Re: Set theory and identity theory

On Mar 14, 7:39 am, G. Frege <nomail@invalid> wrote:
On Fri, 14 Mar 2008 07:13:25 -0700 (PDT), MoeBlee <jazzm...@xxxxxxxxxxx>
wrote:

I don't think so. Note that here "x = y" just "means" that Az(zex <->
zey).

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'.

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 contains 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).

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.

So what puzzles me now is when someone says something like "consider a
model of ZF", how do I know whether the person intends that '=' is
treated as from identity theory and with the ordinary fixed semantics
so that the model must map '=' to the identity relation on the universe
or whether the person is taking '=' as defined, thus without the fixed
semantics, so that '=' might not map to the identity relation on the
universe?

Right! Hence 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 specifcally do not. So I'm wondering whether the
authors that do not expect that it is implicit that they too are using
a fixed semantics for '=' even though their '=' doesn't come from
identity theory?

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 explictly mention that the semantics for '=' is the fixed one.

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.

(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.

MoeBlee


.

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