Re: Set theory and identity theory

On Thu, 13 Mar 2008 16:04:12 -0700 (PDT), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:


Or we can dispense with identity theory and the axiom of
extensionality and have instead the pure logical axioms and/or rules
and the rest of the axioms of Z plus the following axiom:

Axiom: Az(zex <-> yex) -> Az(xez -> yez).

So this axiomatization has only one primitive - 'e'. We'll call this
axiomatization Z2.

Then we conservatively extend the theory axiomatized by Z2 by adding
this definitional axiom (which is actually the ordinary axiom of
extensionality):

Def: x=y <-> Az(zex <-> zey)

Now this axiomatization has two non-logical symbols 'e' and '='. We'll
call this axiomatization Z3.

Yes. I discussed that approach with george some time ago.


[...]

But what about the interpretation of '=' when we've used
axiomatization Z3? In this case, since we are not "coming from"
identity theory, I would think we wouldn't give '=' the special
dispensation of a special semantics. In this case, '=' is a defined
symbol, and it gets interpreted however it may be interpreted. But
then is it the case that for any model of the consequences of Z3, it
turns out ANYWAY that '=' gets interpreted as identity on the universe
of the model?

I don't think so. Note that here "x = y" just "means" that Az(zex <->
zey). Now consider a universe (for our model) which contains "decorated"
sets (or "colored" sets if you like. Just assume that we have objects
with "elements" which in addition have a color). Then we might have two
_different_ decorated sets a, b which just have the same elements. In
this case we would have

Az(zea <-> zeb).

And hence

a = b

would be satisfied. BUT a and b would NOT be identical (in the usual
sense of the word).


(I was unable to prove it when I sat down to the problem
last night.)

I guess the REASON why you COULD NOT succeed is clear now? No?


Additionally, with such theories as the consequences of Z1 and the
consequences of Z3, we get the indiscernibility of identicals (which
is expressed by the axiom schema of identity theory, which is a
theorem schema of Z3). But, as is famous, it is not possible in first
order to express the identity of indiscernibles (not even in a meta-
language for a fist order theory).

This is demonstrated for Z3 (at least) above. While we might claim that
the "=" of FOPL= actually refers to /identity/; after all we treat it
this way in our semantics of FOPL=. (No?)


So, to recap, my questions are:

(1) Is it the case that for any model of the consequences of Z3, it
turns out ANYWAY that '=' gets interpreted as identity on the universe
of the model?

Imho the answer has to be /no/. (See argument from above.)


What about when the model is standard (i.e., interprets
'e' as the membership relation on the universe)?

This does not exclude "colored"/"decorated" sets, does it? Of course we
are not dealing with "usual" sets any more.


F.

--

E-mail: info<at>simple-line<dot>de
.

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