484l3iehugtil]

What do you think about taking the median instead of the mean? (Me using statistics terms instead of probability terms.) It sounds like the median gives you a much better idea of the short-term reward: you have 50% chance to get above the median and 50% chance to get below.

[@andrewmichel2525]

For Gabriel's horn you were supposed to calculate the surface area of the area of revolution, not the area beneath the curve of 1/x. The surface area integral is the integral from 1 to infinity of ((1/x)sqrt(1+(1/x^4)))dx which you can bound below by the integral you did, which diverges.

[@seyerus]

I was just about to point this out.

[@dawica]

My thinking on the Ross-Littlewood paradox is that if the balls are removed in order, whenever any ball of number ‘n’ is removed, the vase will still contain ball n+1. So even though for all values of n, ‘ball n’ will be removed, it is impossible for the vase to be empty.

[@n484l3iehugtil]

While I think your point is valid, I think the question is asking about the state of the vase after all actions have been taken. The paradox is saying that every ball will have been removed at some action before the end, thus one cannot claim to have a ball still left in the vase.

[@dawica]

@@n484l3iehugtil And I understand that. My thinking is along the line of Cantor's Diagonal Argument for proving the real's are uncountable. Simplifying for brevity, it says that if you list "all" reals in a numbered list, a new real can be constructed from the nth digit of every number n, adding 1 (without carrying, so 9s roll back to 0). Since this number differs from all others in the list by at least one digit, it must not be in the set, and thus the set does not contain all reals. So even when starting from the premise that the set contains all real numbers, it can be shown that it doesn't. Similarly, even with the premise that "all" balls will be removed, the last ball removed will always have its successor still within the vase, and thus the vase can never be empty. Another more mechanical proof is that with each step, you add 10 balls and remove 1, so the number of balls at each step is 10x - x, and the limit as x approaches infinity for that is infinity.

[@theshaggycreeper220]

⁠​⁠​⁠@@n484l3iehugtil I get what you’re saying but, (and I may be wrong) at least in their example there is always 9 times as many balls as have been taken out so no matter what number of steps you take even if it’s an infinite number of times there will always be at least 9 times as many balls in the vase as have been taken out. The mathematical format of what you appear to be stating is: “The limit as n approaches infinity of (10n-1n)” Which would normally create a indeterminate form of (infinity-infinity) but, it can always be factored which then just gives n(10-1) or n(9) with the limit as n approaches infinity therefore = infinity

[@nisonatic]

The problem with the paradox is in the idea of a super-task. It doesn't matter if the steps take 0 time, an infinite number of steps are never finished. Imagine you're driving around a roundabout, what direction are you facing when you get to the "end" of the roundabout? It doesn't matter how fast you drive, or if you're doing extra stuff, it's a circle. What does this program do if it's run on a machine that's infinitely fast? 10 GOTO 10 20 PRINT "Hello" It just hangs. Putting balls in a vase doesn't change anything: 10 balls = 0 20 balls = balls + 10 30 balls = balls - 1 40 GOTO 20 50 PRINT balls You never get to line 50. It's just an infinite loop with busywork.

[@bqazo]

@@nisonaticthe point of the thought experiment is to assume it happens and explore the outcome, not to explore whether or not its possible. thats like when someone asks u "what would u do in a zombie apocalypse" and then u respond "zombies dont exist".

[@roxrockman]

I'm so high right now that this actually makes sense!

[@Alph413]

Good one!!

[@no_dogs]

A true nerd, high and still watching maths videos

[@giddycadet]

the probability of hitting any individual point on a dartboard changes quite a lot when you remember that the point of the dart itself ALSO has an area to it, and thus its own infinite number of points

[@n484l3iehugtil]

☝🤓 Well for the sake of theoretical argument, assume the dart's point is infinitely thin.........

[@Lisa-dd1uq]

Infinity is infinite, even if you subtract infinity from infinity there will still be infinty left over. The amount of points on the dart changes nothing.

[@giddycadet]

@@Lisa-dd1uq it's not a matter of subtracting, but dividing. and in analyzing this particular problem physically there is actually no need to resort to infinites - both the relevant areas (dart and board) can be measured in real units with physical tools

[@Lisa-dd1uq]

@@giddycadet it doesn't matter if your subtracting or dividing, there is an infinite number of infinitys in infinity

[@giddycadet]

@@Lisa-dd1uq ok, think of it this way: if there are an infinite number of points on the dartboard, and an infinite number of points at the tip of the dart, then you say that the chance of ever hitting one specific point on the dartboard is zero. but if the numbers are both infinite, then the chance of hitting any individual point on the tip of the dart must ALSO be zero - even though geometrically we assume that ALL points on the tip of the dart are hit when it enters the dartboard.

[@billpines06]

How does this affect the local trout population

[@komos63]

negatively

[@garyb6219]

I asked some trout that very question and the answer they gave me was, disturbing.

[@commitfelonyfeline]

As a trout, yes

[@Brindlebrother]

Infinitely. 🐟

[@kennymartin5976]

The real question about Gabriel's horn is "does it go doot?"

[@mitabpraga7487]

The key to resolving any apparent paradox is one's ability to spot the bullshit. Once the bullshit is removed the paradox ceases to exist.

[@randyzeitman1354]

Exactly. "Infinite number" .... no such thing.... the strawman of a veridical paradox.

[@Aera223]

lol

[@ChineduOpara]

😅

[@alexritchie4586]

Perhaps the real bullshit is holding hard onto the notion that the Universe in general gives two shits about what human epistemology considers a paradox, and will adjust itself accordingly to satisfy our limited capcities of observation, comprehension, and rationality.

[@Yupppi]

Yet sometimes it's good to recognise what kind of paradox you're dealing with and recognise the limits of your models to make a useful decision. Way too often we use our models without considering their limitations and tell people to shut up when they point out the limitations and how they could lead to problems if ignored.

[@fulltimeslackerii8229]

The hotel one isn’t even a paradox because how could it ever be full in the first place?

[@Xhu666]

It’s only countable infinity, so if an infinite number of buses filled with an infinite number of people arrive at the hotel looking for a room, it’s uncountable infinity because the hotel will always need to provide +1 room than the number of tenants to maintain vacancy, and there are two layers of infinity stacked against the hotel’s singular plane of infinity.

[@benjaminhill6171]

​​@@Xhu666I don't think that answers their question. @fulltimeslackerii8229 The paradox isn't whether or not it's physically possible to have an infinite hotel, or for it to be completely filled. The paradox is about what happens if you could. This is a purely mathematical paradox because it's just a way to visualize how you can fit more numbers into an infinite set without changing its size. Also, this is the kind of paradox that is only called a paradox because it's unintuitive, not because there's actually a logical contradiction - because there's no contradiction here.

