Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


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


Re: Ordinals describable by a finite string of symbols
Posted:
Jul 10, 2013 9:07 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!) >
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 Booleanvalued 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.



