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fom
Posts:
1,969
Registered:
12/4/12


Re: CON(ZF) and the ontology of ZF
Posted:
Feb 19, 2013 3:53 AM


On 2/17/2013 12:53 PM, Zuhair 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.
I think you need to make a change here, Zuhair.
If you try to define that bounding set as the intersection of all sets satisfying that property, you will collapse to something similar to my axiom,
AxAy(xcy > xe(GC(TC({y}))))
would you not?
(whoops! You did not see this because I did not put it in the original posting. I formulated an unreviewed, unpublished paper in 1994. That had been my size limitation attempt.)
Lets call your bounding set a Zuhair successor abbreviated to ZS.
Then, in addition to what you already specify, you want something like
{x}e(GC(TC({y})))
so that ZS(x) is independent of x relative to the transitive closure operation. By independent, I mean always requiring something bigger than what x can generate through its singleton.
Think of the set universe dynamically. There is an involution streaming down from the choice function on the set of Zermelo names through the Zermelo names
... {{{{x}}}} > {{{x}}} > {{x}} > {x} >
It is at the singleton where the transitive closure operation starts branching if x is not simple.
I know that sounds funny, but, it is just the iterative application of the inverse transformation for the transformation one would describe by
{x} is the name of x {{x}} is the name of {x} {{{x}}} is the name of {{x}} {{{{x}}}} is the name of {{{x}}}
where I am using braces instead of the quotes a logician might prefer.
Your bounding set cannot be captured by TC({x}).



