Seems like a key issue here is whether it’s allowed that two distinct sets compare as equal (C++ programming analogy: two instances of the std::complex type, at different memory locations, which compare as equal because they represent the same complex number)

In normal set theory (and hence OP’s Formulation 1) I believe this is impossible by definition, c.f. Suppes, Axiomatic Set Theory, section 1.2:

‘=‘ is taken as the sign of identity. The formula ‘x=y’ may be read as ‘x is the same as y’, ‘x is identical with y’…it is understood that equality means sameness of identity.

Or more concisely here:

If two terms are equal (i.e. they refer to the same object)

If the OP’s Formulation 2 actually is intended to allow two distinct sets to compare as equal, then it would imply the possibility that:

• $a \in X$ and $b = a$ but $b \notin X$

which would be a very weird new kind of set theory.

But if that’s not the intent, then the two formulations seem equivalent to me (though Formulation 1 seems a lot clearer)

Seems like a key issue here is whether it’s allowed that two distinct sets compare as equal (C++ programming analogy: two instances of the std::complex type, at different memory locations, which compare as equal because they represent the same complex number)

In normal set theory (and hence OP’s Formulation 1) I believe this is impossible by definition, c.f. Suppes, Axiomatic Set Theory, section 1.2:

‘=‘ is taken as the sign of identity. The formula ‘x=y’ may be read as ‘x is the same as y’, ‘x is identical with y’…it is understood that equality means sameness of identity.

Or more concisely here:

If two terms are equal (i.e. they refer to the same object)

If the OP’s Formulation 2 actually is intended to allow two distinct sets to compare as equal, then it would imply the possibility that:

• $a \in X$ and $b = a$, but $b \notin X$

which would be a very weird new kind of set theory.

But if that’s not the intent, then the two formulations seem equivalent to me (though Formulation 1 seems a lot clearer)

Seems like a key issue here is whether it’s allowed that two distinct sets compare as equal (C++ programming analogy: two instances, at different memory locations, of the std::complex type which compare as equal because they represent the same complex number)

In normal set theory (and hence OP’s Formulation 1) I believe this is impossible by definition, c.f. Suppes, Axiomatic Set Theory, section 1.2:

‘=‘ is taken as the sign of identity. The formula ‘x=y’ may be read as ‘x is the same as y’, ‘x is identical with y’…it is understood that equality means sameness of identity.

Or more concisely here:

If two terms are equal (i.e. they refer to the same object)

If the OP’s Formulation 2 actually is intended to allow two distinct sets to compare as equal then it becomes possible that:

• $a \in X$ and $b = a$ but $b \notin X$

which would be a very weird new kind of set theory.

But if that’s not the intent, then the two formulations seem equivalent to me (though Formulation 1 seems a lot clearer)

Seems like a key issue here is whether it’s allowed that two distinct sets compare as equal (C++ programming analogy: two instances of the std::complex type, at different memory locations, which compare as equal because they represent the same complex number)

In normal set theory (and hence OP’s Formulation 1) I believe this is impossible by definition, c.f. Suppes, Axiomatic Set Theory, section 1.2:

‘=‘ is taken as the sign of identity. The formula ‘x=y’ may be read as ‘x is the same as y’, ‘x is identical with y’…it is understood that equality means sameness of identity.

Or more concisely here:

If two terms are equal (i.e. they refer to the same object)

If the OP’s Formulation 2 actually is intended to allow two distinct sets to compare as equal, then it would imply the possibility that:

• $a \in X$ and $b = a$ but $b \notin X$

which would be a very weird new kind of set theory.

But if that’s not the intent, then the two formulations seem equivalent to me (though Formulation 1 seems a lot clearer)