How Big are All Infinities Combined? (Cantor's Paradox) | Infinite Series
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- Äas pĹidĂĄn 25. 06. 2024
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Infinities come in different sizes. Thereâs a whole tower of progressively larger "sizes of infinity". So whatâs the right way to describe the size of the whole tower?
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The Geometry of SET
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Talking about the sizes of infinite things is tricky in part because the word âinfiniteâ is often used in two distinct ways -- to refer to the sets themselves, and also to refer to the sizes of those sets. In what follows, letâs try to keep as sharp a distinction as we can between infinite sets and infinite set *sizes*, because doing so will let me highlight an especially paradoxical feature about infinite sizing that I don't think gets enough coverage. The technical term for a âsizeâ, infinite or otherwise, is âcardinalityâ, and I should probably use a term like ânumerousnessâ or ânumerosityâ rather than âsizeâ because the idea it tries to generalize is the notion of "how many". Still, Iâm going to say âsizeâ a lot in this episode just because itâs easier.
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At 10:44 you have not yet proven that the cardinality has to be greater than aleph_0. All you have shown is that it does not behave as the ordinal omega, it might behave as omega+1 in which case its cardinality is still aleph_0.
Good video by the way :-)
Good point. That was sloppy logic on my part. Dammit! Ironically enough, in the episode 2 weeks from now, I actually make a very similar point to the one you just made... and yet somehow, I let a bonehead thing like that creep into this video. Ugh! Anyway, that's why the audience is so useful here -- thanks for pointing out the logical lapse. I'm upvoting this so that (I hope) other people see it.
Well, as we don't actually have a set here it is okay(cheat the system)
yeah, I thought it was not right
@@yuvalpaz3752 ××× ×× ××Ş× ×¨××× ×ת ××
Mahlo cardinal:M
Weakly compact ordinal:W
Grahams number:g(64)
TREE:TREE(3)
This is probably one of those Infinite Series videos I just can't quite grasp my first few times through - without the follow up. To be fair, I also had a hard time with VSauce's Counting Past Infinity one, too.
Which is to say, now I'm having a lot of fun!
I understand stuff like this and Iâm only 11, 5th grade is too easy for me!
It's turtles all the way down..
Ah, but since the turtles are not rational animals, should we assume they are dense and real instead?
I found 1.4142135623730950488⌠of a turtle. Where do I go now?
Is that a JOHN Greene reference?
all the way up
Brief history of time?
This is like numberphile if they didn't assume we were all 12. You guys deserve more subs.
"The cardinality of the totality of these cardinalities"
Such poetry
*Sigh.* I am going to need more than the beer in my hand to figure this out.
That's why mathematicians drink coffee
Ellobats it's an empty set so empty set so 0 cardinality.
I drank a beer. One beer leads to another. By induction, the whole case was finished.
As per ErdĹs's definition: "a mathematician is a machine to turn coffee into theorems". Note he never said whether the machine halts...
A beer in the hand is worth two in the keg!
This guy finally helped me understand spacetime. Glad he popped back up
All I keep hearing is "Olive Knot"
Lucky you. I keep hearing "all of naught".
It is so good to finally have popular mathematics channels which cater to the more advanced layperson! Thanks guys!
Just found this channel and it's like ... FINALLY not being talked down to
Very good episode. And shoutout to the Polynomial Method!
My guess is that theyâre infinitely big.
agreed
My guess is that you are a troll... but I will bite today. What they? The infinite sets!? Of course THEY are infinite... Lol. The problem here is essentially that some infinites have different sizes and when you ask how many sizes of infinity there are you get a paradox.
And infinitely empty.
But which infinity?
is it infinitely infinite
i loved this episode, good job gabe
12:48 "The next time I see you"? What a euphemism.
0:56
"I know the right way to do things but I'm not going to because the wrong way is easier."
Like a true physicist Gabe! ;)
So, does the set of all possible infinite sets contain itself?
I'm going for a walk..
Go home, Bertrand. You're drunk!
first prove that it's a set at all, then we can talk...
It's not clear that that exists, much like "the largest natural number" does not exist. After all, given such a set S ("of all possible infinite sets"), one could form a set which consists of the union of S and {S}, which would be a new infinite set, meaning that S was NOT the set of all possible infinite sets in the first place.
the set of all possible infinite sets *is* itself. it doesn't need to 'contain' it.
what gabe doesn't understand here about unioning sets together, is that unioning a set with it's powerset, does not necessarily create anything new. the power set of A already contains all the members of A, so the union of A with its powerset is exactly the same set as just it's powerset.
unioning a set of chain power sets (for example A, P(A), PP(A), etc), all together, is exactly the same size of whatever the largest powerset is, in the chain.
if that chain is infinitely big, i don't think this changes anything.
i'm not really sure there is a paradox here, just a misunderstanding of how set union functions.
