Re: Equal Sets and Identical Sets

On Mar 14, 4:12 am, Alan Smaill <sma...@xxxxxxxxxxxxxxxx> wrote:

why would you expect that to be the case?
That's more than asking that the defined equality satisfies
the axioms for identity, after all.

Right. But something is still lingering unclear and admittedly hazy in
my statement of it:

The identity of indiscernibles is ordinarily thought not to be
statable even in a meta-language of a first order theory. I understand
that. But in the case where there are only finitely many primitives,
we don't have to "quantify" over all properties, but rather we can
mention just the finite number of primitive properties specifically.
For example, in set theory we have:

Az(zex <-> zey) -> x=y

also

Az(xez <-> yez) -> x=y

So indiscernibility just on either of those two bases alone is enough
to entail identity.

So, I was wondering whether that "strength" would carry over somehow
to showing that the theory (that includes the abovementioned theorems)
could only be satisfied by a model that maps '=' to the identity
relation on the universe.

But, okay, maybe I'm barking up the wrong tree. Then it would not be
the case that satisfying such a theory entails that '=' maps to the
identity relation on the universe.

Though, of course, where the semantics for '=' is fixed as for
ordinary identity theory, we do get '=' mapped to the identity
relation on the universe.

So the question I then have is this:

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"?

Thanks for your help. It's appreciated.

MoeBlee




.

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