Moyo wrote:Lets do this slowly
say we have a set A = {a;b;c}
In what way is that element "a" the element "a".
Why ! but by the axiom of identity.or in other words a=a . AKA (a,a).
Oh I see your point. We need sets to define relations, and we need the identity relation in order to talk about equality of sets.
I think the answer is the law of identity,
https://en.wikipedia.org/wiki/Law_of_identity, which is a principle of logic. In other words we have the law of identity even before there are sets. That's my understanding of how this works; but in the end I think you raise a pretty good point. We need the law of identity before we write down the axioms for sets. Logic is logically prior to set theory.
In that sense I suppose we'd have to call equality a special kind of relation, one that precedes set theory itself and is not defined by a set-theoretical relation. I am not aware of how philosophers and logicians think about this.
(ps) I understand what's going on now. From the Wiki article
https://en.wikipedia.org/wiki/Axiom_of_extensionality:
The axiom given above assumes that equality is a primitive symbol in predicate logic.
So the '=' symbol is not something defined within set theory. Rather, equality exists even before we write down the axioms of set theory.
Now when we write down the rules for what sets are, we start with Extensionality; which says that if we have two sets A and B with the property that for all x, x ∈ A if and only if x ∈ B, then we say that A = B.
In other words equality is given by logic; then we say what it means in set theory for two sets to be equal.
This avoids your infinite regress.
It's interesting that when we say for example that in the real numbers, 5 = 5, is that the logical equality? Or is it the equality relationship that we could define
within set theory by considering the collection of all ordered pairs (x,x) where x is a real number?
I think it's not clear to me at all; but in practice, it doesn't seem to matter. In any event, we avoided the circularity by pushing '=' down into logic and making it available for use in set theory.
You raised a very good point but it seems the logicians and set theorists have already thought of it and prevented the problem.
(pps) Found this discussion ...
http://math.stackexchange.com/questions ... gical-symb The consensus is that one can distinguish between equality as a logical symbol; and equality as a predicate when restricted to a specific set or domain.
