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Topic: Ordinals describable by a finite string of symbols
Replies: 3   Last Post: Jul 11, 2013 8:42 PM

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Registered: 12/4/12
Re: Ordinals describable by a finite string of symbols
Posted: Jul 10, 2013 9:07 PM
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On 7/10/2013 6:40 AM, Aatu Koskensilta wrote:
> fom <> writes:

>> What is expressed by both, however, is that the universe of discourse
>> must be expressed by a set -- an object of the theory.

> What is expressed by the axioms M and SM is that there exists a set
> with certain properties. Neither says anything whatever about the
> universe of discourse or how it must be expressed. In any case, for
> (relative) consistency and independence results by forcing, the use of M
> and SM is always eliminable, as Cohen himsels explains in _Set Theory
> and the Continuum Hypothesis_. (G. H. Moore, in /The Origins of
> Forcing/, reports Moschovakis in a letter urged Cohen to do away with
> the "ridiculous assumption", that there exists a standard model of set
> theory!)

I suppose we disagree over the relationship
between "definite totalities", "model domains",
and "universes of discourse".

I thank you for pointing this out. I am familiar
with notions of forcing as described in Jech.
Forcing interpreted using Boolean-valued models
requires a ground model.

Are you saying that the modal universe of models
for set theory is meaningful without some ground
model described in some way?

I concede my weakness with the particulars of this
eliminability. But, I reject the solutions to the
foundational problems based upon redefining truth.

That is what I believe Cohen did. I had found a
paper by Cohen in which he admitted as much. I
provided a link for Ross Finlayson some time
ago. But I am pressed for time at this moment.

But, as always, I shall pursue your references
to the best of my abilities to remedy any error
I may have propagated with my statements.

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