No set can have itself as an element
11.03 The strongest version of Choice I've ever seen. You were savage man.
In the argument that |A| =/= |P(A)| when contructing the set B, it might have been better to mention, that such a set actually exists thanks to axiom schema of specification. But that might be a wee bit too technical. And with that contradiction, my guess is that constructing "a complete list of all infinite cardinalities" requires unrestricted comprehension and therefore leading to the paradox.
Watch the video he suggested that we should watch, if we haven't already. That should probably prepare anyone better for this one.
Do you know what classes are in set theory? if so, this is the solution to the paradox: A is not a set, it is a class, so the axiom does not let's us construct B.
Yeah I feel like there's a part missing where he proves the set B actually isn't empty for infinite sets. What if 0 maps -> to {} and then the next natural numbers (1) maps to all subsets you can make with 1 (injection so skip previously covered subsets). The following unused natural numbers maps to the subsets you can make with {1,2} while not breaking the injection criterion ([2,3] -> [{2}, {1,2}]). After that you take subsets up to 3 with the following natural numbers like so: [4,5,6,7] -> [{3}, {1,3}, {2,3}, {1,2,3}] and so on. This gives an bijection from N to P(N), which proves that |N| = |P(N)|.
I realise the flaw in this construction however. No member of N maps to an infinite subset (like {5,6,7,...} or {6,7,8,...}) and I think no mapping could for similar reasons as to why you can't map naturals to reals.
I can make a injection from Reals to P(N) and prove |P(N)| >= |R|.
Something like this: Take any real number X and a list L = []. If X is positive append 1 to L, otherwise append 0. The number of digits before the decimal point in X is then appended to L. After that take the whole infinite list of digits in X and append them in order to L. Lsum is another infinite list where each member in it is Lsum[p]=Lsum[p-1]+L[p] (out of bounds is 0 for Lsum). This makes Lsum a list of increasing natural numbers from which L can be derived by subtracing the number before it. The list Lsum can be made into a subset of the natural numbers trivially (and converted back to a list trivially and sorted to get back the position information). This algorithm should be able to represent any real number with a set.
Is |P(N)| = |R|? I haven't figured out yet but I'll post this anyways and maybe I'll get back to it?
If B should happen to be the empty set, then that's just fine. It would still be a set in the power set that is not the image of any element.
nom654 Can you explain and not just affirm that it is so? B is a set of items not reachable by G. If B is empty then G is surjective unless B is not an exhaustive list of items not reachable by G. So if there are other ways to construct unreachable items and B can be empty then the proof given in the video would still be inadequate. Are you referring to my proof that |P(N)| >= |R| to claim it?
Well, the number of finite numbers is too big to be finite, so I think that the amount of infinite numbers is too big to be infinite.
Meaning it's finite? ("infinite" literally means "not finite")
lol Interesting reasoning
You are on the right track. The set of all possible infinite sizes is indeed too big to be infinite. So it isn't infinite; it simply ... isn't.
Formally, the class of all cardinal numbers is not a set; it is a proper class. This means that statement "x is a cardinal number" can be defined with a formula, but a set of all cardinal numbers can be proven not to exist; the assumption of its existence would yield a paradox.
Using the Axiom of choice you can define a cardinal so large that it cannot be reached using any amount of operations on infinite cardinals. These are called Inaccessible cardinals (the same way infinity is inaccessible using any amount of operations on finite numbers). They are the first level of Large Cardinals above infinite. The contraction of U=P(U) is the largest possible cardinal and is written 0=1 as is it so large it is by definition a contradiction.
websupport1.citytech.cuny.edu/faculty/vgitman/images/diagram.jpg
The 0=1 marking at the top of that diagram, then, implies the existence of so many things that finite numbers are no longer distinguishable in size?
(Edit: thatâs my interpretation, at least)
There is a slight inaccuracy concerning the order of quantifiers at 8:05
"There is at least one subset of A, namely B, that no function that maps elements to subsets ever outputs . "
This sentence as it is, is incorrect, because B depends on the function. Better say "for any function, there is a set B such that ...".