[@zackdrake8735]

Also where did they even afford that many rooms 😭

[@josephoyek6574]

​@@benjaminhill6171why dont the new guy just go to the last unoccupied n+1 room instead of makimg everyone move

[@benjaminhill6171]

@@josephoyek6574 Great question! All the rooms are already full at the start, so there actually isn't an empty room at any "n+1" place. That's why it's unintuitive: there's no room at the start, but just reordering the people opens up a room. It's an analogy to show that if you add things to an infinite set, that set doesn't get any bigger.

[@binbots]

Math is just counting an infinite amount of zeros. All negative and positive numbers cancel each other out. Zero flips the number line from negative to positive (-0+). Infinity flips it from positive to negative (+0-). Which is how we can fit infinity into a finite space.

[@theshaggycreeper220]

I mean correct me if I’m wrong mathematically but the Ross-Littlewood paradox is just the infinite sum of 10-1 (or just 9) with n=any natural number because no matter what on each step, the net number of balls being added in = 9. At any given point in time there is at least 9 times as many balls as have been taken out. Of course graphically it just goes to infinity which can be proven with a limit, I don’t quite get the paradox of “Well at some point every ball has to have been taken out of the jar” as mathematically that’s just impossible. Even if you tried to make it a limit formed as “the limit as n approaches infinity of 10n-1n” (as to of make it indeterminate by (infinity-infinity)) you can just factor it out as n(10-1) which after taking the limit still approaches infinity

[@benjaminhill6171]

The problem is that you can't necessarily factor infinite numbers like that. The algebra concept you're using is factoring for finite numbers. Infinite numbers are not guaranteed to be so well-behaved, so it's not a valid operation. This is a paradox because the solution is completely indeterminate - if you assume that any number is the answer, it leads to a contradiction. If you assume there are balls left, then they must have numbers, and once the supertask is finished all of those numbers must have been removed, so there can't be any balls left. But if you assume that there are none left, that's also a contradiction because there must be at least nine left because that's the net number added in the "last" step. Anything you assume leads to a contradiction.

[@mistahmatrix]

@@benjaminhill6171 agreeing with benjamin, since the limit of the function '10n - n' can be said to go towards infinity, we plug in infinity to get '10*inf - inf', however any number * infinity is still infinity, so we get 'infinity - infinity', which is indeterminant.

[@pedramhashemi5019]

Your production quality is getting better and better! keep up good work :)

[@Koroistro]

I find the St. Petersburg paradox very interesting, it highlights (as many other paradoxes here do) how "right" results are wrong when we ignore other variables applicable to that context. For example looking at the variance. While the EV is theoretically infinite we can plot every price of the game to the odds of turning a profit, and how much total capital we would need to ensure a profit in the long-run.

[@nicezombie8054]

You could calculate with a logarithmic function how much it would do to you where you to win or lose and that would then be based on the money you own and you would get a more accurate result, this assumes that doubling your money has the same absolute impact as halfing your money

[@taherkorbi595]

<a href="#" class="seekto" data-time="237">3:57</a> the lamp will be off because you broke it with infinite toggling

[@crispyandspicy6813]

Hilbert's Hotel's Murphy's Law: Even though they can accomodate an infinite amount of guests they never seem to have enough money to fix the ice machine

[@luismuller6505]

Regarding the St. Petersburg Paradox, I think the reason for why people would not pay an infinite amount of money to play this game is that if the probability of tails is even slightly less than 50%, then the expected value immedeatly becomes finite. For example if the probability of tails was 47.9%, then paying 25 Dollars would already put you at a disadvantage as a player.

[@LoganPost-c6p]

Although this is mathematically true, I disagree that this is the primary psychological reasoning. Borrowing from economics, a person might intuitively aim to maximize expected “utility”. Bc of diminishing returns, utility is a sub-liner function of money and thus under-weights rare, high-value outcomes. This generalizes to other behaviors as well

[@its_lucky252]

it's actually because the increase of payout is proportional to the chance of the payout

[@BryanPorten-Willson]

@@LoganPost-c6pthank you for saying this. money usually doesn’t add never-ending value/happiness to one’s life. therefore, there’s no paradox in not investing in this game

[@AscendedCup]

Pretty sure it's because most people don't have infinite money laying around.

[@landsgevaer]

I surmise that it is because the chance of losing is much bigger than the chance of winning. If I had to pay $1000, my chance of net winning money would be approx 0.1% which feels negligible (unless I made an off-by-one-error, but it would still be very small). The fact that I would win a lot if I win does not outweight that (you could insert utility here, but I suspect people are simply sensitive to probability of winning in addition to expected payout).

[@deim3]

We call everything paradox now, eh? Paradox is something, that contradicts itself by its own rules and cannot be resolved. "This statement is false" is a paradox. The "if a turtle runs 1/10 of a distance a rabbit runs in the same amount of time, so the rabbit will never catch the turtle" is not a paradox. Firstly because it doesn't contradict itself, it contradicts your intuition. Secondly because it can be resolved with sufficient understanding of calculus.

[@akultechz2342]

The Thomson lamp paradox is as equivalent as to figure out whether sin(infinity) will be -ve or +ve or 0

[@lancesmith8298]

Or in other words, if infinity is even or odd

[@akultechz2342]

@@lancesmith8298 i could say its even since infinite/2 = infinity However if u take deeper look, the two infinities here arent same, infact they dont corelate at all so i guess i could also say its odd, that is if infinty was real at the first place!

[@FishSticker]

Fun counterexampke to ross little wood, add 10 balls, numbered, remove the smallest number divisible by 3, repeat. After the super task is complete there will be infinite balls, every ball not a multiple of 3

[@ultimazilla9814]

when you've been walking down the hall in Hilbert's Hotel for the past 10^6875 years to reach your room and an announcement plays saying "please move to the room number that doubles yours"

[@UltimaDJS]

<a href="#" class="seekto" data-time="170">2:50</a> this one doesn’t make sense to me because even if the new real number is not found anywhere in the current real numbers index there will always be another natural number to reference that index. In this wise I can see it being paradoxical but not in the way the problem is stated. In my interpretation of the paradox it’s more like how can a set that can be infinitely broken down also itself be considered infinite logic would dictate the real numbers must be a larger infinity but because there will always be a natural number index that can increase with every real number in that index both must be the same magnitude, or maybe I’m misunderstanding something. <a href="#" class="seekto" data-time="435">7:15</a> I also am having a problem with the gabriel’s horn not because the proof isn’t understandable but because despite pi being represented as a solid number solution it has infinite places after the decimal meaning that itself must represent an infinite volume as it’s the same a saying 3 + 1/10 + 4/100 + 1/1000 etc, with every decimal point being the equivalent of adding a fraction of volume equivalent to number over place if that makes sense, so though it would appear to be finite and is indeed useful in finite calculations it is still infinite. Or once again I may be misunderstanding something, I’m really not any sort of mathematician. <a href="#" class="seekto" data-time="508">8:28</a> this one doesn’t seem like a paradox to me since every iteration is the equivalent of saying “balls in vase=balls in vase+9” because every iteration, even if you pull out the index value ball, you are still leaving 9 balls in its place with every index. So the real problem would look like “balls in vase=balls in vase + 10 - 1” which is just “balls in vase=balls in vase + 9” because no matter what index we are at we are still adding 10 and removing 1 so it will never and could never empty completely.

