Are there Infinities of Different Sizes? Of Course Not! Cantor was Wrong (The Disbeliever, Part 7)

This video highlights some of the many issues with Cantor’s arguments that there are infinities of different sizes. It is the seventh in a series of ten videos in which a ‘disbeliever’ (of supernatural things) reveals the reasons behind his disbelief. He complains about the basis of mathematics, how the Ancient Greeks messed up, why infinity is an invalid concept, how maths could be much better without the mysticism, the flaws in Cantor’s logic, how science got corrupted by maths, why quantum computers are rubbish, and how you should deal with a disbeliever.

[aarondavidson]

Agreed. I always said if you created a new number then the list wasnt full in the first place.

[Simon-tsfu]

I think the real issue is the failure to properly distinguish objects from processes. The claim about different sizes of infinity is presented as being about sets of objects but is actually about comparing processes. I cannot offer a solution but the source of all mathematical obscurantism is the notion of the infinite set. A set is intuitively thought of as explicitly exemplified objects, which cannot be infinite. An “infinite set” is rather a finitely defined generative process.

[jeffjo]

1:36 "By cantor's logic we can simply ignore the not-matched problem."

What whole numbers are "not matched?" Please, enlighten me by finding one. But be warned that whatever number you name, call it n, I will point out is matched to 2*n.

The issue here is that you can't see past the method you learned in Kindergarten for determining size. Which is to measure/count/whatever a thing from its beginning *_to_* *_its_* *_end._* But the infinite set of whole numbers has no end, so you can't do it. And that is the failure here - the definition of "as many" that you want to use cannot be applied, so your argument fails.

In fact, one definition of an infinite set is one that can be put into a bijection with a strict subset of itself.

Next, I'll point out that "12345" is not "a number." It is a character string; it can be interpreted as the *_decimal_* *_representation_* of a whole number. Or a street address, which only looks like a number. Or an access code for entry into a secure facility. So now I'll ;let you in on a little secret: Cantor did not apply his diagonal argument to numbers. He said this specifically: "There is a proof of this proposition ... which does not depend on considering the irrational numbers." He applied it to infinite length strings using the two characters "m" and "w". The examples he used were:

S1 = (m, m, m, m, … ),
S2 = (w, w, w, w, … ),
S3 = (m, w, m, w, … ).

He used different names, but this corresponds more closely to the notation in Wikipedia. And my point here is that you can't try to apply it to natural numbers, as you try at 4:04 "To a disbeliever the concept of infinitely many leading digits is ... absurd." Yet Mathematics uses "infinitely many" things in several ways.

But I'll continue using real numbers, since the proof works with them.

4:52 "In order to go down a diagonal we would have to use leading zeroes trailing zeros or both and the result would not be of the same type as the numbers in the list."

Patient: "Doctor, Doctor, it hurts is I do THIS!"
Doctor: "So don't do THAT!"

The reply to this claimed failure is "don't do that." CDA only works if you start with infinite-length strings, whether they be "m"s and "w"s or the decimal representations of real numbers. But you have to include all those pesky trailing digits. So the diagonal will also be an infinite-length string. By definition it *_has_* to be "the same type." Yours is a strawman argument.

5:30 "Note that as we change this example to allow more decimal places such as two decimal places then three decimal places and so on the size of the diagonal result grows exponentially."

Doesn't it bother you that, in this argument, you are quite literally saying "We can't use this proof that the set of decimal numbers is larger than the natural numbers, because it grows faster and so we can't make the diagonal?" That is, we can;t prove it to be true because it is true?

5:54 "But now let's assume that all real numbers from zero up to but not including one are listable."



This is where you really go wrong. I know that you were taught that this the start of Cantor's Diagonal Argument, but it isn't. CANTOR NEVER ASSUMES HE CAN MAKE THIS LIST. What CDA proves, translated to use numbers, is this proposition: ""If S1, S2, …, Sn, … is any simply infinite list of real numbers in [0, 1], then there always exists a real number S0­ in [0, 1], which cannot be connected with any real number in that list."

The words "all, " "complete, " "full, " or whatever you want to use never appear in this. Examples of such lists are trivial: Sn=SQRT(1/n) is one. And yes, we can, and need to, include 1 in the set since .