That's what I meant in that whole section. In other words, I didn't mean "there is at least one subset of A..." in the mathematical sense of first-order quantifiers; rather, I meant the example here with one A, one B, and one G to be representative, with a tacit "Now generalize to all G", ergo my diction when I say "But we didn't specify what G was" yadda yadda. It's funny, even though I'm now doing a math channel, it doesn't always occur to me that a substantial fraction of the viewership reads things as mathematicians tend to read them. ;)
This actually gives me an interesting idea for an episode. There are some fun open-ended questions to which mathematicians/computer scientists tend to give one class of answer while physicists/engineers on the other give different answers, and the nature of that split seems to be feature nontrivially in difficulties that early undergrad STEM students who are of one bent vs the other face in their introductory calc and linear algebra courses. Some very interesting stuff at the interface of math and math pedagogy. Maybe we could attach this to a poll and get a rough snapshot of the Infinite Series audience like this...
I think Scientia Egregia put things too gently. I believe the statements you made will be misleading or confusing to ordinary viewers who try to follow your reasoning carefully (as well as being technically incorrect). It's as if someone said: Pick any integer g. there is an integer strictly greater than g, call it b. Now, since > is antisymetric g is not greater than b. But there's nothing special about g, the same holds for any other integer. So "there is at least one integer, namely b, such that no integer is greater than it." (Exactly parallel to your sentence. Do physicists read that sentence in an innocuous sense?)
How do you know it exists ? I mean how can you prove it's not the empty set ?
Abdel Armstr Then every output of G would contain its input, and therefore never be empty - so then the empty set would never be the output of G.
even if it is an empty set, the proof stays correct, because empty set is also an element of P(A).
This videos answers a question I've had for more than 6 years.
The comment section is superb. By the way Gabe, all of us are supporting you guys. You really do us a favour by uploading these videos. I wish you could come back.
10:30 I knew it was gonna get this crazy when you started :)
The issue with this argument is set-theoretic. The set of all infinite sizes here is ill-defined. A subtle point about the axioms of standard set theory is that the way you can construct sets is limited. When you build a set with quantifiers, you have to create it as a subset of a set that already exists. So you can create sets such as "the set of real numbers, x, such that x > 0" but you can't create something like "the set of all sets." This restriction on how sets can be built is how Russell's paradox is avoided: that paradox arises when you have a collection of all sets and consider the elements of this collection that contain themselves. Similarly, the set of all infinite sizes isn't created from a coherent base set, so is not well-constructed.
New to the channel. Very interesting topics. Plus Tai-Denae is suppa cute. đ
very good project
Doctor: "Stick your tongue out."
Patient: "Aaaaaaaaaaah-leph naught."
The definition of comparing cardinalities isn't quite right. There can exist a mapping from A to B that is injective but not surjective when the sets have the same cardinality. For example, the identity map from N to Z is injective but not surjective, even though N and Z have the same cardinality. The important thing is that, if |A|
Or for example the function f(x) = x + 1 on the natural numbers, which is also injective but not surjective.
I think of it as "is there a bijection"? Partly because I still keep mixing up injective and surjective.
Agreed, take f(x)=2x over the natural numbers, which is an injection but not onto. The rules at @3:27 would have you conclude that |N| < |N|
He said all injective functions would not be surjective, not that there exists such a function (I just rewatched at your timestamp).
Hey all, Gabe here. Did I inadvertently misspeak? I kind of thought people would follow that what we're going for is "Can you find an injection that's also surjective, i.e. can you find a bijection?", not "Is every conceivable injection also onto?". But I get how the language in the graphic circa 3:27 could be confusing -- it inadvertently suggests that the cardinality comparison tests involve "check surjectiveness of the specific injective map you found in step 1", as opposed to "see whether *any* injective maps you can find in step 1 are also bijective". Fair point, and I should have been more lexically careful.
PBS Infinite Series yeah I believe the graphic is the only problem
Yeah.... I'm gonna need you to go ahead and make an ELI5 for this video. That'd be greaaaat.
Man such a cliffhanger ! When are you going to release the next episode ?
Best topic since infinite series birth. One virtual beer for everybody involved :)!
This video is excellent! Maybe too good for CZcams!
My 3rd year university math classes suddenly make a whole lot more sense. I should've spent more time just thinking about consequences of what I was shown. If only I'd had more time.
Wow a mind blowing video in my favourite topic :) power sets of power sets. When Vsauce was good, it made an also awesome video about infinity. I am looking forward to your next video!
This is essentially equivalent to the Burali-Forti "paradox", although the Burali-Forti paradox as usually stated uses ordinals. If such a U (the cardinality of all infinities) existed, it would be Aleph(the largest ordinal) and P(U) would be Aleph(some strictly greater ordinal, which is the largest ordinal +1 assuming GCH) and so the supposed largest ordinal cannot exist as it would have to be larger than a strictly greater ordinal.
The way to solve it is simply that the "set" of all infinities cannot be an actual set. It's known as a proper class, which is essentially some collection of sets that isn't itself a set and thus it cannot have a cardinality.
