fom
Posts:
1,969
Registered:
12/4/12


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 7, 2013 3:44 PM


On 7/7/2013 8:02 AM, Julio Di Egidio wrote: > > I have tried and reread the thread: I fail to see all the connections, > but that's good. :) >
It is in Kant that one finds precisely the remark that compares the relationship of a universal judgement to a singular judgement as being that of infinity to unity.
Apoorv attaches the introduction of infinity into mathematics as related to the universal quantifier.
Not grasping the sense of the quote (which is fine  who wants to read this anyway?), he asked additional questions that introduce even greater complexity.
I believe you have read "Tractatus LogicoPhilosophicus"
Wittgenstein rejects Leibniz' principle of identity of indiscernibles and proposes that the sign of equality is eliminable on the basis of each object having a unique name.
Modern model theory "extends languages" with constants without definitions. This is an algebraic approach (in the sense of universal algebra), and, is thus relevant to the perspectives of Skolem.
But, it is in the nature of definitions that symbols are introduced with respect to properties. This is an application of Leibniz' principle of identity of indiscernibles.
So one is confronted with the relationship of uninterpreted symbols of a language as treated in model theory and the defined symbols which Apoorv hopes to use to reasonably restrict notions in set theory.
Does that help?
It just gets worse...