The point is that CDA is only a lemma used to prove that [0, 1] is a bigger set, not the proof of that proposition by itself. In Cantor's words: "From [the proposition I cited above] it follows immediately that the totality of all real numbers in [0, 1] cannot be put into a list S1, S2, S3, ..., Sn, ... otherwise we would have the contradiction, that a number S0 would be both in the set [0, 1], but also not in the set [0, 1]."

[KarmaPeny]

About 1-to-1 correspondence between the natural and real numbers..

I was thinking about how maths concepts might be described in terms of 'explicit procedures' when I encountered a maths forum comment that said "Do you consider 'f(n) = n + 1' a bijection between ℕ and ℕ+, where ℕ = {0, 1, 2, 3, ...} and ℕ+ = {1, 2, 3, ...}?"

In my view, the expression "f(n) = n + 1" denotes a programming function that takes a natural number as input and increments it by one. To me, it's simply a compact representation of a code snippet or algorithm translatable into various programming languages. While mathematicians may perceive it differently, perhaps as an infinite mapping between two infinite sets, I struggle to see it as such.

Likewise, I don't view '√2' as a fixed value situated on an imaginary number line; instead, I regard it as representing a code snippet or algorithm that would perpetually continue if executed. While mathematicians may dispute whether a mathematical term like '√2' equates to a code segment housing a 'square root of two function', I anticipate they might acknowledge some form of connection between them. Essentially, I hope they would concede that one could be 'mapped' to the other.

Now, let's contemplate a set of well-defined symbols capable of constructing functions related to real numbers. For instance, some symbols could define a 'square root of 2' algorithm, while others could depict a pi algorithm, and so forth. This task could be accomplished using a programming language or possibly by using existing mathematical symbols.

Here's where it gets intriguing. With only a finite number of symbols ('x', say), there's a finite limit to the number of 'real number functions' achievable with 'x' symbols. Consequently, we can establish a one-to-one correspondence between each 'real number function' (formed using 'x' symbols) and natural numbers. As we increase 'x' to accommodate more 'real number functions', we can systematically continue to 'count' and thus map them to more natural numbers.

Since 'x' grows 'without bound', no real number can elude our encoding into a function. Thus, for any conceivable specification of a real number, there will exist a mapping to an individual natural number. Hence, it seems we've uncovered a one-to-one correspondence between natural and real numbers.

[En_theo]

Glad to see that I'm not the only one to have noticed all these contradictions. But if we talk about it with people on forums, or even just ask these questions, we get insulted or mocked or they suddenly give "links" that "prove" they're right. Ofc, it leads to an incredibly complicated article that does not prove what they say.

[michaelharding]

I've spent the past week or so diving further and further into this topic and I really appreciate this video because every single other video just accepts the diagonal argument as fact without question.

I've come across some very interesting conclusions that still haven't seen acknowledged once and I'm curious what your thoughts are.

The only thing Cantor's diagonal proves is that all real numbers appear to not be able to be listed on a 2D chart. However, a list of numbers can take any shape. For example, all integers or naturals are listed on a 1D line, but when listing all rational numbers, mathematicians "get creative" by listing them on a 2D chart so that they can count diagonal and prove one to one correspondence. It may just be that we need an infinite 3D grid to list all of the real numbers, in which case cantor's diagonal no longer has the properties it did when it "proved" larger infinities. Upgrading from a 2D infinity to a 3D infinity does not change its cardinality because all the natural numbers can be mapped within any number of dimensions. It's also worth noting that the unit size gets multiplied by infinity with each dimension (infinite points on a line, infinite lines in a plane, and infinite planes in a space, infinite spaces in a timeline), but once again that doesn't affect cardinality because you're counting the number of things, regardless of how many things are contained within each thing.

Another interesting observation is that the diagonal itself doesn't actually have to be a diagonal. It can take any shape as long as each selected cell does not share a row or a column with any other selected cell. Meaning you could theoretically construct ANY real number using the diagonal, even a real number you know is on the list.

