Re: Equal Sets and Identical Sets

G. Frege <nomail@invalid> writes:

On Fri, 14 Mar 2008 06:58:31 -0700 (PDT), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:


Suppose we take '=' as defined in our set theory, not primitive from
identity theory. Then in our interpretation we only have to interpret
'e', since '=' is a defined symbol. But then '=' might get mapped to
some relation other than the identity relation on the universe. But
usually, in casual discussion, we're not so specific about whether we
took '=' from identity theory or whether we took it as defined in our
set theory. So in such a context, we don't know whether by "a model
for set theory" we mean one in which '=' must map to the identity
relation on the universe or one in which '=' very well might not map
to the identity relation on the universe. That ambiguity now is
puzzling me. How do I know which of the two options the author intends
when he says something like "consider a model for ZF"?

Just a wild (and uneducated) guess. Maybe that difference is not
"essential". Maybe those models are "isomorphic" (in a sense).
Or wait... equivalence classes come to mind... In the model for ZFC with
defined "=" we might consider equivalence classes of all those objects
in the universe for which Az(z e a <-> z e b) holds. Then there should
be an isomorphism between the set of these equivalence classes and, etc.

that's right -- for a given model (of any theory) in which "=" is not
interpreted as the identity, but satisfies the identity axioms, there
is an elementarily equivalent model where "=" *is* interpreted as the
identity formed by taking equivalence classes.

It won't in general be isomorphic to the original model, though,
because many objects map to the same equivalence class.


F.

--

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

--
Alan Smaill
.

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