details re Z-R |- PA consistent
in [Math]
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From: herbzet on 29 Jul 2010 22:07 Aatu Koskensilta wrote: > It follows from the existence of a large cardinal that ZFC is consistent > but it is consistent with ZFC that ZFC is inconsistent. Officer, arrest that man!
From: Ross A. Finlayson on 2 Aug 2010 15:02 On Jul 29, 9:46 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> writes: > > > But wouldn't a large cardinal be a model of ZFC? * > > No, no ordinal is a model of ZFC. V_kappa for a large cardinal kappa is > a model of ZFC. > > > With large cardinals, it would only be consistent that there are no > > models of ZFC with ZFC being inconsistent thus that both are true. > > It follows from the existence of a large cardinal that ZFC is consistent > but it is consistent with ZFC that ZFC is inconsistent. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Aatu, I agree that you consistently maintain that it is not for ordinals that the ordinal containing the transitive closure of the theory's regular well-founded elements, that being the model, simulacrum in all detail, here though you are saying it's now the cardinal, where that would be over all the language classes with the regular models. Now before it even had to be all the interpretations, identical under various operations (i.e., each of the two set- theoretic ones) then is simply a matter of model detail. You know I think that I keep saying that I think that the ordinals there are sufficiently the models where the theories generally have a simple maintainence from the natural counting models, eg then simple associative models, with natural ubiquitous ordinals. Thus I so frame theory. Here I so frame theory: no, an axiomless system of natural deduction. I repeat that. Warm regards, Ross Finlayson
From: Ross A. Finlayson on 2 Aug 2010 15:07 On Jul 29, 7:07 pm, herbzet <herb...(a)gmail.com> wrote: > Aatu Koskensilta wrote: > > It follows from the existence of a large cardinal that ZFC is consistent > > but it is consistent with ZFC that ZFC is inconsistent. > > Officer, arrest that man! Ah Herb, hello of course generally, yes via description via, formally, an axiomless system of natural deduction, there are different and less the paradoxes with the null axiom theory, than for example ZF set theory, with the paradoxes about the irregular or non-well-founded consequences of collections of the objects (eg that that of sets is a set and that that of ordinals is a set and an ordinal), yes still the other axioms of ZF seem to follow as theories from the consequences of the operations, of these elements each defined uniquely by the other elements to which they relate in part-hood or set-hood (neighborhood or parenthood). Yet where the topology and graph-theoretic links are so defined there, it could be either way, in topology and links. Now as ordinals, basically they're ascribed particularly concise structure, that works with other strictly formal set-theoretic machinery in the establishment of for example induction, and as well with generally some theorems in the middle everything that follows from mutual induction. Now it is a result of Lowenheim and Skolem that if there is a model for the theory to witness it consistent, then there is a countable model. So there is basically a machine that works off of the countable ordinals, and for that matter, all the recursive ordinals up to the Church Kleene Omega 1 can be modeled by the finite natural numbers, no infinite numbers. (This Omega 1 CK is larger than any combination of countable ordinals, least non- recursive, still countable, then I digress about translations back and forth preserving model semantics between the recursive and non- recursive, also I digress about the set of recursive ordinals). So there's a countable model for any theory like ZF that's consistent (in model theory which generally uses ZF), in this case ZF having a standard model is a separate theory. Anyways as a countable theory it's no different than having a mapping from the natural integers onto each other with representing each part of the machine in the natural integers, if ZF had a standard model which it is not proven that it does or rather nor is it that it doesn't, there are no cardinals different than ordinals. No I agree that what Aatu said there is consistent and with much the same meaning as what I said there, in his generally careful (besides correct) manner. Heh, Herb you were flying. Still, I hope you would tell us more about that it is pretty funny. Yeah I agree with Herbert too. Warm regards, Ross Finlayson
