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Re: CON(ZF) and the ontology of ZF
Posted:
Feb 18, 2013 12:11 AM


On Feb 17, 1:53 pm, Zuhair <zaljo...@gmail.com> wrote: > On Feb 17, 9:40 pm, fom <fomJ...@nyms.net> wrote: > > > > > > > > > > > On 2/17/2013 12:12 PM, Zuhair wrote: > > > > On Feb 17, 11:42 am, fom <fomJ...@nyms.net> wrote: > > > >> So, returning to the statements in the opening > > >> paragraph, it does not surprise me that Zuhair > > >> may have succeeded in devising a means by which > > >> to show Con(ZF) relative to MorseKelley set theory. > > >> MorseKelley set theory as presented in Kelley > > >> presumes a global axiom of choice. > > > > The theory that I've presented can actually work without the axiom > > > of global choice! > > > I believe this. You represented the forcing methodology > > directly. And, I am now fairly convinced that that > > methodology is implicit to the axiom of induction for > > arithmetic. > > > Think carefully about how I ended that post. I pointed > > to a link explaining the relationship of AC to GCH > > > There is a reason I did that. I do not ascribe > > to the usual model theory for set theory. It is not > > logically secure. Very few people like my posts, > > but this is one attempt at explaining myself on > > "truth" for set theory. > > > news://news.giganews.com:119/5bidnemPpsnq13zNnZ2dnUVZ_sOdn...@giganews.com > > > > this is done by replacing axiom of Universal limitation by > > > axiom of direct size limitation. > > > > To reiterate my theory. It is too simple actually. > > > > Language: FOL(=,e) > > > > Definition: Set(x) <> Ey(x e y) > > > > Axioms: > > > > 1.Extensionality: (Az. z e x <> z e y) > x=y > > > 2.Class comprehension: {x Set(x) phi} exists. > > > 3.Pairing: (Ay. y e x > y=a or y=b) > Set(x) > > > 4.Hereditary limitation: Set(x) <> Ey. Set(y) & AzeTC(x).z=<y > > > 5.Size limitation: Set(x) & y=<x > Set(y) > > > / > > > > where x =< y <> Ef. f:x>y & f is injective > > > and TC(x)={yAs. x subset_of s & s is transitive > y e s} > > > > This proves MKchoice. However it might be stronger than MKchoice? > > > MK+global choice proves all the above axioms. > > > In general, I am unfamiliar with MorseKelley. I have > > read through the appendix of "General Topology", and > > that does have only global choice. So, I am not certain > > of your distinctions here and cannot even begin to > > address the question. > > There are many versions of MK, those differ by altering size > limitation, for > example this can be altered exactly as in this theory, so instead of > Universal > kind of size limitation (which is the usual in MK) we can use the one > I wrote > as the last of the axioms here, this will deprive MK from choice as it > did here. > Anyhow > > Zuhair
Zuhair,
Whenever someone says they can solve what seems to be a very difficult problem, but never really describes exactly how it can be done because of the complexity of it, I like to ask them: Ok, can you give a detailed solution to a small representative subset of the problem to illustrate your complete solution?
In this case, I would be interested in a detailed proof that any two axioms are consistent, and a list of the other axioms you think are also consistent with them. Of course the form of the two axioms would have to be such that, on the surface they can be mutually contradictory. For example, one could be of the form ?If xxx then yyy is a set.? and the other of the form ?There is no set such that zzz.?
Can you show how you can prove even a simple subset consistent in detail?
CB



