Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Just another exposition of MK.
Replies: 10   Last Post: Mar 17, 2013 1:50 PM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Just another exposition of MK.
Posted: Mar 17, 2013 2:30 AM

On 3/16/2013 1:08 PM, Zuhair wrote:
> 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)
>>
>> 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?

Date Subject Author
3/16/13 Zaljohar@gmail.com
3/16/13 fom
3/16/13 Zaljohar@gmail.com
3/16/13 Zaljohar@gmail.com
3/17/13 ross.finlayson@gmail.com
3/16/13 Zaljohar@gmail.com
3/17/13 fom
3/17/13 Zaljohar@gmail.com
3/17/13 fom
3/17/13 Frederick Williams
3/17/13 Zaljohar@gmail.com