[@UnluckyLilly]

For pi, yes you are misunderstanding decimals. No matter how many numbers come after the decimal point in pi, it will NEVER be greater than 3. This is obvious when you consider that 1/3 is 0.333 repeating infinity, but we know for a fact that 3/3 is 1. So no matter how many 3’s there are after the decimal point, they will not make 1/3 infinitely large. As for the jar, you’re misunderstanding infinity. Infinity is not a number. You cannot add 9 to infinity and make it bigger, it doesn’t work like that. The reason it’s a paradox is because if you remove the ball labeled 1 the first time you add 10 balls, and remove the 10 the 10th time, infinite times, then you have removed every single number. It doesn’t matter that 10 times infinity is added, because 10 times infinity is still infinity. You can’t do algebra with infinity because it’s not a number. Though paradoxically, if when you add the 10 balls you remove a ball that is even, like 2, then once you’ve removed the infinite amount of balls, there’s still the infinite amount of numbers in the jar that are odd.

[@UltimaDJS]

@@UnluckyLilly still having trouble understanding. If Pi will never be larger than 3 (which it is but I understand your point) then the volume as well could never be greater than x amount meaning as pi can’t represent infinite volume in this circumstance then the volume itself can’t be infinite. The jar explanation still doesn’t make sense. Infinity isn’t a number but a representation of uncountable numbers, possibly unending. So yes adding numbers to the concept of infinity doesn’t change the concept, just as subtracting numbers from it wouldn’t change it, but the uncountable number it’s added to would change by 1. That’s not entirely the argument I want to make anyways but it seems to me like a more grey argument than anything else. I think that it goes along with the larger and lesser infinite concepts you are subtracting a lesser infinity from a larger infinity which would still be infinity. But I’m not all caught up on infinity theory so again I could be mistaken.

[@gokk99]

Hilbert slayed with that hat

[@Terrahex1]

Hilbert's Hotel: we can accept any number of guests if we keep an equal number distracted long enough to kick someone out for them

[@user-po2oi5yq7q]

<a href="#" class="seekto" data-time="488">8:08</a> I think the problem is that you can’t label all balls with just an integer. By contradiction, assume you can label each balls with an integer. After putting in 10 balls and taking out 1 ball infinity many times, there’s infinitely many balls in the vase. However, for any remaining ball, there’s no integer that can represent it, which contradicts our assumption.

[@radupopescu9977]

Good video, BUT click bait title!!! Knowing infinity, it obvious there are infinitely many. Banach-Tarski paradox, or Pythagoras theorem which is also link with infinity (infinitely small)

[@samueljames8654]

My favourite 'paradox' is the continuum hypothesis which is likely the simplest to express unaswerable question in ZFC. Though explaining exactly how it works and why it is the case is a lot of fairly heafty maths

[@NichaelCramer]

An interesting variant (or at least a related) of the Ross-Littlewood (Vase and Balls) paradox: 1] Alice starts building a stack of dollars bill by adding 1000 bills at a time. This continues for an infinite number of turns. 2] Bob’s task is to ensure that there are no bills left in the stack at “the end”. 3] Bob must choose between the following two strategies: Option 1: Each time Alice places her 1000 bills, Bob can remove 999 bills from the top of the stack. Or Option 2: Each time Alice places her 1000 bills, Bob can remove 1 bill from the bottom of the stack. So, in short, which is the winning strategy?

[@benjaminhill6171]

That's really interesting.

[@NichaelCramer]

@@benjaminhill6171 : Right. I’ve always thought that the correct answer (that is, removing 1 bill each time results in 0 bills left in the stack, while removing 999 bills each times results in an infinite number of bills) is pretty much the definition of a counter-intuitive answer. ;-)

[@sunbleachedangel]

"Some infinities are bigger than others" must be one of the most misunderstood phrases ever (not here of course)

[@simonwillover4175]

<a href="#" class="seekto" data-time="504">8:24</a> if you choose to remove the N-th ball, then you will run out, yes. But if you instead choose to remove the ball with the number of the n-th prime, then you will have all of the inifnitely many composite balls.

[@roberthill2462]

An important thing to note about Cantors Diagonalisation proof is that is is a proof by contradiction; it is assumed that there is a mapping between all the real numbers between 0 and 1 and the naturals then using logical steps you show that there is a number between 0 and 1 not in the mapping. From there a contradiction is raised (the value both being in the mapped values and not) and the initial premise is rejected. It’s a bit pedantic but distinct to being able to add it to the mappings and do it again.

[@uBreeze]

For the dartboard paradox, with infinitely small points, when split into a quadrants, how does the premise of a dart being able to hit the precise center or even just the precise divide between two quadrants affect it, wouldn’t that mean there’s a non zero chance of it landing outside just one quadrant, meaning it wouldn’t be precisely 1/4, but infinitely close to 1/4?

[@wiener_process]

The probability of the dart hitting the dividing line is 0, so the probability of the dart hitting the interior of the quadrant is the same as the probability of the dart hitting the interior or the border. 1/4+0=1/4

[@quinnalexander3825]

What I learn from this is that there's a lotta thought snags you get yourself into when you base your line of logic on totally hypothetical equations.

[@jimliu2560]

@<a href="#" class="seekto" data-time="35">0:35</a> If all the rooms up to infinity are “filled”, how can N be moved to N+1…? That is only possible if you switch to different, a larger infinite-hotel……therefore it’s Not the same hotel…so it is just word-trickery…

[@michaelmicek]

I don't understand why you say that the person in room N can't move to room N+1.

[@jimliu2560]

@@michaelmicek Because by definition, all the rooms are occupied ( or listed)….. that is what the first type of infinity means….ie to be “counted or listed” If it’s all listed, then how can you have a unlisted room…? In order to move to N+1, it’s switching to a different hotel , ie a different set..

[@dumblr]

i think it is because the person in room n+1 would be in room n+2 and n+2 in n+3 and n+3 in n+4 thus you cannot find a guest doesn't have a new room

[@jimliu2560]

@@dumblr In order for N to move to N+1, you must have a empty room (or another decimal place; ie 2.11. Vs 2.111) In that case, it’s just word play. (Room)-occupied doesn’t mean it’s “All” occupied, it just means “only the listed rooms” are occupied! If I say there are no numbers between 2.1 and 2.2… You say: 2.12 exists (the extra room). I say: 2.12 was never listed. If every decimal point was not listed, then the hotel was never “occupied” to start with…

[@n484l3iehugtil]

@@jimliu2560 The guests all move simultaneously. They each leave their room at the same time, and enter the next room, which is empty by then.