Gonna have to watch the 2nd half again, I always assumed it was just aleph0 infinities (based on CH being true).
Hard to take in one go, amazing though, maybe I can figure it out, or maybe not!
Welcome back..!
very interesting
The 4 year old in me wants to put my arms out and say "this big"
What I find really fun is that even though the size of the real numbers is larger than the size of the natural numbers, the size of all numbers a mathematician would likely work with (the computable numbers) is the same size as the natural numbers.
Although it's unknown if all the constants in physics fall in that set.
Joshua Hillerup It's even more crazy. Since any mathematical statement must ultimately consist of a finite list of fundamental logical deductions, of which only finitely many exist, one can deduce there are only countably many possible mathematical statements as a whole.
In particular, there are only countably many definable objects as a whole. Even more particular, only countably many real numbers are definable. In measure theory we could literally say "almost all numbers are undefinable" and it would be a well-defined true statement :D
SmileyMPV what does "definable" here mean? If I say "let there be a distinct funky object with these properties for every real number" I've just definined uncountably many funky objects, no?
Any number you give a unique description for, say "the square root of two" requires some finite list of instructions that uniquely define it. So the amount of numbers definable uniquely, rather than in some group eg. the set of real numbers, so there are unaccountably many real numbers for which no definition can be made.
blog.ram.rachum.com/post/54747783932/indescribable-numbers-the-theorem-that-made-me
is a good blog post that goes into it
Uniquely defined. "square root of 2" is the positive solution to x^2-2=0.
Joshua Hillerup Can you be more concrete? Do you mean for example "let a>0"? Because that is not a definition. Also, you can define the set of real numbers, but that does not mean you defined every individual element.
Also, a definition must not leave room for ambiguity. For example "define x by x^2=2" is an invalid definition as multible objects satisfy the equation. But "define x as the positive real number satisfying x^2=2" is a valid definition, because it has been proven there is exactly one such x.
ayy its old spacetime dude. Good to see him again
I believe the phrasing around 1:45 might be a bit misleading. I may be misunderstanding but I think it accidentally implies the existence of a non bijective mapping would prove the cardinalities weren't equal.
Ex, N has the same cardinality as Z. But of course if we mappedevery n in N to itself in Z we would have an injection that isn't onto.
Really appreciate the channel, great series of videos.
I can't see which is the next video... but is so funny.
I have SO MANY answers for THIS video...
In 11:12 as I understand it the set U is used and compared to the former U set even though they are different and contain different sets/subsets
Given the set of all positive integer numbers, calculate S - the sum of all the numbers, then S is larger than any number in the set yet it is a positive integer number... It is easy to define non-constructable/non-computable operations.
It's exactly the same issue you get when looking back at "the biggest positive integer" (biggest element of N).
They are different integer of different "bigness" (=some are bigger than other)
If n is an integer, then I can get a bigger one by doing n+1
If I take all the integer I know, I can still construct a bigger one by taking the biggest and adding one
So if I try to get "the biggest integer" I just get nonsense because from the set of all integers, the biggest integer is in it but I can still build a bigger one.
The only answer is that there are no bigger integer with just these axioms.
brain... hurts...
Watching this channel makes me feel like an idiot.
Brain farts
You can define cardinalities formally by defining cardinal numbers as sets. Each cardinal can be represented by the lowest ordinal with that cardinality. Each ordinal can be represented by a set containing all previous ordinals. So for all cardinals K, |K|=K.
Is the "thing" that is all infinite sets not a set maybe, and if you could describe it as having a cardinality greater that the largest possible cardinality of any set?
Joshua Hillerup I believe this is the problem.
How do we know we've hit every infinite set?
No matter what we do, we can keep unioning and powersetting and we'll get new ones.
So there's no way, even with infinitely many steps to "get everything"
Romaji same way we do it with creating the set of natural numbers. We introduce an axiom saying it exists.
The difference here is the axiom we introduce can't make it be something that has all the properties of sets.
Joshua Hillerup actually, we only have Aleph Nul existing by axiom.
You never actually build infinity, you just say it exists
Romaji it's called the Axiom of Infinity. Without it (or some other such axiom) you only have finite sets.
Joshua Hillerup exactly what I'm talking about
For anyone didnât get the second part of power set has larger cardinality than itself, hereâs my explanation. (His wording were quite hard to understand even I know the proof.)
The way the subset B is chosen is known as âCantorâs diagonalizationâ. It has became a standard proof technique. You can find this process in proving real number is uncountable, and similar arguement in proof of Arzela-Ascoli Theorem. In this case, it is a systematic way to find a element (sbuset) that is not on the list.
