fom
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Re: Ordinals describable by a finite string of symbols
Posted:
Jul 5, 2013 9:42 PM


On 7/4/2013 11:42 PM, apoorv wrote: > > Unless the ' the set of all nameable ordinals' is itself not a sharply > Defined entity.
A short time ago, I began a thread with these links:
http://mathoverflow.net/questions/44102/istheanalysisastaughtinuniversitiesinfacttheanalysisofdefinablenumb/44129#44129
http://de.arxiv.org/pdf/1105.4597v2
In the second link you will find the remark:
"In a model of set theory, this property strengthens V=HOD, but is not firstorder expressible"
You may take that as confirmation that "nameable ordinals" (in so far as the modern paradigm would associate that with the notions of definability used in the paper) are not "sharply defined".
The notion of "sharply defined" would correspond with the idea of "definite totalities" that I discussed with you in an earlier post.
Or, comply more closely with the statement itself, note that firstorder logic involves certain presuppositions regarding denotation (as noted elsewhere, denotation without instantiation is a notion that may be compared with Russellian description theory). This is stated explicitly in the first sentence of the link:
http://plato.stanford.edu/entries/logicfree/
What is involved here, however, is that the instantiation of a denotation in set theory will presuppose a "definite totality" in the received paradigm.
In general, mathematicians might not be committed to firstorder logic. But, it is important to most in the study of set theory.
Recall that the link on Skolem's paradox mentioned that it is really only applicable in a firstorder context.
Thus, we return to the infinite regress you questioned elsewhere.

