Re: ZFC means?

MoeBlee wrote:
> The identity axioms don't preclude being satisfied by a model
> in which '=' is mapped to an equivalence relation and not the actual
> identity relation in the universe.

I should have made explicit that, of course, that only addresses the
approach (vastly less common than the usual FOL= semantics) in which
the semantics don't stipulate that '=' be mapped to the actual
identity relation.

That raises a question in my mind. If we do allow the semantics to map
'=' to an equivalence relation other than actual identity, then what
ramifications, if any, does that have for models of set theory? This
would be the situation in which '=' is a primitive and set theory is a
non-conservative extension of identity theory (without the built-in
semantical rules for '='). That's a question beyond my level of study,
especially of models and inner models of ZF and other theories, so
hopefully it's not ill-premised.

MoeBlee

.

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