Re: Set theory and identity theory

On Fri, 14 Mar 2008 08:08:23 -0700 (PDT), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:


Sure. But since 'e' is the only primitive, all other properties are
"generated" by it.

Yes, of course. In our theory. (But that does not take over to our
model. There sets, i.e. the objects in our model, might have additional
properties, say, colors.)

But if our model theory is itself given within a meta-theory that is a
set theory (which I take it to be), then ain't no properties 'ceptin
those made up from 'e'.

Well, almost looks like a philosophical question now...


Didn't you like my considerations?

Yes, and they are along the lines I've pretty much always understood.
I just want to be more clear that my reasons aren't enough to overide
the considerations you mentioned.

... consider a universe (for our model) which contains "decorated"
sets (or "colored" sets if you like. Just assume that we have objects
which not only contain elements, but in addition have a color). Then we
might have two _different_ decorated sets a, b which just have the same
elements, but different colors. In this case we would have

        Az(zea <-> zeb).

And hence

        a = b

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

Of course I assume that at meta-level (model theory) we DO have an
notion of identity - "the real one". But of course, even this seems not
to be "necessary" from a mathematical point of view.


Yes, I do understand your reasoning. Except, as I mentioned, if the
the model theory itself is in set theory, then epsilon is the only
primitive property even at the meta-level.

Not necessarily. Why do you assert that? We might very well use (some
sort of) "FOPL with identity" at meta-level; and I'd guess most authors
actually just do that (implicitly).

And concerning our set theory at object level:

[...] I actually prefer to develop ZFC in the framework of /FOPL
with identity/. (And you will see that MOST authors also prefer that
approach).

It seems most authors do (but it's not entirely clear in many cases),
but some authors specifically do not.

Older texts, I guess?


This ambiguity is real since it's usually the case that we
talk about set theory without being so specific about where our '='
came from, whether from identity theory and its fixed semantics or
from definition from the sole primitive 'e'.

Not in my case. I actually always (unconsciously) assume that we are
working with ZFC in the framework of /FOPL with identity/. Actually the
cases (textbooks) where this is not the case are rather rare, I'd guess.

They're not usual, but not so very rare I don't think. Anyway, my
point is that we may need to check our tacit assumptions in this
regard, especially since it is not very common for a set theory author
to explicitly mention that the semantics for '=' is the fixed one.

Well, if we are talking about textbooks concerning /axiomatic set
theory/ the authors usually are very specific concerning the framework
of logic they are using.


IMHO (even not taking into account the considerations from above)
there's a REASON why considering "identity" a logical primitive (in FOPL
with identity).

I agree that there is a good reason for taking set theory as an
extension of identity theory. But still it's a matter of preference.

Well... bad taste... :-)


(And of course if we define "=" in set theory, we TRY something
analogous; but -as you know- set theory is a first-order theory, and
hence not powerful enough to characterize identity, etc. ->Skolem.)

Do you know where I can get a FORMAL working out of the result about
not characterizing identity? I don't dispute the result; I do
understand why the identity of indiscernibles is not GENERALLY
expressible for first order; but I wonder whether the consideration of
a finite number of primitives provides some kind of special case.

Ufff... I always wanted to get a copy of Quine's "Set Theory and Its
Logic". Maybe that might be a good staring point?


F.

--

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

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