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 <fomJUNK@nyms.net> 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

follows:

"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.