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Re: CON(ZF) and the ontology of ZF
Posted:
Feb 17, 2013 1:53 PM


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



