Math Forum Discussions

fom

Posts: 1,968
Registered: 12/4/12

Re: Just another exposition of MK.
Posted: Mar 17, 2013 8:16 AM

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On 3/17/2013 6:04 AM, Zuhair wrote:
> On Mar 17, 9:30 am, fom <fomJ...@nyms.net> wrote:
>> On 3/16/2013 1:08 PM, Zuhair wrote:
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>>> On Mar 16, 9:33 am, Zuhair <zaljo...@gmail.com> wrote:
>>>> Define: Set(x) iff {x,..}
>>
>>>> Extensionality: x C y & y C x -> x=y
>>
>>>> Comprehension: {x| Set(x) & phi}
>>
>>>> Pairing: x C {a,b} -> Set(x)
>>
>>>> Generation: Set(x) & y C H(x) -> Set(y)
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>>>> where H(x)={z| m in TC({z}). |m| =< |x|}
>>
>>>> Size: |x| < |V| -> Set(U(x))
>>
>>>> where TC, U stand for transitive closure, union respectively defined
>>>> in the customary manner; C is subclass relation; | | =< | | and | | <
>>>> | | relations are defined in the standard manner.
>>
>>>> The theory above minus axiom of Size is sufficient to prove
>>>> consistency of Z. With the axiom of Size it can prove the consistency
>>>> of ZF+Global choice, and it is equi-interpretable with MK+Global
>>>> choice.
>>
>>>> Zuhair
>>
>>> Another reformulation along the same lines is:
>>
>>> Define: Set(X) iff {X,..} exists.
>>
>>> Extensionality: X C Y & Y C X -> X=Y
>>> Class comprehension: {x|Set(x) phi} exists.
>>> Pairing: X C {a,b} -> Set(X)
>>> Subsets: Set(X) & Y C X -> Set(Y)
>>> Size limitation: |X|<|V| -> Set(H(TC(X)))
>>
>>> C is sublcass relation.
>>> TC(X) is the transitive closure of X.
>>> H(X) is the Class of all sets hereditarily subnumerous to X.
>>
>>> Possibly (I'm not sure) pairing is redundant.
>>
>> How does your pairing axiom assert the
>> existence of pairs?
>>
>> Are you assuming that the monadic
>> Set() predicate is satisfied for
>> every finite and transfinite
>> enumeration?
>
> I don't know what enumerations has got to do with pairing here. My
> axiom is saying that any Class of two or less elements is a set, and
> that's all what we need for pairing and building relations.

Typically, the axiom of pairing asserts the
existence of pairs.

AxAyEzAw(wez <-> (x=w \/ y=w))

Since your axiom is asserting exactly what
you have characterized it as asserting, it
seems the usual existence assertion must
be arising elsewhere. Since I was uncertain
about the exact meaning of the enumeration
predicate, I thought it might have to do
with that.