Cantor's first set theory article: Difference between revisions

Revision as of 15:36, 27 November 2020

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== A misconception about Cantor's work ==

[[Akihiro Kanamori]], who specializes in set theory, has stated: that "Accounts of Cantor's work have mostly reversed the order for deducing the existence of transcendental numbers, establishing first the uncountability of the reals and only then drawing the existence conclusion from the countability of the algebraic numbers. In textbooks the inversion may be inevitable, but this has promoted the misconception that Cantor's arguments are non-constructive."<ref name=Kanamori4>{{harvnb|Kanamori|2012|p=4}}.</ref>

Cantor's published proof and the reverse-order proof both use the theorem: Given a sequence of reals, a real can found that is not in the sequence. By applying this theorem to the sequence of real algebraic numbers, Cantor produced a transcendental number. He then proved that the reals are uncountable: Assume that there is a sequence containing all the reals. Applying the theorem to this sequence produces a real not in the sequence, contradicting the assumption that the sequence contains all the reals. Hence, the reals are uncountable.<ref name=Ewald840_841/> The reverse-order proof starts by first proving the reals are uncountable. It then proves that transcendental numbers exist: If there were no transcendental numbers, all the reals would be algebraic and hence countable, which contradicts what was just proved. This contradiction proves that transcendental numbers exist without constructing any.<ref name=Kanamori4/>