LudovicoVan
Posts:
3,223
From:
London
Registered:
2/8/08


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 7, 2013 6:23 PM


"fom" <fomJUNK@nyms.net> wrote in message news:H76dnfhczfi8WETMnZ2dnUVZ_judnZ2d@giganews.com... > 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"
Yes, of course, and even studied a bit about it: I just have zero memory, but for the bottom lines...
> 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.
Wittgenstein also rejects Russellian theory of definite descriptions (if that's the exact name). For the Wittgenstein of TLP the world is the totality of facts, but these are the facts of language!
> 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.
Well, I have no qualms with an algebraic development of mathematics, but not for foundations: no "primitive" is purely syntactical when the language is a language; equivalently, no proof or argument is ultimately purely syntactical.
> 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.
Indeed, going round in circles.
> 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...
Well, I'm loving it. I just wish I could offer more precise quotes and references, as I guess you'd appreciate it, but, hey, we do what we can... :)
Cheers,
Julio