The entire argument is absurd. To suggest that there is a greater cardinality than aleph null is to suggest that you run out of natural numbers before accounting for every item in a set, which is impossible regardless of the properties of that set, given that the entire point of infinity in the first place is that it never ends.

I do think it's fine to distinguish infinities based on the number of dimensions they take up, but they all have the same cardinality.

I also think it's okay to distinguish countable and uncountable in the sense that uncountable can exist between two finite end points given that it has infinitely divisible unit sizes. But a countable infinity requires at least one direction to not have an endpoint. These are interesting and distinct properties, but not ones that result in distinct cardinalities.

[alex-myhp]

I thought that exact same thing about proof by contradiction when I first learned about it! This is the first time in my life that I’ve found someone other than me asking the question of which assumption is the contradiction disproving! Having to prove that some number is irrational again and again has been driving me to anger. I’ve asked countless people to explain it to me but I’m starting to think that either they don’t understand what I’m saying or they’re pretending not to. How can you expect students to use proves that they don’t understand. It’s completely pointless to write out of proof if it doesn’t prove the statement to the person writing it! What utter nonsense!

[johnscovill]

Since each whole number can be multiplied by 2 to get a specific and unique even number, it follows that every whole number has a matching even whole number. Because they always match, the size of the set of even numbers is the same as the size of the set of whole numbers.

[ThePallidor]

When I was a student, Walter Royden's standard textbook Real Analysis contained this admonition: "All students are enjoined in the strongest possible terms to eschew proofs by contradiction."

Apparently it has been removed in the ever-tightening vice of cultural enforcement of semantic nonsense as a fixture of mathematics.

You're quite right that a contradiction only tells you that *something* was wrong with your proof; it doesn't say what. So each proof by contradiction builds on the assumption that all that came before was correctly proven. No error checking. No collapse indicating a prior error. Free rein to "prove" anything you want as long as you can get one foundational error baked into the accepted body of math. The only restrictions are now cultural, not logical.

[mind-blowingmath]

5:20: It is always the case that the diagonal argument produces a result of a different type
So you just agreed that diagonal argument indeed produces a result which was not in original set.

Infinitely many leading digits - that's called p-adic numbers:)
(although calling this numbers is confusing, but yeah - you just proved that there are "more" objects with infinitely many leading digits compared to just whole numbers)

Speaking of numbers with infinitely many trailing digits, there are some very specific examples of that, for example: sqrt(2)
And even 1/3, represented as decimal would have infinitely many trailing digits:

[dialectphilosophy]

I’ve always understood the Cantor cardinality to be more about denumerability than actual “size”. You say in the beginning that it’s absurd to say that there’s as many even numbers as whole numbers — which of course is obviously true — but I don’t think that is what Cantor cardinality actually measures.

I take the cardinality of the set of whole numbers vs. real numbers to mean this: if you pick any two whole numbers you can always count the number of whole numbers between them. Similarly with integers, even numbers, etc. But pick any two real numbers, and there’s no way to count the number of real numbers between them. That’s obviously a consequence of how real numbers are defined, so I never really understood why a big deal had to be made of it, but I also can’t claim to have fully studied the subject.

[wernerhartl]

Thanks for excellent video.

You can apply the logic of CDA to a list of the natural numbers. For each number in the list there is a number not in the list up to that number, for every number in the list. Therefore there is a natural number which is not in the list of natural numbers.

[martind]

So, how many natural numbers are there?

[navi_w]

the reason why the set N has a cardinality less than R even tho it's cardinality equals infinity is bcuz there isn't a bijective function from N to R so their cardinalities aren't equal (the cardinality of N is less than the cardinality of R bcuz there is an injective function from N to R) Cantors argument shows that if you assume the existence of such function then you'll reach a contradiction

[innertubez]

Thanks - great video. I think proof by contradiction does make sense to me in cases such as the infinitude of prime numbers or the proof that not all numbers are rational. And both show what is wrong when there is a contradiction. But for some reason I’ve always had suspicions about the sizes of infinity argument. I’m starting to see more counterarguments that make sense.

[rockapedra]

Infinity is evil. Everytime we play with it, stupid conclusions result. We should stick to "a really big number" instead of infinity so this nightmare goes away.