Starting from A with size 3. Let A={1, 2, 3}, and f : AâP(A). Say f(1)={1,3}, f(2)={}, f(3)={1}. For clarity letâs express this in the following way.
f(1)={1, ,3},
f(2)={ , , },
f(3)={1, , }.
Now to construct the subset B, we look at the diagonal of the right part of the array. If the element is in the diagonal, we exclude it. If it is not, we include it. In this case, we end up with B={2,3}. Note that thereâs no element in A that can map to B through f. No matter what f you start with, you always end up with a subset B that is not on the list, since B is differ to any f(x) by at least one element.
Now for a general set A, finite or infinite, countable or uncountable. We choose B in similar way. Given a function f : AâP(A) (remember P(A) is a collection of sets, hence values of f are sets), if x is an element of f(x), then we exclude x. If x is not an element of f(x), we include it. Then there doesnât exist x such that f(x)=B, since B is differ to f(x) at x. Hence whichever f we choose, we will always miss an element in P(A), and f is never surjective.
"That sounds highly problematic" pls im dying
Miss you on Spacetime, guess I need to subscribe here as well....
8:40
- A child
- NO
@12:25-12:35--It seems kind of like how The number of finite numbers Cab itself be a finite number and so it must therefore be an infinite number, The only problem is what does it mean to go beyond the infinite?
Could have included the Schroeder-Bernstein theorem but amazing video!
Originally had it. Ended up in the script jetsam. As-is, this video ended up longer than we can normally accommodate given the budget.
PBS Infinite Series I thought that was the reason. Can't wait for the next video!
@@beenaalavudheen4343 where's the follow-up video? I've been searching for it
@@pbsinfiniteseries fantastic video, please share where's the follow-up, thank you
Itâs the All. To count you have to be an observer, youâre trying to observe yourself. Itâs pure infinite consciousness. No matter how hard you look it with continued to paradox itself. We lack the language capabilities to reach understanding.
I thought of another infinite set that I'm not sure what the size is: "The set of all possible type matchups". Because for every new type, there needs to be as many new types added that include that type, as the total number of types already is. So the set infinitely explodes.
Infinities are so much fun!
Yes! THE BIGGEST POSSIBLE NUMBER IS âABSOLUTE INFINITYâ, SO NOW IâM GOD!
a set containing all possible set is larger than all of his subsets, but a set containing all possible set must contain itself, and has therefore a cardinality larger than itself
How many times does one have to dislike an ad to never see it again? As far as I'm concerned, Veridesk is the worst
What a question.
What a title.
Singularity intersection-connection of infinities requires a total unique infinity. (?)
This lecture is an observation of the topological significance of the integration of sets of time rates, typical of a spectrum analysis, or reading pulses of resonance within pulses, etc etc, the natural sequences of dimensional everything.., mathematically speaking.
A closely related result for ordinal numbers is the Burali-Forti paradox.
My best guess on what's going wrong :
Let E be a set. It turns out that P(E) contains E: E is an *element* of P(E). However, when trying to take their union, you don't get anything different from P(E) itself: every single element of E is already contained in P(E). Thus, the union of all the powersets is already on that list. So at the limit, the union U would be the omega_0-th set from the list (omega_0 being the first infinite ordinal).
Careful: the union of all sets in P(E) is precisely the set E. But we don't do a union of all sets in P(E); we do a union of all sets in E and then do a powerset (set of all subsets) of the result.
Of course, Aleph(Aleph-0) is not the greatest cardinal; the set of all subsets of this set has even greater cardinality (by Cantor's theorem).
moreover, it is far from trivial, that if you have f:X->Y and g:Y->X injections than there exists a
X->Y bijection (in other words if you have an injection from A to P(A) and not have a surjection than A
Agreed, and I didn't want to get into Cantor-Schroder-Bernstein stuff in this video, which is why I stuck to speaking just about maps in the A-to-B direction. As for your other comment, other people have also raised it over the past few days and I've replied there -- I did not mean to suggest that, once you've found one injection f (which establishes |A|
@11m41s: It's like kids...
"Name a number bigger than infinity."
"Infinity + 1 !!"
Great, we are now talking about Dedekind-finite infinities (an infinite set which doesn't have a countably infinite subset).
maybe the collection of all infinite sizes is not a set but a class, and we haven't defined cardinality for classes?
Huy Savage Yes, in NBG set theory (an extension of ZFC) the paradox is resolved by simply concluding that the class of all cardinal numbers is a proper class.
Thank you so much for this comment, you've untied a knot in my mind.
4:26 [dead face] that's all fun stuff
I came up with this by using the cartesian product instead of the union. Both work, but the union is more to the point.
