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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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fom

Posts: 1,968
Registered: 12/4/12
Re: Ordinals describable by a finite string of symbols
Posted: Jul 1, 2013 7:50 PM
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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 Lowenheim-Skolem
theorem.

If you wish to invoke your statement in support of
your claim, please tell us why the downward Lowenheim-Skolem
theorem is applicable, but, the upward Lowenheim-Skolem theorem
is not.

Indeed, if you wish to invoke the downward Lowenheim-Skolem
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/paradox-skolem/






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