Paradoxes of set and property

DOI: 10.4324/9780415249126-Y024-1
Version: v1,  Published online: 1998
Retrieved September 21, 2020, from

3. Cantor: side-stepping the paradoxes

In 1883 Cantor had introduced infinite ordinal numbers (see Set theory; Cantor’s theorem). He distinguished sharply between finite ordinals, infinite ordinals and what he called the ‘absolutely infinite’, which included the set of all ordinals. In 1897, when Hilbert asked whether every set can be well-ordered, Cantor answered yes. Cantor’s argument indirectly included the paradox of the largest ordinal and explicitly relied on the absolutely infinite. Yet he did not regard this as paradoxical, since he had previously made the relevant distinction between the infinite and the absolutely infinite. (E.H. Moore independently found this ordinal paradox and wrote about it to Cantor in an unpublished letter of 1898.)

Replying to Hilbert, Cantor made the notion of the absolutely infinite more precise. In print two years earlier, he had defined a set as a ‘gathering together’ of distinct objects. Now he called a set ‘completed’ if it is possible, without contradiction, to think of all its members as gathered together into a whole; any other set, such as that of all ordinals, was ‘absolutely infinite’. In his Paris lecture of 1900, Hilbert affirmed that each of Cantor’s infinite cardinal numbers is consistent, but denied that there exists a set of all cardinals. Nevertheless, Hilbert gave no indication that set theory was threatened.

In 1899 Cantor communicated to Dedekind the proof that every set can be well-ordered (see Cantor 1991). Once more, the paradox of the largest ordinal was involved when Cantor showed, as part of the proof, that the collection of all ordinals is absolutely infinite. Now Cantor distinguished between what he called consistent collections and inconsistent collections; only the former were sets. (A ‘consistent’ collection was what he had earlier called ‘completed’.) Writing to Hilbert shortly afterwards, Cantor saw Dedekind’s foundation as threatened, since Dedekind assumed that every well-defined collection was consistent. Cantor had pointed out to Dedekind that the collection of everything thinkable, which Dedekind had used to prove the existence of an infinite set, was inconsistent.

Citing this article:
Moore, Gregory H.. Cantor: side-stepping the paradoxes. Paradoxes of set and property, 1998, doi:10.4324/9780415249126-Y024-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
Copyright © 1998-2020 Routledge.

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