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## Main Question or Discussion Point

I have a question about the potentially self-referential nature of cantor's diagonal argument (putting this under set theory because of how it relates to the axiom of choice).

If we go along the denumerably infinite list of real numbers which theoretically exists for the sake of the example, then by definition after some finite amount of time we will reach the number which cantor is constructing by going along and making his nth digit differ from the nth number in the list. At this point, the digit at this position is altered. But then, by definition, we did not in fact reach the number cantor was constructing after all. Is it not possible to imagine that, for example, every time this occurred the number being constructed was moved 'further down the line' on the denumerable list, so to speak, ie for example's sake to position n+100 where n is the number currently being analysed. Thus at any given time Cantor's construction is in the denumerable list, but it just keeps getting altered so that it never gets reached. Since in any finite amount of time Cantor will never complete his number, does his construction then actually prove that there is in fact a number which is not in the denumerable list? This self-referential conundrum seems to lie more along the lines of an example of Godel's incompleteness theorem applied than as an actual proof for the non-denumerability of the reals. Am I missing something that makes Cantor's argument a little more solid?

sincerely,

jeffceth

If we go along the denumerably infinite list of real numbers which theoretically exists for the sake of the example, then by definition after some finite amount of time we will reach the number which cantor is constructing by going along and making his nth digit differ from the nth number in the list. At this point, the digit at this position is altered. But then, by definition, we did not in fact reach the number cantor was constructing after all. Is it not possible to imagine that, for example, every time this occurred the number being constructed was moved 'further down the line' on the denumerable list, so to speak, ie for example's sake to position n+100 where n is the number currently being analysed. Thus at any given time Cantor's construction is in the denumerable list, but it just keeps getting altered so that it never gets reached. Since in any finite amount of time Cantor will never complete his number, does his construction then actually prove that there is in fact a number which is not in the denumerable list? This self-referential conundrum seems to lie more along the lines of an example of Godel's incompleteness theorem applied than as an actual proof for the non-denumerability of the reals. Am I missing something that makes Cantor's argument a little more solid?

sincerely,

jeffceth