Re: Set theory and identity theory

On Mar 13, 4:56 pm, G. Frege <nomail@invalid> wrote:
On Thu, 13 Mar 2008 16:04:12 -0700 (PDT), MoeBlee <jazzm...@xxxxxxxxxxx>
wrote:

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).

Sure. But since 'e' is the only primitive, all other properties are
"generated" by it. So, I was thinking (not in a rigorous, but rather
in heuristic or speculative way) 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.

But perhaps not.

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 idenity 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? 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'.

MoeBlee

.

Relevant Pages

  • Re: Set theory and identity theory
    ... identity relation on the universe. ... treated as from identity theory and with the ordinary fixed semantics ... so that the model must map '=' to the identity relation on the universe ...
    (sci.logic)
  • Re: Equal Sets and Identical Sets
    ... Then in our interpretation we only have to interpret ... some relation other than the identity relation on the universe. ... took '=' from identity theory or whether we took it as defined in our ... equivalence classes come to mind... ...
    (sci.logic)