apoorv
Posts:
53
Registered:
4/11/13


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 3, 2013 1:33 PM


On Tuesday, July 2, 2013 5:20:30 AM UTC+5:30, fom wrote: > On 6/30/2013 12:24 PM, apoorv wrote: > > > > > > > > > Like all theories in a countable language, ZFC has a countable model. > > > In this model, the set of all ordinals, which does not exist, must be countable. > > > > Do you grasp the presuppositions you make in order > > for this to be true? > > > > This is an algebraic view (universal algebra). > > > > A signature of symbols and their arities are given > > *through analysis*. > > > > That signature includes a parameter to be interpreted > > as a domain of discourse. > > > > Typically, domains of discourse are taken to be > > sets. > > > > Either, set theory is logically prior to universal > > algebra and the application of the algebraic view > > is invalid, or, the model theory of set theory is > > a modal system. > > > > Note, also, that there is an upward LowenheimSkolem > > theorem. > > > > If you wish to invoke your statement in support of > > your claim, please tell us why the downward LowenheimSkolem > > theorem is applicable, but, the upward LowenheimSkolem theorem > > is not. > > > > Indeed, if you wish to invoke the downward LowenheimSkolem > > theorem, what part of your argument rejects the fact that > > the unspecified model that has been presumed is not an > > uncountable model? > > > > > In view thereof,there is one countable limit ordinal , which cannot be assumed > > > To exist.? > > > > Let us leave this part alone for the moment. > > > > But, dullrich's advice is worthy. Here is a > > link for you, > > > > http://plato.stanford.edu/entries/paradoxskolem/ Thanks for that link. Dullrich's argument is that ' the first ordinal that has no description' is not a valid description, And cannot be formalised. Suppose we work in the language of set theory. We let ordinals alpha have the definitions Ax x e alpa <> phialpha x & x e beta , where beta is some uncountable ordinal, and phialpha Is a formula in the language of set theory. Now the set of wffs in set theory is countable . We define F = { (phialpha,alpha) : phialpha is a formula in language of set theory } Range F = { alpha: (phialpha,alpha) e F} I.e Ax x e Range F <> (phix,x) e F and x e beta as Range F is countable. Then Range F e Range F , since Range F is an ordinal and is defined by a formula of the requisite form. Maybe the above argument is again most likely flawed; But then nameless undefinable ordinals also is very counter intuitive. Apoorv