[@HyperSarcasticAvocado]

I feel like hilbert's paradox works because we say that it works. How can a countable set of things be infinite? If you go backwards and empty the hotel using n-1 do we eventually get to 1 person occuping the 1 room in the infinite hotel making infinity =1?

[@Linuxdirk]

I love that the video just starts without any of the common distractions! 👍

[@tonik2558]

I think your explanation of Cantor's diagonal argument doesn't hold up. Why can we map the reals to the naturals in the first place (what if we run out of them)? Also how did you map them? And why shouldn't we be able to do the Hilbert's hotel thing and move all the reals one space down to make room for more? These things can be explained/worked around, but you left out the parts of the proof that do that. The parts you did keep don't really follow a logical flow either. Why are we even constructing a new number? You explain it, but only after making it...

[@elderfrost9892]

I'm not sure which bit of it you're questioning, but the point is that you can't map the naturals to the reals, no matter what way you try to link the numbers together. The reason for not explaining the way they are mapped is because it has to be arbitrary, or the proof has to work no matter what way they're mapped. if you said "well 1 will align with 0.0000001, 2 with 0.0000002" and so on, obviously that won't work for listing all real numbers, but you've only proved it for that specific arrangement, not any arbitrary arrangement. The point of the diagonal is that you *always* can add one more real number, even an infinitely long one to an infinitely long list. if your list is infinitely long, you've listed all the natural numbers already, and theoretically that should also mean you've listed all the reals. but if you can add another, or even infinitely more, that means you don't have all of them, so the mapping doesn't work. In regards to the hilbert hotel thing, yes, you can keep adding numbers to the beginning of the list! But, since you're doing the exact same checking and number creation every time you add a number, you're not actually going to make any progress in listing all of the real numbers, and you're not going to hit a point where you have all of them and the diagonal strategy doesn't work, even after an infinite number of reals added. Hopefully that makes a little more sense, I feel like I said number and infinity a few too many times :P

[@Archanfel]

Proper video on this topic should be infinitely long. It is impossible to explain every infinity paradoxes in finite amount of time. Or it create new paradox.

[@ExternusArmy]

Can someone tell me how Cantor’s argument is legitimate? If you match all numbers between 0 and 1 to all natural numbers, the diagonal argument is invalid. The new diagonally constructed number is contained somewhere within the set of numbers between 0 and 1 by definition and thus has already been matched to a cardinal number somewhere. You haven’t proved anything since this would involve repeat numbers which results in uncountablility.

[@fortune500b]

The new number cannot be in the set, since it differs from every number in the set in exactly one place by definition. Thus there will always be a real number not mapped to a natural number, and therefore there must be more real numbers than natural numbers

[@ExternusArmy]

@@fortune500b thanks for the answer. How can the new number not be within the set when the set is defined as all numbers between 0 and 1? By definition, even after the diagonal process, the number was already within the set. I understand the method but it’s like saying you’ve created a number that isn’t within the set of all even numbers by starting with 10 and subtracted 2. 8 was always in the set, you’ve just created an alternate method of obtaining a number that was always already part of the set. I understand that I’m probably not the first person to bring this up but I never found an answer for this concern.

[@fortune500b]

That’s why the proof is a proof by contradiction. Cantor is saying it’s impossible for every possible real number to be in a set because you can always generate a new one. The set of real numbers is said to be “uncountably infinite” while the natural numbers are “countably infinite.” It’s counterintuitive and hard to imagine how one infinity can be bigger than another but that’s math 😂

[@ExternusArmy]

@@fortune500b yeah but you can always generate a new natural number too. The argument that it’s not counted for is invalid to me because that diagonal number already exists since the set containing all numbers between (0,1) by definition contains the number so long as it is between 0 and 1. That’s why I used the analogy of the number 8. Just because we generated it the number through a formula doesn’t mean it wasn’t already in the set. The number must’ve been mapped already and so we’ve created a duplicate which is the source of the uncountability.

[@fortune500b]

​@@ExternusArmy The proof relies on a contradiction and two things that cannot be true at the same time. _If_ we were able to enumerate the entire set of every possible real number between 0 and 1, then the newly generated number must be in that set. But where is it? It's not the first number, since it's different by one digit in the first position, not number 2 since it differs in the second position, for any element N it is different in the Nth place. It could not be equal to any other element in the set. Therefore we can always create a new number that is not in the set. So which one is true? How can the set contain every possible number between 0 and 1, and also we always have a way to generate a new number between 0 and 1 that is not equal to any other element in the set? The conclusion is it's impossible to create such a set, and thus there are more real numbers than natural numbers.

[@oneMeVz]

<a href="#" class="seekto" data-time="195">3:15</a> your method to generating a new number will trend to numbers of only 8's and 9's

[@michaelmicek]

Yeah, you're supposed to stop the proof as soon as you've got one number not on the list, contradicting the hypothesis.

[@n484l3iehugtil]

The proof method works just as well in base 2 as it does in base 10. In fact, I think base 2 makes it easier to see.

[@opinionhaver574]

I used to have a paradox but then I realized I only needed one.

[@DistrarSubvoyikar]

xD

[@blogomfox]

<a href="#" class="seekto" data-time="399">6:39</a> smoke alarm battery ..

[@IOverlord]

Another paradox: Why do I feel infinitely feel more smart when I'm actually infinitely more dumb lol

[@gemstonegynoid7475]

Those who are truly smart understand there is always something more to learn. Those who arent smart will think theyve got an understanding of everything already.

[@UndefinedFantasticCat]

Ross-Littlewood paradox is just Grandhi's series 2.0: instead of series alternating between 1 and -1, this one alternates between 10 and -1 in both cases the sum cannot be defined properly, Ross-Littlewood just needs an extra step to see that

[@michaelmicek]

Grandhi's series alternates between 0 and 1. It is said to diverge, but not to infinity. The numbers in Ross-Littlewood do diverge to infinity.

[@UndefinedFantasticCat]

@@michaelmicek and yet you can group them in a way that shows an unidentified sum, which is why paradox exists

[@herrhartmann3036]

The spot you choose on the dartboard may be infinitely small. But the dart itself is not! Therefore, the dart does not actually hit any one spot. Instead, it hits a small area. And you can totally calculate the probability of your chosen spot lying inside that area.

[@Benjamin1986980]

@@herrhartmann3036 i did that in another comment thread. Using a dart point area of a hundredth of a square millimeter and a regulation 13.25 in diameter dartboard, there are roughly 9 million potential dart hits in that don't overlap. Now, if you are counting overlapping points as different, we have the issue of atomic width. After all, a dart cannot go through a foam atom. It can only go between two, and so this too is calculable and finite (though I'm not doing it).