What's the difference between the cardinality of a list of sets and the cardinality of a list of those sets' cardinalities? Each set only has one cardinality, right?
The "size" of this tower is not a set (if you suppose it is, you'll get a contradiction, that size is "too big" to be considered a set), it's a proper class.
It's a proper class in ZF(C), but it is actually a set in alternative set theories (e.g NF set theory).
The notion of proper class is not defined in ZF. ZF only deals with sets.
The objects in ZF are indeed only sets, but you can still talk about classes in this theory, check the book Introduction to Axiomatic Set Theory ( G. Takeuti ), chapter 4.
I was afraid it was going to be a cliff hanger... My guess is that there could be another concept beyond the concept of infinity that might describe the total 'size' if the infinite sets
Noticing a major issue with the statements of the cardinality comparison rules. They say |A| B and |A| < |B| if the injection is not onto. That should be |A| < |B| if *every* injection/function from A to B is not onto. Otherwise you have |naturals| < |integers|, since the natural map is not surjective, which is not right.
If all chosen cards are from the same deck, then you can only choose one before you obtain a set. Even if not from the same deck, once you choose the second card, you have a set of two cards.
This question has been driving me crazy.
I've always thought of it in terms of density. The infinite set of whole numbers is very dense, much more dense than the set of prime numbers, but not as dense as the set of irrational numbers. Density being a measurement of the average distance between two adjacent members of the set.
What's the density of the rational numbers? How does that density compare to that of the real numbers?
Oops! I didn't mean to use the term irrational numbers in my first comment. I meant fractions. The set of fractions is much more dense than whole numbers. Sorry about the confusion.
The question wasn't really related to that. My issue is with the general construction that is density, because it has no means of making a distinction between rationals and reals.
I think you're using the word "density" to talk about how "tightly packed" certain sets are vs. how "spread apart" other sets are. In this sense, density is a relative term. In order for it to make sense, you need to be working in a universe (i.e., every set you're dealing with is a subset of some set U - so you're always working inside U) and that universe has to have some sort of topology or measure on it. To describe the density of a subset S of U, you would need to compare S to U itself. When people bring up the notion of density when talking about comparing sets, they are usually only thinking about subsets of the real numbers. So you're comparing each subsets to the real numbers in order to think about density.
But the goal here is to be able to compare arbitrary sets - for any two sets S and T, how do they compare? We don't just want to compare subsets of the real numbers. Here, your notion of density breaks down. There is no universal set in which all sets are contained. So we can't compare each set to a universe.
For example, consider the power set of the real numbers, which I will denote P(R). This is the set of all subsets of the real numbers. So one element of P(R) is the set of irrational numbers. Another is the set of natural numbers. Another still is the set {Ď,e}. Now, we can look at subsets of P(R). For example, you could let S be the subset of P(R) which consists of all elements of P(R) whose members are all irrational. In other words, S is the set of subsets of R in which every element of each subset in S is irrational. Now, how tightly packed is S in P(R)? I have no sort of intuition for how any such notion could/should be defined.
7:23 to be in B or not to be in B?
RIP Infinite Series!
9:13 "the cardinality of the totality of these cardinalities"
There is a simpler argument to show that |A| â |P(A)|:
Map all elements of A to the subset of A that only contains that element. Now, we have "used up" all of the elements of A, but we haven't mapped anything to the empty set.
This is not good enough to show that |A| â |P(A)|. This is because |A| = |B| is _there exists_ a bijection (one-to-one correspondence) between A and B. It doesn't matter if you can create a particular function (or even infinitely many functions) which aren't bijections. In order to show that |A| â |B|, you have to show that *no bijection can possibly exist* between A and B. For example, you could have functions from the natural numbers to the even natural numbers. Or to the prime numbers. Or to the odd natural numbers. Or to the natural number multiples of 5. All of these functions are injective but not onto the entire set of natural numbers. But clearly, the set of natural numbers has the same "size" as itself. So just because a particular function exists which isn't a bijection does not imply that no such bijection exists.
After a little consultation to my old Michael J. Schramm, I think the contradiction presented at the end (edit: i.e. Cantor's contradiction) is tied to Russell's Paradox. Just as R={S s.t. S not in S} is not a set in traditional set theory, so is X={the collection of all cardinalities} since both the statements "Cardinality(Union(selections from X)) is [not] in X" lead to a contradiction.
Looking forward to the next episode!
Cantor's paradox is a stronger result than Russel's paradox. Russel's paradox only proves that there can't be a set of all sets (i.e. given any set X, it is possible to create some set Y which is not in X); Cantor's paradox proves that no set can contain sets of all cardinalities (i.e. given set X it is possible to create set Y which has a strictly greater cardinality than any set in X), from which it immediately follows that there can't be a set of all sets.
