Cantor's theorem and countable additivity of Lebesgue measure

In this book, with more than 300 pages, the author claims that Cantor was wrong:

Transfinity, Wolfgang Mückenheim

https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

In particular, the author lists in more than 100 pages authors that had concerns about Cantor's argument. He even cites Sharon Shelah who wrote

Of course, the most severe skeptics will even deny the mathematical content of Cantor’s theorem ($$2^{\aleph_0} > \aleph_0$$). To these we have nothing to say at all, beyond a reasonable request that they refrain from using the countable additivity of Lebesgue measure.

CARDINAL ARITHMETIC FOR SKEPTICS, Shelah

https://arxiv.org/pdf/math/9201251.pdf

Is there any chance that indeed Cantor's theory is inconsistent? In the paper above, Shelah refers to concerns about "consistency" in Cantor's theory - how far may these concerns eventually reach?

• You want us to list all the errors in a 300 page book! Jul 25 '20 at 17:25
• Shelah's comment is actually quite witty. I imagine its humour was lost on Muckenheim. Jul 25 '20 at 17:26
• Shelah does not have concerns about the correctness of Cantor's theorem. The comment you quoted is about the people (like Mückenheim) who are unable or unwilling to understand Cantor's proof. Shelah says that he has nothing to say to those people. If they don't understand even Cantor's quite easy proof, they certainly won't understand Shelah's far more advanced work. Jul 25 '20 at 17:56
• The value of that book cannot be understated. There is nothing of substance there beyond the author's implicit conviction that mathematics must be relevant to the necessarily-finite physical universe, and the author frequently distorts the meaning of the people he quotes (I'm quoted there, for example, without my statements being remotely understood). Jul 25 '20 at 17:57
• you find a huge number of authors that also had concerns --- To me this is a huge red flag that one might be reading crank literature. In this case, however, I know who Mückenheim is, and don't need to look for red flags. However, if I didn't know who he was, I would definitely be thinking "crank possibility" based on the huge number of author quotes. Jul 25 '20 at 17:58

Well, the answer to the question "What is wrong in that book?" is "practically everything." Moreover, the mathematical errors (as opposed to situations where he misunderstands quoted arguments of others) are not at all original to Mückenheim, but rather are the general cranky arguments against the infinite in mathematics - with seemingly one exception, which may be instructive and which is the reason I'm writing this answer.

(To clarify: the prospects for inconsistency in various set theories is actually an interesting topic, but Mückenheim's book does not form a serious contribution to it. If you're interested, one relevant term is "consistency strength.")

First, let me briefly summarize what's not original. The bulk of Mückenheim's book is a reiteration of the standard arguments-from-incredulity, that set theory displays "bad" features and is therefore clearly inconsistent (although Mückenheim either misunderstands or deliberately misuses the technical term "inconsistent" - he conflates formal inconsistency and inconsistency with physical reality). For example, there is the "paradox" of the banker who at day $$n\in\mathbb{N}$$ gains $$10$$ dollars but spends $$1$$ dollar, and yet winds up "at the end of the day" completely broke based on which dollars they chose to spend. The "paradoxes" of this general flavor are completely resolved once we uncover the implicit assumptions that the relevant set-theoretic operations are well-defined and continuous in the appropriate senses, which they aren't; basically, the justification for the arguments against these situations boils down to trying to lift results about finite sets to infinite sets without justifying their continued validity.

The following error, however, does seem original to Mückenheim. (See here if you can view deleted posts.) Consider two different set-theoretic implementations of the natural numbers: as the von Neumann numerals $$0_V=\{\}, 1_V=\{\{\}\}, 2_V=\{\{\}, \{\{\}\}\}, 3_V=\{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}, ..., (i+1)_V=i\cup\{i_V\}, ...$$ versus the Zermelo numerals $$0_Z=\{\}, 1_Z=\{\{\}\}, 2_Z=\{\{\{\}\}\}, 3_Z=\{\{\{\{\}\}\}\}, ..., (i+1)_Z=\{i_Z\}, ...$$ Now take an appropriate "set-theoretic limit of the natural numbers" in each sense: we have $$\limsup_{i\in\mathbb{N}}i_V=\{i_V: i\in\mathbb{N}\}\not=\emptyset$$ but $$\limsup_{i\in\mathbb{N}}i_Z=\emptyset.$$ Aha! says Mückenheim, we have here a contradiction. Well, no, we don't - what we have is two different implementations which behave differently with respect to a set-theoretic operation. But that set-theoretic operation is not meaningful at the level of the structure being implemented itself! This is basically the same error as looking at two programs which compute the same function and being confused about how one is longer than the other: "the length of the program" is not a property of a bare function.

So this mistake reveals the need to distinguish between the thing being implemented and the choice of implementation, and more importantly between operations/relations defined on the level of the thing being implemented vs. the implementation framework. There are indeed interesting things to say about this (the relevant logical term is "interpretation") ... but Mückenheim doesn't. However, since this does appear to be an original confusion and is vaguely related to something interesting it seems worth mentioning.

• Re: non-originality of errors, I can't help but mention this famous passage: "This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new." Jul 25 '20 at 18:46
• Here's a possible analogy to the argument you discuss in the second half of your answer. Let $x$ be a real number that has a terminating decimal expansion, and thus $x$ has both a terminating decimal expansion and a non-terminating decimal expansion (e.g. $2.43000\ldots \;= \; 2.42999\ldots).$ Define $x_T(n)$ be the $n$'th digit to the right of the decimal point of the terminating expansion of $x,$ and let $x_N(n)$ be the $n$'th digit to the right of the decimal point of the non-terminating expansion of $x.$ (continued) Jul 25 '20 at 18:57
• Then $\lim_{n \rightarrow \infty} x_T(n) = 0$ and $\lim_{n \rightarrow \infty} x_N(n) = 9.$ So decimal expansions involve a contradiction, and thus cannot exist. Jul 25 '20 at 18:57
• Different representations impact implementation but not the thing being implemented. However, can we build an argument similar to Cantor's diagonal that is representation independent? Is the Cantor's diagonal argument representation independent? Jul 25 '20 at 20:58
• @DanielS. Crap, that was a really nasty typo! I meant to say "totally implementation-independent." My apologies, that was about the most misleading typo I could have made. "The $n$-th decimal digit of $r$ is ambiguous, as pointed by Dave above, right?" Only if you define it wrong - we can specify the "right" decimal expansion as the one which doesn't eventually consist only of $9$s (and we can do this purely algebraically - no reference to implementation here). Jul 26 '20 at 7:47