[@samsonreed3150]

hey i just thought about this and the diagonal one can use the same argument but backwards, meaning there are both more real numbers than natural numbers but also more natural numbers than real numbers

[@johannesvanderhorst9778]

Some more infinity paradoxes Consider "the set of everything." By the diagonal argument (on subsets rather than on digits) one can prove that it doesn't coontain the power set of "the set of everything", hence "the set of everything" doesn't contain everything. Solution: modern set theory gives axioms that give a restriction to what can be a set, denying "the class of everything" as a set. So it has no power set. According to (modern) set theory, cardinal numbers and ordinal numbers are both sets. A cardinal number is defined to be an ordinal number alpha that is equal to the lowest ordinal number beta such that alpha can be mapped injectively into beta. Then the smalles transfinite ordinal number often is denoted w (in fact we call it omega.) It is also a cardinal number, and as a cardinal number it will often be denoted Aleph_0. But, w^w is countable while Aleph_0^Aleph_0 is uncountable. Explanation/solution: when doing ordinal numbers, alpha^beta is an ordered set. But when doing cardinal numbers, alpha^beta is just a set (and we forget about order.) So w^w differs from Aleph_0^Aleph_0. Banach - Tarski Paradox. It is possible to 'cut' a solid ball into a finite amount of pieces and reassemble those pieces such that you assemble two balls, each being a perfect copy of the original ball. However, each of those 'pieces' is in fact an infinite scattering of points, rather than a solid piece of the ball. Achilles Paradox. Consider a race between Achilles (who was able to run very fast) and a turtoise (who was quite slow), with the turtoise starting ahead. Each time Achilles covers the distance between him and the turtoise, the turtoise also moved a little bit. Hence Achilles has to cover another distance to catch up the turtoise. This repeats infinitely many times. "So Achilles can't win the race." It took quite the time before this paradox was finally solved. First we had to define a notion of "converging limits" before finally reason well why Achilles is able to come ahead of the turtoise. The point where Achilles will pass the turtoise is the limit point of the set of points where either is at any of the mentioned moments. These are the moment the race started, the moment Achilles has reached the starting point of the turtoise, the moment Achilles has reached the point where the turtoise was at the moment Achilles reached the starting point of the turtoise, etc.

[@wizardswarrior]

Because I’m bored and slightly petty, I want to “disprove” the dartboard paradox. First what is the surface area of the average dartboard, roughly 254.47 square inches Next, what is the size of the point of a dart needle, the tip has an approximate size of 7.11mm, or roughly .27 square inches. Assuming the point is equivalent to a single dart needle’s width, there is a chance of hitting that point roughly once out of 943 throws

[@hevn3690]

The diagonal argument is bs in my opinion. If you are able to generate and add more real numbers to the list you should also able to always give them a natural number index, since you never run out of natural numbers, it's the same infinity, infinite is infinte, it is only reached faster with real numbers than with natural numbers, but the "size" should be still the same in my opinion and understanding. What you do on the right side on the list you can always do on the left side. If you add something to right you can also add an index to left. Since the last real number can also never be reached since there are infinitely many of them, it should also not be possible to generate a new real number with this method, since you would never reach the end of the new "over-infinite" number you are trying to generate. When the number itself has to be infinite in digits to be called something "new" which is not indexable with natural numbers, than the argument is still bs, because then you are also allowed to come up with endless digit natural numbers, which will always give an index to any real number.... so I don't get the explanation to the argument. In my opinion it does not work and is flawed. Maybe someone can give a better explanation why this is believed to be indeed true, that there are a greater infintiy of real numbers than natural numbers.

[@InYourFace7861]

The assumption we make is that the sizes of the two sets are equal. Then we pair the elements from the sets one to one. The contradiction is that there DOES exist a rational number which is not in any of the pairs. The key insight is that both sets are assumed to be "complete" from the beginning. When we "generate" a new real number we aren't adding anything, we are simply showing that there is a real number which is not in the list, showing that the set of real numbers we started with is incomplete.

[@spaceman688]

You have two fixed set (the theorem is in regards to reals and naturals, and you cannot just ad new numbers to any of them). We assumed a fixed bijection. If they have the same cardinality then we must have a bijection in hand. We show that the bijection is in fact not a bijection. Contradiction, and we are done. What you described is closer to ordered sets, rather than set cardinality.

[@Areus_]

All natural numbers all already paired with a real number. You can’t do the “n + 1” trick because that number already exist in the list, and that’s by definition. The argument simply says that if you put all natural number in a one-to-one list, there would be real numbers left, and you can’t put that number in the list because all natural numbers, up to infinity, are already paired with another number.

[@seyerus]

Fun Fact: Cantor went mad and unalived himself.

[@DjVortex-w]

I hate it when the word "paradox" is used to describe something that's merely a bit unintuitive to some people. That's not what "paradox" means.

[@MuffinsAPlenty]

When was the last time you looked up the word "paradox"?

[@piercexlr878]

Paradox does in fact have a definition as being unintuitice

[@darrennew8211]

There are several meanings for / types of paradox.

[@British_Barbarian]

The dart board paradox is an easy 50/50, either the dart hits the infinitely small target spot or it doesn't thus 50/50

[@ZoveRen]

<a href="#" class="seekto" data-time="710">11:50</a> I'm from there

[@NicoTheMooTheShroom]

i usually hate math videos but these are always great before bed

[@zombieregime]

The dart one is a prime example of "snarky maths nerds hiding obvious mcguffins in the verbage used to describe the 'problem' and working against finding a realistic solution" Instead of stating the obvious "hur dur its amazing" trash 'it could land infinitely anywhere on the board (completely ignoring the grain structure of the cork the board is made of) you could easily state it in the much more realistic, not masturbating over your ti-82, way of 'the dart tip diameter is 3mm diameter. What is the probability it will land within the 3mm area you believe it will.' My purposefully introducing infinity the answer trends towards infinity, therefore it being locked to an infiniti is not that special. It's like saying I have an infinite amount of eggs how many eggs do I have.... It would be amazing if I had two eggs, it would not be amazing if I had an infinite amount of eggs. Other than actually possessing an infinite amount of eggs....

[@jackkalver4644]

The lamp is half as bright as it can be. This is true for ever-so-fast frames, and so it must be true for continuous perception.

[@landsgevaer]

Afaik, the paradox is not what does the lamp look like to you towards the end if the switching, but is the lamp on or off *at* (or after) the 2-minute mark.

[@giddycadet]

they call that PWM brightness control

[@darrennew8211]

Nah. The trick is the problem doesn't specify what happens at the 2-minute mark. It says "Before two minutes, this is how you manipulate the switch. What do you do at the two minute mark?" Just like Zeno's paradox: you never consider what happens when Achilles actually catches up, but only the infinite number of slices before that.