Thanks for the clarification and confirmation that this is an issue of "This kind of set actually can't exist! [in traditional set theory]" I was looking for a problem with the logic when applied to a set of all cardinalities or defining a set like that for example, but couldn't find one and figured an analogy to Russel's seemed likely.
What exactly do you mean by Cantor's paradox is a stronger result than Russel's? Usually I would interpret this to mean "Cantor's implies Russel's, but Russel's does not imply Cantor's", but neither works with the ZF axioms so either could imply anything. Is one "stronger" in the sense I am bringing up if you work with some alternate (sensible) set of axioms?
Once again: from Russel's paradox it follows that given any set X there is some set Y which is not in X (i.e. the set of all sets cannot exist). From Cantor's paradox it follows that given any X set there is a set Y with cardinality strictly greater than any set in X (i.e. no set can contain sets of all cardinalities). Of course, such a set can't be in X, because every set has the same cardinality as itself (or, using the negative formulation, the universal set would contain sets of all cardinalities, because it contains all sets, so Cantor's paradox precludes its existence as well).
ok, what is the practical use of all these different sizes infinities? Can they be useful in physics or engineering?
Many thanks for the enlightenment. For me, the first two types of infinities is important for philosophical reasons :-) And in trying understanding tailor series, the Basel problem and things like that. But I have only taken highschool math.
I think... we might need some set which are greater than |A| but less then |P(A)|, ie contium since |p(a)|= 2^|a| for finite set, perhaps a smaller function could be used.
This is the generalized continuum hypothesis. In general, it cannot be proven that such a set (given an infinite set A) exists; on the other hand, it cannot be proven that such a set doesn't exist.
It's not even that subtle. The axiom schema of specification can only produce subsets of a given set, thus there's no reason the "set of all cardinals" should exist. In fact, this very argument proves, by contradiction, that it doesn't.
There is an infinitesimal of numbers between 01-2.0,but also an infinity of numbers between 1 and 2 even though the infinity between 1 and 2 is obviously larger. Get it y'all
When I extend the resolution of the paradox inwards, I would expect Natural numbers to be a non-onto subset of Integers.
I'm imagining: #N U #{-1,-2,-3,...} = #I
* U #{0} applied as necessary
Additionally, what happens when you map the Integers to the power set of the natural numbers?
Sadly, my understanding of mathematics is less complete than I wish it to be.
Thereâs a number called âAbsolute Infinityâ, which means no number before it can be grouped any amount of times to get it.
I think that what you're talking about is the (strongly) inaccessible cardinal, which cannot be obtained by taking either a powerset of a set smaller than itself; nor by taking a union of a collection of sets smaller than itself, where the collection is also smaller than the cardinal number itself.
Absolute infinity is the class of all sets, which is a proper class and can be proven not to exist as a set. (On the other hand, a set of all sets with a rank less than an inaccessible cardinal - assuming such a cardinal number exists - is a model of set theory. So what is a proper class - class of all sets - from the point of view of the inner model, is a set - von Neumann universe up to an inaccessible cardinal - from the point of view of the outer model.)
Gabe from SpaceTime, hi :D
I believe the answer to this question is 42.
my only gripe with this video is with the term "1-1 correspondence" in relation to bijective. Not to be confused when one sees a "1-1 mapping" which is an injective map.
The distinction of "correspondence" over "mapping" should not be over looked.
wow! really cool, I have to check out deeper but I definitely think that when you take the union of all the infinite sets ( so when you construct U ), you just create the same thing as if you took P of the ÂŤ biggest Âť infinite (here aleph infinite) so you just create aleph infinite+1 wich is equal to aleph infinite.
The thing is that you add one thing to someting infinite and says that it is larger, but when you talk about infinite adding 1 doesnât change anything.
To convince ourself we can just look what would be into U : there will be all the sets of sets of sets... of numbers . This is just as P(P(P(...N))) when the numbers of P is ÂŤ equal to infinity Âť . So when you take P(U) you dont change anything becaus you just add one P() in front of the infinitely many ones. so P(U) = U
sorry for being a bit confuse but I kind of changed my mind while wrigting, and I have to say that everything about this is quite hard to wrap your mind around.
Yeah, this episode is really hard. Not intended to be easily digested on just one viewing, for sure.
Cantordinal numbers be transfinite though?
is that a pun?
is the pope catholic?
Is the cat alcoholic?
2:42 No, just because there is an injection A -> B doesn't prove that |A| < |B|. You also need to show that there does not exist a bijection A -> B. Otherwise we could prove that |N| < |Z| < |Q|.
