On 12/5/2013 1:04 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. > > We are randomly choosing a Real Number Sequence, and Not choosing a Sequence whose elements are in a random distribution. >
These sequences are not random. They are random in relation to constructive methods.
WM's ( and Herc's ) idea that the given sequence can be "conditioned" in some manner that restricts the sequence to constructive mathematics can only apply to a set of measure zero.
The arbitrariness of the given sequence precludes one from being able to apply epistemic reasoning.
Where WM is probably correct is that if the notion of "given sequence" is epistemically bound in the first place and if one does not admit a completed infinity, then one is given a sequence in that set of measure zero and one cannot implement the Cantor diagonal.
Unfortunately, I do not know enough about the formalism to know if that can be made rigorous.
The counter-argument is that the epistemically bound sequence cannot be completed by virtue of rejection of a completed infinity. Hence, what is given is either plural or vague.
But, that argument carries no weight for someone who accepts the syntactic grounding of numbers as strokes in an alphabet. All notions of identity resolve to syntactic stipulations and, in the final analysis, to inscriptional identity based upon the shape of letters in the constructive alphabets.