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From: MoeBlee on 7 Jul 2010 13:03 On Jul 7, 7:59 am, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Tue, 6 Jul 2010 13:43:56 -0700 (PDT), Transfer Principle > <lwal...(a)lausd.net> said: > > In that case, would ZFA still be a "set theory," as it refutes > > Foundation/Regularity? > > It would not be for those who would consider the cumulative conception > to be the only legitimate conception of set. But that would be a silly, > doctrinaire way to think. The well-founded and non-well-founded > universes simply reflect two related but different conceptions of set. > Both lead to rich and interesting theories. To try to determine whether > one or another conception corresponds more closely to some ordinary > intuitive notion of set might be a mildly interesting semantic or > anthropological exercise, but it doesn't seem to me to be a > philosophically or mathematically significant one. My own (amateur) view is somewhat along those same lines. A consistent (formal) theory has its (formal) models, which are "abstract situations". So, proving theorems in these theories is a discovery of what is or is not the case in certain abstract situations. In one model something may be true that is not true in another model, since the models are different abstract situations. In the "situation" of <S O>, where S is the set of natural numbers and O is the standard ordering on naturals, O is not not dense. But in the "situation" <S O> where S is the set of real numbers and O is the standard ordering on reals, O is dense. One doesn't have to say one or the other is "the situation of "true reality"" or whatever. Merely, that they are different abstract situations. One doesn't need to say ZFC describes the "situation of true reality" but ZFA does not. Rather they describe different abstract situations from one another. MoeBlee |