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The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”)

It is true that if we represent the integers each by a different sign (or figure), we will have a (infinite) set of signs; and if, in that set, we wish to highlight with special signs, the numbers that represent evens, then we will have a “second” set that will be part of the first; and, being infinite, both sets will have the same number of elements, confirming Cantor’s argument. But he is confusing numbers with their mere signs, making an unjustifiable abstraction of mathematical properties that define and differentiate the numbers from each other.

The series of even numbers is composed of evens only because it is counted in twos, i.e., skipping one unit every two numbers; if that series were not counted this way, the numbers would not be considered even. It is hopeless here to appeal to the artifice of saying that Cantor is just referring to the “set” and not to the “ordered series”; for the set of even numbers would not be comprised of evens if its elements could not be ordered in twos in an increasing series that progresses by increments of 2, never of 1; and no number would be considered even if it could be freely swapped in the series of integeres.

Is there an axiomatic system that his argument makes sense but not the usual ZF? The only possible link I found was this: The mathematical argument is this: Analysis is in contradiction with set theory. See pp. 256ff of https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

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    $\begingroup$ That link is to a well-known crank, with a long history of railing against Cantor; you can still see him doing so in sci.math. $\endgroup$ Jun 29, 2021 at 1:22
  • $\begingroup$ @arturomagidin What site is this? Does anyone show me the link? $\endgroup$ Jun 29, 2021 at 10:05
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    $\begingroup$ @Lambertmacuse: the link at the end of the post. $\endgroup$ Jun 29, 2021 at 11:41
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    $\begingroup$ As to the rest of the post, it reminds me of a quote I've seen attributed to Henri Lebesgue: "As far a philosophy and mathematics are concerned, it is my view that philosophers should not attempt to do mathematics, and mathematicians should not attempt to do philosophy. A view, moreover, that has been expressed forcefully to me by many philosophers." $\endgroup$ Jun 29, 2021 at 11:44
  • $\begingroup$ Please don't deface questions, especially ones with accepted answers. $\endgroup$ Jul 19, 2021 at 3:44

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As Arturo suggests in the comments, WM is a very well known crank and one should not take his words seriously on set theory. You can even find some questions of his here where users (including me) may have engaged with him on occasion. He is very adamant that he cannot be mistaken. And this is despite the fact that Cantor's theorem is incredibly straightforward to prove.

Let's review the proof. Given a set $X$ and its power set $\mathcal P(X)$, take any function $f\colon X\to\mathcal P(X)$, then consider $C_f=\{x\in X\mid x\notin f(x)\}$. If $f(x)=C_f$, then $x\in C_f$ if and only if $x\notin C_f$, which is a contradiction.

So, what did we use?

  1. We assumed that $X$ is a set and that $\mathcal P(X)$ is a set, otherwise there's no meaning to this. So we didn't use the Power Set axiom.

  2. We assumed that $f$ is a given function. So what would that be? In most reasonable foundations of mathematics, functions are objects of the theory, not the meta-theory. So it's not that $f$ is some abstractly defined function, it is a concretely existing function (which may or may not have a definition).

  3. We used $f$ to define a subset of $X$. So we need to be able to "separate" subsets from $X$ using $f$ as a parameter. Here is where it gets tricky. Any reasonable set theory, and indeed a foundation of mathematics, should allow you to separate some subsets, and you should be allowed to use parameters for that, because that's how we normally do mathematics, and foundations of mathematics is not about restricting, it's about giving us the breadth to do what we need to do.

  4. We use the fact that $x\in C_f$ or $x\notin C_f$ is true. In other words, we appeal to the excluded middle.

Now, one can argue that $x\notin f(x)$ is not a good formula to separate a subset of $X$ with. In Positive Set Theory and in New Foundations we can't quite use these formulas. This is why these set theories do have a universal set, $U$, and also the Power Set Axiom, so $\mathcal P(U)\subseteq U$ is true there.

Another way is to look at constructive mathematics and topos theory. I am far from knowledgeable in this subject, so I won't say much here. But there are models of topos theory in which there is an injection from the real numbers into the natural numbers. Whether or not these are suitable to serve as a foundation of mathematics, or are just used to highlight certain necessities, I do not know.

One common thread in all of this is that to disallow Cantor's theorem, you need to somehow throw in a lot of logic into your foundation of mathematics which is then inescapable when you try to write down a proof. Is the formula allowed? Did I use the law of excluded middle? These are not necessarily bad questions to ask yourself, but to quote a famous mathematician, a good foundation is one that stays in the background and you don't feel it when you're doing your actual mathematics.

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  • $\begingroup$ Can you make any sense of the long quote (which isn't from WM)? I can't... $\endgroup$ Jun 29, 2021 at 11:59
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    $\begingroup$ Yes, it reads "I don't actually understand mathematics, but I somehow still feel comfortable making claim that contradict what every freshman in mathematics proved in the last century". $\endgroup$
    – Asaf Karagila
    Jun 29, 2021 at 12:01
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    $\begingroup$ So, Philosophy 101, then... $\endgroup$ Jun 29, 2021 at 12:34
  • $\begingroup$ You mean that's how they torture mathematicians in Room 101? $\endgroup$
    – Asaf Karagila
    Jun 29, 2021 at 13:45
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    $\begingroup$ @ArturoMagidin I think counting Olavo de Carvalho (who hasn't finished high school and started as an astrologer) as philosophy 101 may be a bit too mean even towards philosophers. $\endgroup$
    – Vsotvep
    Jun 29, 2021 at 13:48

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