CANTOR DISPROOF <<<<<<<<<<<<<<<<
in [Logic]
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From: herbzet on 3 Jul 2010 03:44 Transfer Principle wrote: > Chris Menzel wrote: > > MoeBlee said: > > > > ZF does not prove there exists an infinite yet Dedekind finite set, > > > right? Rather, it is undecidable in ZF whether there exists such a > > > thing. Same with sets without a choice function, right? > > > Of course, so all that can be said is that ZF simply cannot prove that > > there are no such sets. It is misleading to say that there *are* such > > sets that "satisfy" ZF (whatever exactly "satisfy" is supposed to mean). > > I mean that there's a model of ZF with such sets. Of course, > such a model would be a model of ZF+~AC, not a model of ZFC. > > Note that Hughes was the first to use the word "satisfy" in > this manner: No, it was me. [herbzet] > Even in a not-too-distant from standard set theory like NF, it > seems fairly evident to me that objects which could conceivably > satisfy NF are somewhat different from objects that would satisfy, > say, ZFC. > > So what does [herbzet] 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, That is, a model in which the object exists. That said, I agree with your understanding of my use of "satisfy". > but not one of ZFC, I don't think there are objects that at one and the same time can satisfy (make true) the axioms both of NF and ZF(C). I could be wrong about that. > but maybe [herbzet] had something else in mind. > > Thus, a D-finite T-infinite set "satisfies" the axioms of ZF > in that it satisfies Extensionality (i.e., it's determined by > its elements), satisfies Powerset and Union (since it has a > powerset and union), and so on. But maybe there's a better > word than "satisfies" to indicate that an object adheres to > each of a list of axioms. It seems like a good word to me. > All I wanted to know is whether objects whose existence is > refuted by ZFC but can exist in other theories should still > be called "sets." Good question, matter of convention I guess. > 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. -- hz
From: Chris Menzel on 3 Jul 2010 04:49 On Fri, 2 Jul 2010 20:46:04 -0700 (PDT), Transfer Principle <lwalke3(a)lausd.net> said: > On Jul 2, 3:58 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: >> On Fri, 2 Jul 2010 14:40:34 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com> >> said: >> > ZF does not prove there exists an infinite yet Dedekind finite set, >> > right? Rather, it is undecidable in ZF whether there exists such a >> > thing. Same with sets without a choice function, right? >> Of course, so all that can be said is that ZF simply cannot prove >> that there are no such sets. It is misleading to say that there >> *are* such sets that "satisfy" ZF (whatever exactly "satisfy" is >> supposed to mean). > > I mean that there's a model of ZF with such sets. Of course, such a > model would be a model of ZF+~AC, not a model of ZFC. > > Note that Hughes was the first to use the word "satisfy" in > this manner: > > Hughes: > Even in a not-too-distant from standard set theory like NF, it seems > fairly evident to me that objects which could conceivably satisfy NF > are somewhat different from objects that would satisfy, say, ZFC. Not sure if the etymology is relevant here, but ok. > 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. So maybe something like this: An object s satisfies a theory T in a model M of T iff there is a formula A(x) such that M |= A[x/s] (that is, M satisfies A(x) when s is assigned to 'x'). So suppose M is a model of ZF. Then we can say that s in M is "not an object of ZFC" if, for some formula A(x), ZFC |- ~ExA(x) but M |= A[x/s]. > Thus, a D-finite T-infinite set "satisfies" the axioms of ZF in that > it satisfies Extensionality (i.e., it's determined by its elements), > satisfies Powerset and Union (since it has a powerset and union), and > so on. Ok, but you've dropped reference to any model here. The fact that, M |= "s is infinite but D-finite" for some model of ZF doesn't mean that there are, in fact, any such sets (assuming a realist view of sets). By the Löwenheim-Skolem theorem, we can construct models of ZF+"there is an infinite, D-finite set" out of, say, the finite von Neumann ordinals -- none of which, of course, is D-finite. > All I wanted to know is whether objects whose existence is refuted by > ZFC but can exist in other theories should still be called "sets." I think it's clearer and less misleading to talk about theories instead of objects, for the reasons just noted. A better way to put the question, I think, is to ask at what point we would say that a given theory is no longer a theory of *sets*. > 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.
From: Nam Nguyen on 3 Jul 2010 10:04 Chris Menzel wrote: > A better way to put the > question, I think, is to ask at what point we would say that a given > theory is no longer a theory of *sets*. Would you share with us what that threshold point might be? Thanks.
From: Jesse F. Hughes on 3 Jul 2010 17:40 Transfer Principle <lwalke3(a)lausd.net> writes: > Note that Hughes was the first to use the word "satisfy" in > this manner: > > Hughes: > Even in a not-too-distant from standard set theory like NF, it > seems fairly evident to me that objects which could conceivably > satisfy NF are somewhat different from objects that would satisfy, > say, ZFC. > > 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. -- Jesse F. Hughes "He was still there, shiny and blue green and full of sin." -- Philip Marlowe stalks a bluebottle fly in Raymond Chandler's /The Little Sister/
From: MoeBlee on 3 Jul 2010 17:58
On Jul 2, 3:58 pm, Chris Menzel <cmen...(a)remove-this.tamu.edu> wrote: > On Fri, 2 Jul 2010 14:40:34 -0700 (PDT), MoeBlee <jazzm...(a)hotmail.com> > said: > > > On Jul 2, 12:12 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > >> at what point beyond which a theory differs from ZFC such that we > >> should no longer call the objects which satisfy them sets? > > > I don't know. But we can adopt certain definitions, such as: > > > x is a set <-> (x=0 or Eyz y in x in z)) > > > That will work as long as the theory defines '0' appropriately. > > And surely extensionality is essential to our conception of set. As far as I can tell, it is, Chris. MoeBlee |