Date: Jul 11, 2013 8:42 PM
Author: fom
Subject: Re: Ordinals describable by a finite string of symbols

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!)

Thank you for a very correct statement.

I looked up the remark in Cohen's book.

The paragraph in question begins as

"If one does not care about the construction
of actual models, ..."

I am less interested in relative consistency
and independence results than I am in the
model theory of set theory. That is probably
clear from my other reply. But, I had been
somewhat rushed.

The "eliminability" of which you speak is
precisely associated with relative consistency
and independence results. Your remark is
clear and exact. I just missed it yesterday.