Controversy over Cantor's Theory

In this article, "Cantor's Theory" refers to the naive set theory

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introduced by Georg Cantor in the latter

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part of the nineteenth century. The "anti-Cantorians" are

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people who claim that Cantor created a fantasy world. The

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"Cantorians" defend Cantor's Theory. This terminology seldom

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appears in the mathematical literature, but it has become

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almost standard in Usenet discussions of Cantor's Theory.

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The anti-Cantorians claim that while infinite sets and

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power sets of those infinite sets are undeniably useful

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abstractions, what Cantor did was to take an argument

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(the diagonal argument), which is perfectly valid

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in concrete mathematics, and recklessly apply it to

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the abstractions of the infinite, ultimately producing something

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which may have philosophical value, but no scientific value.

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The Cantorians (which includes almost all modern pure

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mathematicians) claim that since Cantor's Theory can be

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formalized in an (apparently)

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logically consistent way (e.g. ZFC), there

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is no room for debate about Cantor's Theory.

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