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我把wiki上关于“康托尔理论的争议（Controversy over Cantor's theory）编辑如下。

**Cantor's diagonal argument**

Cantor's first proof that infinite sets can have different cardinalities was published in 1874. This proof demonstrates that the set of natural numbers and the set of real numbers have different cardinalities. It uses the theorem that a bounded increasing sequence of real numbers has a limit, which can be proved by using Cantor's or Richard Dedekind's construction of the irrational numbers. Because Leopold Kronecker did not accept these constructions, Cantor was motivated to develop a new proof.

In 1891, he published "a much simpler proof ... which does not depend on considering the irrational numbers." His new proof uses his diagonal argument to prove that there exists an infinite set with a larger number of elements (or greater cardinality) than the set of natural numbers N = {1, 2, 3, …}.

**Reception of the argument**

Initially, Cantor's theory was controversial among mathematicians and (later) philosophers. As Leopold Kronecker claimed: « **I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there.** » Many mathematicians agreed with Kronecker that the completed infinite may be part of philosophy or theology, but that it has no proper place in mathematics. Logician Wilfrid Hodges (1998) has commented on the energy devoted to refuting this « harmless little argument » (i.e. Cantor's diagonal argument) asking, « what had it done to anyone to make them angry with it? » Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.”

Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a **potential existence**, rather than an **actual existence**. « Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already » . Carl Friedrich Gauss's views on the subject can be paraphrased as: « Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics. » In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets.

Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others. Hilbert predicted: « No one will drive us from the paradise which Cantor created for us. » To which Wittgenstein replied « if one person can see it as a paradise of mathematicians, why should not another see it as a joke? » The rejection of Cantor's infinitary ideas influenced the development of schools of mathematics such as constructivism and intuitionism.

Wittgenstein did not object to mathematical formalism wholesale, but had a finitist view on what Cantor's proof meant. The philosopher maintained that belief in infinities arises from confusing the intensional nature of mathematical laws with the extensional nature of sets, sequences, symbols etc. A series of symbols is finite in his view: In Wittgenstein's words: « ...A curve is not composed of points, it is a law that points obey, or again, a law according to which points can be constructed. »

He also described the diagonal argument as « hocus pocus » and not proving what it purports to do.

**Objection to the axiom of infinity**

A common objection to Cantor's theory of infinite number involves the axiom of infinity (which is, indeed, an axiom and not a logical truth). Mayberry has noted that « ... the set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them—indeed, the most important of them, namely Cantor's Axiom, the so-called Axiom of Infinity—has scarcely any claim to self-evidence at all … »

Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl wrote:

... classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory … »

**Reference :**

AN EDITOR RECALLS SOME HOPELESS PAPERS WILFRID HODGES

https://gwern.net/doc/math/1998-hodges.pdf

the wiki entry on “Controversy over Cantor's theory” as follows:

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory

https://blog.sciencenet.cn/blog-2322490-1454183.html

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