Talk:Cantor's first set theory article: Difference between revisions

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:Corollary 2. The real numbers cannot be written as an infinite sequence. That is, they cannot be put into a one-to-one correspondence with the natural numbers.

:Observe the flow of reasoning: Cantor's second theorem holds for any sequence of reals. By applying his theorem to the sequence of algebraic reals, Cantor obtains transcendentals. By applying it to any sequence that allegedly enumerates the reals, he obtains a contradiction—so no such enumerating sequence can exist. Kac and Ulam reason differently [20, p. 12-13]. They prove Theorem 1 and then Corollary 2. By combining these results, they obtain a non-constructive proof of the existence of transcendentals. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:RJGray|RJGray]] ([[User talk:RJGray#top|talk]] • [[Special:Contributions/RJGray|contribs]]) 01:56, 14 September 2018 (UTC)</small>

So Cantor does give a method of constructing transcendental numbers. By the way, Kac and Ulam present the same non-constructive proof that Stewart uses. <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:RJGray|RJGray]] ([[User talk:RJGray#top|talk]] • [[Special:Contributions/RJGray|contribs]]) 01:56, 14 September 2018 (UTC)</small>