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Topic: Ordinals describable by a finite string of symbols
Replies: 16   Last Post: Jul 9, 2013 7:59 AM

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apoorv

Posts: 53
Registered: 4/11/13
Re: Ordinals describable by a finite string of symbols
Posted: Jul 3, 2013 1:33 PM
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On Tuesday, July 2, 2013 5:20:30 AM UTC+5:30, fom wrote:
> On 6/30/2013 12:24 PM, apoorv wrote:
>

> >
>
> >
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> > 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).
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>
>
> A signature of symbols and their arities are given
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> *through analysis*.
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>
>
> That signature includes a parameter to be interpreted
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> as a domain of discourse.
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>
>
> Typically, domains of discourse are taken to be
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> sets.
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>
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> Either, set theory is logically prior to universal
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> algebra and the application of the algebraic view
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> is invalid, or, the model theory of set theory is
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> a modal system.
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>
>
> Note, also, that there is an upward Lowenheim-Skolem
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> theorem.
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>
>
> If you wish to invoke your statement in support of
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> your claim, please tell us why the downward Lowenheim-Skolem
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> theorem is applicable, but, the upward Lowenheim-Skolem theorem
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> is not.
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>
>
> Indeed, if you wish to invoke the downward Lowenheim-Skolem
>
> theorem, what part of your argument rejects the fact that
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> the unspecified model that has been presumed is not an
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> uncountable model?
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>
>

> > In view thereof,there is one countable limit ordinal , which cannot be assumed
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> > To exist.?
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>
>
> Let us leave this part alone for the moment.
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>
>
> But, dullrich's advice is worthy. Here is a
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> link for you,
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>
>
> http://plato.stanford.edu/entries/paradox-skolem/

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




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