[@benjaminhill6171]

​@@darrennew8211Uh... you just contradicted yourself. You said that what we care about is what you do when you *reach* the 2 minute mark, but then you said you don't care what happens when Achilles actually passes? That doesn't follow.

[@darrennew8211]

@@benjaminhill6171 No. I said Zeno's paradox has the same root cause as the lightbulb thing. In Zeno, you say "first go half way, then half of that, then half of that, ..." and you never actually consider the moment you reach the turtle. You're just looking closer and closer at the part *before* you get to the turtle. In the lightbulb, you say "turn it on, then turn it off, then turn it on, then turn it off" and you never specify what you should do when you get to the end of the task. You specify an infinite number of steps to be performed in a finite length of time *before* two minutes, looking closer and closer, but never actually looking at the two minute mark. What does the person flicking the switch do last? Nobody even says what that is. The difference is in Zeno's paradox, we resolve it by invoking the calculus of limits. In the lightbulb paradox, we don't have a limit because it's bouncing back and forth, so there's no limit. But it remains the case that the problem doesn't specify enough information that you can deduce either answer.

[@adamruzenec-freerunning9838]

I think most of the paradoxes arises because of limitation of conceptual thinking. Reality simply doesnt fit into our conceptual boxes.

[@TomMurphy-gy4dm]

lol "infinite hotel at capacity" there is so much humor packed into that little phrase how and when was its construction finished? what it is made of? how do you tell the guy in the last room to move? how far away is he? at what point would the message take longer than a human lifespan to get to him? how do they heat the whole building? Why would you even move if you were far down the list and were told to move? Who is going to force you? How far is the kitchen? How is it at capacity? why would anyone agree to rent a room where you will be forced to move every time a new customer shows up? how can you leave? don't I live there already, since an infinitely large object is omnipresent by definition?

[@elderfrost9892]

we just say that the kitchen, boiler, and all of those problems are at the other end, then we don't have to deal with them. if the heating breaks down, it'll take an infinite amount of time before we notice :P

[@benjaminhill6171]

@TomMurphy-gy4dm It's... not a joke. Hilbert's Hotel doesn't care about practical things like you listed. It's not a thing that's supposed to exist in the real universe, but in a perfect mathematical one. You just assume it exists, and it does. Nobody constructed or worked to fill it to capacity - it just *is*. That's the basic assumption. Besides, it's all just an illustration for the fact that an infinite set can have more elements added to it without changing its size. This has nothing to do with actual hotels and logistics.

[@TomMurphy-gy4dm]

​@@benjaminhill6171 Well on the odd chance you reply again I'm gonna mute you. You ever wonder why you have no friends?

[@benjaminhill6171]

@@TomMurphy-gy4dm So I can't even talk about educational things on an educational video without being blocked? I obviously couldn't detect the sarcasm in your comment.

[@HyperSarcasticAvocado]

How can people exist if all finite matter in the universe has been turned into hotel rooms? The hotel thing is just dumb thought experiment where you're told what to think. Its basically just what one random dude imagined. I imagine that since the infitie rooms are occupied there isn't any more space, even if you built a new room, its already filled and the new guy has to sleep outside. See, I just made my imagination into a paradox too weeeeeeeee

[@robertlee8519]

I'm surprised that people don't give more pushback on the hotel paradox, because it can be solved algorithmically too. You just have to get creative with your algorithms.

[@joshuapowers4623]

Yeeeaaa, so the only two of these I even kinda understood were the hotel & dart board ones. And that was only because I could follow the logic of it rather than the mathematics of the proof.

[@densedecisions4568]

<a href="#" class="seekto" data-time="569">9:29</a> "Can you go down an asymptote?" Depends how far down until it counts

[@NonCrazyNorGamer]

A Glimpse Into Every Infinity Paradox Is more of an accurate title. How can you call this an explanation?

[@simovihinen875]

There are those who don't accept the axioms leading to things like different sizes of infinity, and various infinity paradoxes. Having heard out one of these "atomists", I feel attracted to the idea that instead of infinity, we should be talking about "arbitrary" or "indefinite" values.

[@benjaminhill6171]

That's fair, you can choose any axioms you like, and by definition it's impossible to say that you chose the wrong axioms. But I'd argue that the premise of this video chose the axioms for us, since we're dealing with the standard mathematical idea of infinities. You can't reasonably come into this and tell the content creator that they're using the wrong axioms. If you use different axioms, all of these paradoxes cease to exist, and this video is meaningless.

[@mantasr]

I think mathematicians don't understand infinite the same way normal people do.

[@ballom29]

St petersburg paradox happenned in a game of Magic the Noah. One player had the option to go gambling, and the rules of Magic the Noah being...very vagues and up to interpretations... he asked if he could go in debt. So he simply kept loosing and increasing each time the amounth of money he was gambling, until he enventually won and earned back all the gambled money + interrests

[@airproci]

The dartboard paradox is not a paradox if the point is infinitly small. Because the dart sooner or later will hit the area where the point it is. Is a paradox if Also the point of the d'art is infinitly small as well

[@yumusakgeyamyami]

i feel so dumb when i think that those paradoxes could actually have a solution. like, simply the vase one. if you dont put more than one ball in the vase, there are no ways that it will be empty. i mean, i feel so dumb and so smart at the same time but i cant make sure if im right or wrong. it feels stupid, sorry.

[@XoIoRouge]

The white background kills the video. Blinding AF bro. Have you never heard of dark mode?

[@asherblair-jm8lj]

A paradox in time travel is that if you went back in time to do something then you wouldn't have gone back in time to do that thing so then you would not have went back in time to do it do you would and so you wouldn't ect...

[@jimboSleeeeiiice]

I mean, I dont have the slightest clue about any of these, but with Gabriel’s horn it’s weird to me that they would consider finding the area beneath the line paradoxical to transforming the line into a geometric shape and finding its volume.

