Re: Set theory and identity theory
- From: G. Frege <nomail@invalid>
- Date: Fri, 14 Mar 2008 15:39:26 +0100
On Fri, 14 Mar 2008 07:13:25 -0700 (PDT), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:
Yes, of course. In our theory. (But that does not take over to ourSure. But since 'e' is the only primitive, all other properties are
I don't think so. Note that here "x = y" just "means" that Az(zex <->
zey).
"generated" by it.
model. There sets, i.e. the objects in our model, might have additional
properties, say, colors.)
Of course. Same with me (here).
So, I was thinking (not in a rigorous, but rather
in heuristic or speculative way)
Maybe my grasp of the English language isn't that good... :-/
that perhaps the fact that the sole
primitive 'e' now "controls every other property down the line"
provides us a with a kind of "end run" around the problem that we
can't state the identity of indiscernibles in first order (not even in
the meta-theory), so that that would lead to some way to show that our
axioms of set theory are only satisfied by '=' getting mapped to the
identity relation on the universe.
Didn't you like my considerations?
But perhaps not.
.... 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).
Right! Hence I actually prefer to develop ZFC in the framework of /FOPL
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?
with identity/. (And you will see that MOST authors also prefer that
approach).
Not in my case. I actually always (unconsciously) assume that we are
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'.
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.
IMHO (even not taking into account the considerations from above)
there's a REASON why considering "identity" a logical primitive (in FOPL
with identity).
Of course, in a system of 2OL we actually might _define_ identity (for
objects/things):
a = b :<-> AF(F(a) <-> F(b))
(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.)
F.
--
E-mail: info<at>simple-line<dot>de
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