On 12/5/2013 2:28 PM, Michael F. Stemper wrote: > On 12/05/2013 01:04 PM, Zeit Geist wrote: >> On Thursday, December 5, 2013 12:49:37 AM UTC-7, fom wrote: >> >>> No. It also uses the fact that the listing may be >>> >>> arbitrarily given, >>> >>> http://en.wikipedia.org/wiki/Algorithmically_random_sequence#Interpretations_of_the_definitions >>> >> >> Be careful here, the "arbitrarily chose" sequence does Not need to be >> a "random" sequence. >> It is just a sequence of Real Numbers who only properties is that it >> IS a sequence of Real Numbers that "supposedly" contains all Real >> Numbers. > > There is no need for the assumption that it contains all reals. We can > prove that no sequence of reals contains all of them without having to > first assume that it does. >
I had to think about that.
Cantor's proof had been directed to an audience who thought of "infinity" as a monolithic concept. Proving that any list asserting to put the set of reals in correspondence with the naturals was, in fact, not a complete list demonstrated that "infinity" could be viewed as a plural notion subject to logical analysis.
Your statement reflects Hilbert's formalism. In his papers he specifically mentions how the ideal formal axioms relegate the class-based constructions of Dedekind and Cantor to definable structures within the theory. Thus, they lose the import they have outside of a formalist theory.