[@valinorean4816]

you missed Bernadette's book and Metuselah's diary and continuum hypothesis and Freyling's axiom

[@jazzabighits4473]

Several of these aren't paradoxes. Hilbert's Hotel isn't a paradox, it's just wrong, as it implies that a building with an infinite number of rooms can be "full". If it was infinite, it would never be full. Cantor's diagonal argument isn't paradoxical, it's just not necessarily intuitive. It makes perfect sense mathematically. Thompson's Lamp isn't a true paradox, it's just explaining a mathematical limit, some series diverge, some don't. Saying 'it's always off before being on and then always on before being off' is insufficient, because you MUST know the starting state of the lamp (either off or on). Gabriel's horn is a normal object, nothing special about it. It's like if you have a normal ice-cream cone, you could say that the shape starts cylindrical and then converges to an infinitesimally small point. That doesn't really mean anything though, an ice cream cone doesn't have infinite surface area (or volume, obviously) just because it has a decent apex or corner. The dartboard paradox isn't a paradox at all. The dartboard doesn't hit a "zero area" point, it hits a very small point. There may be 10 billion different points the dart can hit, but that means its 1 in 10 billion chance to hit any point. The tip of the dart isn't infinitely small, you can see its width and things that are thinner (e.g. the width of a hair), so this one isn't even close to a paradox. Not to mention, you can't meaningfully divide space smaller than a Planck length, so there aren't infinitesimal regions for the dart to hit. It's not a paradox that people would only pay 20 bucks to play that coin flipping game because while you could theoretically get a series of infinite heads, you are much more likely to not get anywhere near that. The expected values for the game are also powers of 2 (win 2 dollars, win 4 dollars, etc), given that fact, I'd say that it's reasonable to not want to pay more than 2^4 (16) or 2^5 (32) dollars. Unless you really believe you're going to flip a coin and get heads more than 5 times in a row

[@benjaminhill6171]

First, there are multiple kinds of paradoxes, and I agree that most of the examples here are just the unintuitive kind, not the actual contradiction kind. Now, Hilbert's Hotel is not a real hotel, it's a mathematical one. In this mathematical hotel, you don't start with it empty and try to fill it, you start by assuming it's full and see what happens then. This is another example of an unintuitive paradox, not a contradiction. It's just to show that adding elements to an infinite set doesn't change the size of the set. Hilbert's Hotel isn't "wrong". For the lamp, I disagree that the starting state matters at all, but I agree that it's just an example of a diverging sequence. Another example of an unintuitive paradox, no contradiction. For the dart board, these are mathematical darts we're talking about, and a mathematical dart board. The dart tip is infinitely thin, and the board is perfectly continuous, not made of atoms like a physical one would be. In this idealized situation, the probability of hitting any given point on the board is truly zero, but again that's just unintuitive and not a contradiction. I think I agree with the other things you mentioned.

[@duffman18]

The Hilbert's Hotel one needs to be actually explained. Because you seem to be saying something to the effect of: Q: Why is the hotel infinitely large? A: Because it is. Like, you've told us the hotel is infinite. So why on earth would you need to move every guest into the next room whenever a new guest turns up? Just add the new guest to the next room. Because you've already told us the hotel is infinitely big. So what's the paradox? Please could you give an explanation for Hilbert's Hotel? Because just saying "the hotel is infinitely large, because it is" isn't an explanation. You didn't explain why the guests need to be moved into the next room along whenever a new guest arrives. If the hotel is infinite then you don't need to move anybody at all, you just put the guest in the next available room. And so they never run out of room, because you already said the hotel is infinite. So that isn't a paradox at all. It's the equivalent of saying something like: Q: How come when you cut a tree into pieces, you get wood? A: Because trees are made of wood. Like... yeah? Duh. And this whole nonsense about having to move guests into the next room along when you don't actually need to because you already stated the hotel is infinite, is like saying "ooh if you cut the tree in this special particular way, cutting it into pieces of a specific certain shape and size, then you get _WOOD_ out of it, from doing that process". Like, yeah? You get wood out of it regardless of how you cut it, because trees are made of wood. And you always have space in the hotel for new guests, regardless of whether you move every guest into a new room or not, because the hotel is infinitely large. So what exactly is the issue or paradox here? There isn't one. So it shouldn't be in this video surely?

[@benjaminhill6171]

This is the kind of paradox that's only unintuitive, not contradictory. Also, the point is that the hotel starts off *full*, with every room occupied, but you can still fit more. It's an analogy to show that if you add things to an infinite set, then the size of the set doesn't change.

[@der.Schtefan]

It should he called "enumerable infinity" instead. It can't be counted, because depending on language understanding you'd die before completion, but it is enumerable.

[@piercexlr878]

It countable in the sense you have an order to count them by

[@michaelmicek]

"Enumerate" can be a synonym for "count", so you'd still have to explain what you mean.

[@Tekkaras]

I understood a couple of these!

[@edunitin5338]

This video is look more entertaining then your other videos. I liked the sarcasm in this video and also the animation you put in which make video easy to understand. Kudos to you 👏

[@LittleYorgee]

Example two just depends on how you organize it

[@hunterthornton9609]

About thompsons lamp, couldn’t the 1/2 just mean it’s a 50/50 chance of it being on or off instead of just half on? Just saying it makes the most sense

[@alcidedragon]

The fact is that they're is no probability here

[@evilskwdirector]

i have no idea what he's talking about it but i can relate

[@Torthrodhel]

I had an idea when I was little but people seem to get oddly emotional over it and downright rude when I've tried to explain it before (although I don't know why). By myself and working more off curiosity than actual knowhow, I had noticed that dartboard thing basically, and rather than concluding that one divided by infinity equals zero but a different kind of zero that doesn't mean "no probability" unlike the ordinary zero that does mean that, I concluded that it therefore can't quite equal zero can it. So I figured to give this conceptual "impossibly small number" a name (I called it "trace value" and for reasons I can't remember labelled it using an "n") and then had a think on how it might be useful, and I came up with the idea that you could multiply something by it in order to ensure a necessarily separate order in an indexing list. For example: say you have three values: A, B & C. You wish to order something by A, but only if their A are equal, then order it by B, but only if they're also equal, then order it by C... no matter what the values themselves might be limited to. So I would express this, as ordering it by: A + Bn + Cn^2 ... (if you imagine the ^2 as a power of two since I don't know how to get that as a symbol on my keyboard). And you could continue to any power the further down the list of "and then by" you got. The result would determine the final order. You'd probably never actually have, nor ever actually need, a way to DO that calculation; nevertheless it'd be a neat way of writing it that is technically correct. As I say, I have no idea why people have got angry when I've expressed this. I know I'm not educated enough to "do things properly" but surely every idea has merit no matter the relative capability (or even just relative standardization) of its expression? Like there still seems to me to be something to that idea. Not sure what, obviously. Thoughts?

[@lawrencebates8172]

There are branches of maths that use infinitesimals, which are essentially this concept of an impossibly small number. There are some situations where that can be useful, it lets you do math-y things with infinity. But the downside is that the systems are harder to work with

[@Torthrodhel]

@@lawrencebates8172 thankyou! I knew there had to be something to it, and look there was. Could be intriguing, could look into it some day. I dunno why me saying it made people so cross before. Maybe I was just around snobs at the time. Ah well, I am reassured now. :)

[@DaenaMichelle]

My head hurts

[@doneidson]

Let's say that an object moves one meter in one second, then another meter in 1/2 second, then another meter in 1/4 second, then another meter in 1/8 and so on. Where would it be after two seconds have elapsed?

[@mitabpraga7487]

E=mc^2 will answer that.

