Yes, this is a reupload. The previous version had a mistake in the conditions for the axiom schema of refinement, which has now (hopefully) been corrected. There are also a few cosmetic changes, such as renaming a few variables.
Honestly, I'm of the opinion that the axiom of choice is obvious, and the fact that it's not a theorem is an indication that the rules of 1st order logic are too weak; it shouldn't be a rule specific to sets, but should instead be added to our rules of logic. You're entitled to your opinion, though. But it certainly is useful, and that's the main reason why we take it as an axiom.
It would definitely make a difference. We want to claim that if every element of a is also an element of b and vice versa, then a equals b. That is, the condition is that for every c, if c is an element of a, then c is also an element of b, and vice versa. But c doesn't have to be an element of either. If we had *and* instead of *iff* here, the statement would be "if every set c is an element of both a and b, then a equals b". That is, if a and b are both sets of all sets, then they are equal. But this isn't saying much, since we mean to rule out sets of all sets anyway.
Awesome explanation!
Yes, this is a reupload. The previous version had a mistake in the conditions for the axiom schema of refinement, which has now (hopefully) been corrected. There are also a few cosmetic changes, such as renaming a few variables.
A wonderful presentation. The axiom of choice is, IMHO, preposterous; but, OTOH, useful, so may as well accept it, right?
Honestly, I'm of the opinion that the axiom of choice is obvious, and the fact that it's not a theorem is an indication that the rules of 1st order logic are too weak; it shouldn't be a rule specific to sets, but should instead be added to our rules of logic. You're entitled to your opinion, though. But it certainly is useful, and that's the main reason why we take it as an axiom.
2:26 Maybe a silly question but why isn't it "c ∈ a AND c ∈ b" instead of " c ∈ a iff c ∈ b"? will it make a difference?
It would definitely make a difference. We want to claim that if every element of a is also an element of b and vice versa, then a equals b. That is, the condition is that for every c, if c is an element of a, then c is also an element of b, and vice versa. But c doesn't have to be an element of either. If we had *and* instead of *iff* here, the statement would be "if every set c is an element of both a and b, then a equals b". That is, if a and b are both sets of all sets, then they are equal. But this isn't saying much, since we mean to rule out sets of all sets anyway.