From: MoeBlee on 2 Aug 2010 15:35 On Aug 2, 2:07 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > Ah Herb, hello of course generally, yes via description via, formally, > an axiomless system of natural deduction, there are different and less > the paradoxes with the null axiom theory, than for example ZF set > theory, with the paradoxes about the irregular or non-well-founded > consequences of collections of the objects (eg that that of sets is a > set and that that of ordinals is a set and an ordinal), yes still the > other axioms of ZF seem to follow as theories from the consequences of > the operations, of these elements each defined uniquely by the other > elements to which they relate in part-hood or set-hood (neighborhood > or parenthood). Yet where the topology and graph-theoretic links are > so defined there, it could be either way, in topology and links. > Now as ordinals, basically they're ascribed particularly concise > structure, that works with other strictly formal set-theoretic > machinery in the establishment of for example induction, and as well > with generally some theorems in the middle everything that follows > from mutual induction. Now it is a result of Lowenheim and Skolem > that if there is a model for the theory to witness it consistent, then > there is a countable model. So there is basically a machine that > works off of the countable ordinals, and for that matter, all the > recursive ordinals up to the Church Kleene Omega 1 can be modeled by > the finite natural numbers, no infinite numbers. (This Omega 1 CK is > larger than any combination of countable ordinals, least non- > recursive, still countable, then I digress about translations back and > forth preserving model semantics between the recursive and non- > recursive, also I digress about the set of recursive ordinals). So > there's a countable model for any theory like ZF that's consistent (in > model theory which generally uses ZF), in this case ZF having a > standard model is a separate theory. Anyways as a countable theory > it's no different than having a mapping from the natural integers onto > each other with representing each part of the machine in the natural > integers, if ZF had a standard model which it is not proven that it > does or rather nor is it that it doesn't, there are no cardinals > different than ordinals. Sure, of course. > Heh, Herb you were flying. Speaking of herb, and of flying high... MoeBlee
From: Ross A. Finlayson on 2 Aug 2010 21:05
On Aug 2, 12:35 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Aug 2, 2:07 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> > wrote: > > > > > Ah Herb, hello of course generally, yes via description via, formally, > > an axiomless system of natural deduction, there are different and less > > the paradoxes with the null axiom theory, than for example ZF set > > theory, with the paradoxes about the irregular or non-well-founded > > consequences of collections of the objects (eg that that of sets is a > > set and that that of ordinals is a set and an ordinal), yes still the > > other axioms of ZF seem to follow as theories from the consequences of > > the operations, of these elements each defined uniquely by the other > > elements to which they relate in part-hood or set-hood (neighborhood > > or parenthood). Yet where the topology and graph-theoretic links are > > so defined there, it could be either way, in topology and links. > > Now as ordinals, basically they're ascribed particularly concise > > structure, that works with other strictly formal set-theoretic > > machinery in the establishment of for example induction, and as well > > with generally some theorems in the middle everything that follows > > from mutual induction. Now it is a result of Lowenheim and Skolem > > that if there is a model for the theory to witness it consistent, then > > there is a countable model. So there is basically a machine that > > works off of the countable ordinals, and for that matter, all the > > recursive ordinals up to the Church Kleene Omega 1 can be modeled by > > the finite natural numbers, no infinite numbers. (This Omega 1 CK is > > larger than any combination of countable ordinals, least non- > > recursive, still countable, then I digress about translations back and > > forth preserving model semantics between the recursive and non- > > recursive, also I digress about the set of recursive ordinals). So > > there's a countable model for any theory like ZF that's consistent (in > > model theory which generally uses ZF), in this case ZF having a > > standard model is a separate theory. Anyways as a countable theory > > it's no different than having a mapping from the natural integers onto > > each other with representing each part of the machine in the natural > > integers, if ZF had a standard model which it is not proven that it > > does or rather nor is it that it doesn't, there are no cardinals > > different than ordinals. > > Sure, of course. > So, you agree that cardinals are ordinals, or indistinct from their initial ordinals, and that they have one. > > Heh, Herb you were flying. > > Speaking of herb, and of flying high... > > MoeBlee What about it? Moe, this is about ZF and Peano Arithmetic, or as the subject reads: Z - R -> PA consistent. I talk about that in what I just wrote, glad you could contribute. Oh, wait, well thanks anyway. Still, that is my impression of Herb, what is that about, Herb? Excuse me, Herb, I by no means meant to wonder except to in knowing the veracity of the connoted words see no reason for undue alarum. Herb, I just didn't get it and it made some sense though, I thought you wrote that because I wrote the statement about the ordinals and Aatu made the same statement as a theorem or fact, with the figuring out the transfer principle over reflexive componentry. So I read that as Herb's exclamatory reaction to what we were talking about: mathematical logic and formal mathematical logic. Only one way up! Warm regards, Ross Finlayson |