[@vincehomoki1612]

About 30m away, as it can't move faster then the speed of light.

[@victorhiggins2118]

Pi isn't a finite value. It's not a measurable quantity.

[@WilliamWizer]

cantor's diagonal can be counter argued too. the digits of an irrational number are countable infinite (evident, since we can count them) if we assume there are uncountable irrational numbers that means the list contains more rows than columns. the number we form using the diagonal can be below the countable infinite first numbers. that means, the number on cantor's diagonal IS on the list. which destroys cantor's proof that the cardinality of the real numbers is greater than the cardinality of the naturals. maybe it is indeed larger but cantor's diagonal doesn't prove it. thomson's lamp can be gaslighted by taking into account plank time. once you reach it, you can't meaningfully reduce it more. that means there's a finite amount of times you can half the time. but, of course, that's just me being a SoB.

[@peterg76yt]

"if we assume there are uncountable irrational numbers" Then you're done and the theorem is proven.

[@MuffinsAPlenty]

"that means, the number on cantor's diagonal IS on the list." The issue is you're not understanding the rules of what a "list" is in this case. Because we want the list to be countable (that is the assumption after all!), then there must be as many entries on the list as there are natural numbers. This means that the list _can_ be arranged in such a way that each element appears in a _finite position_ on the list. The list is infinitely long, but each individual entry of the list is finitely far down the list. You propose that there should be something infinitely far down the list. And this is at odds with what a list is supposed to mean.

[@thomassynths]

The contradiction requires us to show that real numbers are valid mathematical objects to begin with. Key takeaway is that mathematicians want reals to be real so badly that they will ignore obvious holes in

[@creativenametxt2960]

​@@thomassynthsanchient greece moment

[@WilliamWizer]

@@MuffinsAPlenty and here is the small detail you seem to miss. the list NEEDS to be infinitely long since it has, at least, as many entries as natural numbers and natural numbers are countable INFINITE. but what I wanted to point is that cantor's diagonal self-disproves itself. I'm not going to point if real numbers are countable infinite or uncountable infinite. just pointing that cantor's diagonal doesn't prove sh*t. the diagonal is only possible if there's as many digits on a real number as real numbers and we know that a real number has countable infinite digits. this implies that either the diagonal ends before we finish the uncountable list of irrationals (so the reasoning is wrong) or that there's only countable infinite irrationals. which of the two is correct is something I won't bother to argue. it's irrelevant. one of the two is correct and both contradict cantor's diagonal proof.

[@mattmcdonough3282]

"Time to move rooms everyone!"

[@CoseDaTux]

i didn't understand the st.pietrobourg paradox. Is obvius that that if you play a game where in the worst of cases you don't lose nothing and you play it infinite times you end with infinite money.

[@RibusPQR]

I would play the Saint Petersburg game if someone paid me 1/12 dollars.

[@Sirmaadman24]

This has officially fried my brain 🤯

[@jeremyashford2145]

Thomson's Lamp. With that mistreatment the bulb will blow. Lightbulbs are engineered to function finitely. If they functioned infinitely it would be bad for business.

[@RedBlackDish]

Infine hotel is not a paradox because it creates an infinite chain of people switching rooms. Diagonal argument is flawed, because you can put it on it's head and just create a new natural number by putting all the existing natural numbers in series, thus making a new natural number to match the ner real number. Gabriel' horn is not a mathematical paradox, but a well hidden lie, since the horn will have an opening on each side which would make the narrow side an infinite tube with infinite volume. But all of that is unnecessary sine Pi is an *infinite* number that is infinitely big. The lap paradox is also not a paradox: the lamp will be off after 2 minutes pass. I have a definitive explanation of why that is so, but it's mine.

[@ianliu88]

I'm always frustrated with the diagonal argument, because it is not immediately obvious why the same argument doesn't work for rational numbers.

[@yharon8243]

Because the diagonalized number will always end up being an irrational number.

[@creativenametxt2960]

rather because the diagonalized number *can* be irrational a number is rational if and only if it has a period after which the decimal digits repeat (aka 1/3=0.3333... or 0.(3), 1/2=0.5000... or 0.5(0), 1/99=0.(01), etc) so you can end up in a situation where the new number you construct is aperiodic and is not rational

[@creativenametxt2960]

It is also worth noting that the diagonal argument presented in the video is *technically* incorrect because notation of "0.999..." does not represent any real number, either that or a number can be represented in several ways (aka 0.1=0.099999, which would fail the argument) to fix that simply replace all digits by 0, except for zeros, who you would replace by 1, since that guarantees the sequence of digits represents a real number.

[@haitaelpastor976]

Just my thoughts! Heck, it can be applied to natural numbers in fact.

[@creativenametxt2960]

@@haitaelpastor976 fairly certain that if you try to apply it to natural numbers you can end up with an infinite string of digits, which itself is not a natural number and thus the diagonal argument would not be applicable

[@md-sl1io]

in an infinite universe, there are infinite people about to press a button to destroy the universe, and infinite people about to press a button to save the universe, and infinite people about to press a button to prevent that from happenning and so on also, in an infinite universe, the eifel tower and a car are the same %age of the universe, which is clearly not true so therefore there is no infinite universe

[@xXkongergateXx]

im too stupid too understand any of this but here I am watching it anyway.

[@mybonesclack4948]

you’re not stupid for trying to learn new things 😄

[@scarletevans4474]

<a href="#" class="seekto" data-time="525">8:45</a> "Not enough information", is it similar piece of bread with Sleeping Beauty paradox? 🙂

[@1three7]

I think with a lot of these it's just illustrating why our universe isn't actually smoothly continuous. There are minimal sizes for stuff beyond which interactions no longer make sense. These infinities just can't actually exist in our reality

[@oncinaust5178]

Im not a mathematics genius, but the dart board "Infinity" shouldn't be infinite....yes, in theory it is but darts have thickness rather than true points. So if you plug in the numbers it shouldn't be infinite. Also eventually you could loose your dartboard. That may be the more interesting Infinity. "How many times must you throw a dart before you no longer have a dartboard?"

[@gergpoo]

The Ross-little wood paradox blew my mind lol idk how I’ve never heard it before

[@henryptung]

Drawing another parallel to the series summing -1 and 1, I'd say it falls under the modern paradox where rearranging nonconvergent infinite series can result in arbitrary outcomes. Classic example is rearranging an alternating harmonic series to approach any real number. Numbering the balls and adding/removing in order produces a particular correspondence between the add/remove events, and grouping by element orders the sum in a particular way (i.e. a sum of (1-1) terms). You could just as easily say "remove the largest-numbered ball" and get infinity again, or remove only balls greater than N to achieve a final count of N, for any natural number N.

[@darrennew8211]

Fun fact: It showed up in an episode of The Simpsons. :-) I think "Mathologer" covered it.