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I am doing an essay on the knowledge of Mathematics and how we know what we know to be true. I was just wondering if there was an example in mathematics of some theorem that was disproven by a counterexample, but then the counterexample was wrong and the theorem stayed true after all?

Thanks for your time. :)

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    $\begingroup$ This happens to me approximately once per week. $\endgroup$ May 27, 2021 at 2:35
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    $\begingroup$ For the issue of "how we know what we know to be true" I think something in the spirit of what you are asking is a proof of a theorem that is accepted for a reasonable period of time, the proof turns out to be wrong, and later the theorem is proved correctly by another method. After all, why is it important for a supposed theorem to be disproved by a counterexample that winds up being wrong instead of the original proposed proof of the theorem being wrong instead? Something along those lines is the saga of the four-color theorem. It was conjectured in 1852 and "proved" by Kempe in 1879 and T $\endgroup$
    – KConrad
    May 27, 2021 at 3:20
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    $\begingroup$ Perhaps the theorem that the Euler characteristic of a polyhedron is 2 is an example of what you are looking for. The telling in Proofs and Refutations is a very good read. $\endgroup$
    – Zhen Lin
    May 27, 2021 at 5:37
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    $\begingroup$ @ZhenLin. I took a look. What really happens there is that the definition of polyhedron is stretched in such a way that yes the conjecture fails for it, but it's not an instance where a good faith wrong counterexample is given. $\endgroup$ May 27, 2021 at 14:45
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    $\begingroup$ The top answer to Widely accepted mathematical results that were later shown to be wrong? mentions a 1994 Annals paper by Gaoyong Zhang, whose main result can be interpreted as stating that the unit cube in $\mathbb{R}^4$ is a counterexample to the claim that every origin-symmetric convex body in $\mathbb{R}^4$ is something called an "intersection body." This was regarded at the time as demonstrating that the Busemann-Petty problem has a negative answer in dimension 4. But Zhang's paper turned out to be wrong. $\endgroup$ May 27, 2021 at 20:29

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A very famous and important example of a counterexample that was found to be defective occurred in set theory. As Georg Cantor developed the theory of infinite sets, he proved that some infinite sets are larger than others, by showing that there was not a one-to-one function between them. Most famously, he showed that the real numbers $a\leq x\leq b$ are more numerous (have greater "cardinality") than the natural numbers $\{1,2,\ldots\}$.

However, he also proved a more general statement, Cantor's Theorem—that if $X$ is a set (any set, finite or infinite), and ${\cal P}(X)$ is the set of all subsets of $X$ [we call ${\cal P}(X)$ the "power set" of $X$], then ${\cal P}(X)$ has a greater cardinality than $X$. Crudely put, this says that any set has more subsets than it has elements. That is certainly true for finite sets; a set with $n$ elements has $2^{n}$ subsets, and $2^{n}>n$ for all integers $n\geq0$. However, Cantor showed it was true for arbitrary sets, even infinite ones.

There were many objections to Cantor's theory of infinite sets, and one important objection was to the theorem that ${\cal P}(X)$ has a greater cardinality than $X$. This seems to be impossible, because it cannot be true of the set of all sets (usually denoted by ${\bf U}$, for the "universe" all sets). For, if ${\bf U}$ contains all sets, then it necessarily contains all elements of its own power set ${\cal P}({\bf U})$, meaning ${\cal P}({\bf U})$ cannot be more numerous than ${\bf U}$.

The resolution of this apparent paradox is that the supposed universal set ${\bf U}$ cannot exist, at least not as a set. (The universe ${\bf U}$ does exist as a more general kind of object, a proper class.) This was not evident in Cantor's approach, since he did not provide a axiomatic basis for his set theory. (His approach is sometimes therefore known as "naive set theory.") However, with the development of a rigorous foundation for set theory, based on axioms (such the Zermelo-Fraenkel axioms), it became clear that the object ${\bf U}$ is simply not definable as a set in the theory. While this might seem to be a problem, since it means that a rather natural-seeming object cannot exist, it is actually extremely advantageous, since it means that the counterexample to Cantor's Theorem does not actually exist, leaving the theory (so far as we can tell) consistent.

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