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



fom
Posts:
1,968
Registered:
12/4/12


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 11, 2013 8:42 PM


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.



