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?
 A: 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.
