Date: May 18, 2013 7:39 PM Author: fom Subject: Re: A logically motivated theory On 5/18/2013 2:43 PM, Zuhair wrote:

> On May 18, 8:58 pm, fom <fomJ...@nyms.net> wrote:

>> On 5/18/2013 10:40 AM, Zuhair wrote:

>>

>>

>>> The whole motivation beyond this theory is to extend any first order

>>> predicate by objects.

>>

>> Could you please clarify this remark?

>

> I'll expand on that. The motivation of this theory is to 'define'

> objects that uniquely corresponds to first order logic predicates,i.e

> for each first order predicates P,Q there exist objects P* and Q* such

> that P*=Q* iff for all x. P(x)<->Q(x)

>

> We want to do that for EVERY first order predicate P.

>

> The plan is to stipulate the existence of multiple PRIMITIVE binary

> extensional relations, each one of those would play the role of a set

> membership relation after which object representative of predicates

> are defined.

>

> Now the first axiom ensures that no 'distinct' objects can be defined

> after equivalent predicates all across the membership relations, so

> although we have many membership relation but yet any objects X,Y that

> are co-extensional over relations E,D (i.e. for all z. z E X iff z D

> Y) are identical!

> The second axiom (schema of course) ensures that no object represent

> non-equivalent Predicates, and so although we do have 'multiple'

> membership relations (primitive extensional relations) however from

> axiom schemas 1 and 2 this would ensure that each object defined after

> any of those relations would stand 'uniquely' for a single predicate.

>

> The last axiom scheme is just a statement ensuring the existence of an

> object that extends each first order predicate after some membership

> relation.

>

> Those objects uniquely corresponding for first order predicates are to

> be called as: Sets.

>

> The point is that paradoxes are eliminated because of having

> 'multiple' extensional relations each standing as a membership

> relation.

>

> I'm not sure if that would interpret PA, but if it does, then PA can

> be said to be a PURELY logical theory!

>

> Now if (and this is a big if) we allow infinitely long formulas to

> define first order predicates (infinitary first order languages) then

> second order arithmetic 'might' follow. And I think if this is the

> case, then second order arithmetic is also PURELY logical!

>

> This mean that the bulk of traditional mathematics (most of which can

> be formulated within proper subsets of second order arithmetic) is

> purely logical!

>

> However I don't think that higher mathematics can have a pure logical

> motivation comparable to the above, the motivation behind those can be

> said to be 'structural', or 'constructive' in the ideal sense that

> I've presented in my latest philosophical notes on my website and to

> this Usenet.

>

> Zuhair

>

Once again, I think you should take a look at NF by Quine.

Note that NF supports Fregean number classes.

You might find the following link informative:

http://plato.stanford.edu/entries/frege-logic/#6.4

The entire summary at that page is very well written.