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Topic: CON(ZF) and the ontology of ZF
Replies: 5   Last Post: Feb 19, 2013 3:53 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: CON(ZF) and the ontology of ZF
Posted: Feb 19, 2013 3:53 AM
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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 Morse-Kelley set theory.
>>>> Morse-Kelley 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 re-iterate 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)={y|As. x subset_of s & s is transitive -> y e s}

>>
>>> This proves MK-choice. However it might be stronger than MK-choice?
>>> 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}).

















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