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From: Transfer Principle on 6 Jul 2010 16:38 On Jul 3, 2:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > So what does Hughes mean for an object to "satisfy" a theory? I > > interpreted it to mean that there exists a model of NF which > > proves the existence of the object, but not one of ZFC, but > > maybe Hughes had something else in mind. > It would be nice if you would specify *which* Hughes you meant, since > there are two different Hugheses that are posting in similar threads. Apparently, neither. It was actually herbzet who made that comment above. I'm not sure how I got confused and thought that either Hughes made the comment that herbzet made. I apologize for the confusion.
From: Transfer Principle on 6 Jul 2010 16:43 On Jul 3, 1:49 am, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Fri, 2 Jul 2010 20:46:04 -0700 (PDT), Transfer Principle > > Menzel gives some criteria, namely that it should at least adhere to > > Extensionality, and that sets ought to contain elements (except 0) > > and be elements of other sets. > Minimally. But when one starts constructing a more definite picture, > notably, the cumulative conception of sets, a lot more would seem to be > required, e.g., foundation. Foundation? In that case, would ZFA still be a "set theory," as it refutes Foundation/Regularity? I knew that Extensionality was a biggie, since some of WM's old ideas (especially wrt "potentially infinite" objects) were said to be non-sets. Also, I've heard that those posters who wanted to use mereology (zuhair, galathaea, and tommy1729) also needed to come up with names other than "sets" for their objects (yet tommy1729 still insists on calling his theory "tommy1729's _set_ theory). Thanks for the info!
From: Chris Menzel on 7 Jul 2010 08:59 On Tue, 6 Jul 2010 13:43:56 -0700 (PDT), Transfer Principle <lwalke3(a)lausd.net> said: >> > Menzel gives some criteria, namely that it should at least adhere >> > to Extensionality, and that sets ought to contain elements (except >> > 0) and be elements of other sets. >> Minimally. But when one starts constructing a more definite picture, >> notably, the cumulative conception of sets, a lot more would seem to >> be required, e.g., foundation. > > Foundation? Well, yes, IF one insists upon basing one's notion of set on the cumulative conception. > 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.
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 |