3:09 Isn't it true only for finite sets?
Do we need the axiom of choice to pick the sets at 11:03? Is this paradox still valid in ZF?
You can't use the axiom of choice on the class of all sets of a particular cardinality. But assuming AC, there is a way to get a cardinal number for every set: 1) From the axiom of choice it follows that every set can be well-ordered (so that every subset has a minimum); 2) For every well-ordered set there exists a corresponding ordinal number; 3) A cardinal number is an ordinal number for which there doesn't exist a smaller with the same cardinality (it can be proven that ordinal numbers themselves are well-ordered, so the set of ordinal numbers less than or equal to x but of the same cardinality as x has a minimum).
Without axiom of choice, you can't prove that all sets can be well-ordered, but I believe that the paradox still works if you restrict yourself to the sets that can.
On the other hand, we can skip the problem with cardinal numbers and the axiom of choice, and just use this formulation: 1) Assume that set X contains sets of all cardinalities; meaning that every possible set can be 1-to-1 mapped to some set in X. 2) Take X' = union of all sets in X. 3) Take P(X') = set of all subsets of X. By Cantor's proof, P(X) can't be put in 1-to-1 correspondence with X'; nor with some subset of X', and every member of X is a subset of X'. 4) So the set P(X') can't be put in a 1-to-1 mapping with any set in X, contradicting our assumption; therefore, X doesn't contain sets of all cardinalites.
We know that ×_i>i. So we have to find number Îą such that ×_Îą=Îą. That looks exactly like our contradiction. Thatâs why the answer is ×_×_×_..... (where Îą_β means Îą with index β)
That's just a limit (=union) of the sequence of sets Aleph-0, Aleph-Aleph-0, Aleph-Aleph-Aleph-0, ... And this is a well-defined cardinal number, a union of countably many sets (and yes, it is a fixed point of the Aleph function). If we label this set as Ί, then P(Ί) certainly has a greater cardinality.
But an inaccessible cardinal cannot be proven to exist in ZFC. Worst, you can't even prove that its existence is consistent with ZFC (as long as ZFC itself is consistent).
My guess is that U or P(U) somehow contains itself, therefore it is not an actual set.
3:40 This is actually because of how a finite set is defined.
âGod made the integers, all else is the work of man.â (Leopold Kronecker) I think thereâs countable infinity and uncountable infinity. Anything after that is a mathematical game. The question may be the validity of the idea of a power set on an uncountable infinity. How can there be an uncountably infinite input to a power set function?
Sure there can: the set of all sets of real numbers. (The set of all real numbers is uncountable, so its own powerset, or a set of its subsets, has an even greater cardinality than the set of real numbers.)
Wait, what's to stop B being the empty set?
Surely the empty set would have to be possible as the power set of a single element set or the empty set would be itself, right?
If you're talking about the subset B from 6:51, and you want to handle the case where the original set A contains only element -- say, A = {x} -- then B could very well be the empty set, and that's ok. Watch... a singleton set like A has only two subsets: A itself, and the empty set. So P(A) = { {}, {x} }. A function G from A to P(A) either maps x to {} -- in which case B would have to be {x} (yes, same as A in this case, which is cute) -- or it maps x to {x} -- in which case B is in fact the empty set. Either way, though, the map G isn't onto, so you can't have a bijection b/w A and P(A).
Oh right, ok. Thanks for clarifying. I honestly just missed that the empty set was part of the power set.
Ay 10:30 we show that |P(U)| is a strictly larger cardinatlity than any in the list "|N|, |P(N)|, |P(P(N))|, ...", where U is the union of {N, P(N), P(P(N)), ...}.
This is fine. But we could have easily just shown that |U| itself is a strictly larger cardinarilty too, than any in our first list. This is because, for any S in U, P(S) is in U too, which implies the cardinality of U is strictly greater than S.
True, but I think the argument is a little bit slipperier and has more caveats, ergo I opted against it for presentation purposes. Here's what I mean. To lay it out that way, we'd need to remember that things in U aren't the sets N, P(N), P(P(N)), ... per se -- the things in U are all the *elements* of all those sets. Now in the late 19th century, elements of N aren't necessarily thought of as sets in the Zermelo or von Neumann sense -- the math world isn't there yet. But elements of all the power sets are of course sets, so say S is one of those elements of a power set. That means S is a subset of one of the previous sets in the chain. Since every set is a subset of itself, you could always pick S to be one of the actual power sets in the chain. Carrying the above argument forward one extra step, then P(S) is in U as well, which means |U| >= |P(S)| > |S|, where the 1st inequality is due to the rule that A subset-of B ==> |A|
You're right! Your approach makes for much smoother teaching ^_^