Good talk! Interestingly, Leibniz's law (presented @22:17) originally only goes in one direction - it's the law of "identity of indiscernibles", meaning the direction of implication is from having all the same properties to identity, but not the other way around. The idea of "indiscernibility of identicals" constitutes the other direction of implication - and in logic/analytical metaphysics, there are experts arguing that one may be true, but not the other. A small gripe: The biconditional on the slide "Proposition (Equality Preservation)" (@22:15) doesn't seem to make sense. As stated, it says that if we pick any two random (not necessarily distinct) elements of A and any two random (not necessarily distinct) elements of B, then whenever the elements of A are identical, the elements of B will also be identical and vice versa. First, there is no mention of a mapping and second - this is just obviously not true. Say A is the set {'a', 'b', 'c'} and B is {1, 2, 3} - I can pick ('a', 'a') from A and (1, 2) from B (or ('b', 'c') from A and {2, 2} from B) - and naturally neither direction of implication holds. It should read: Let f denote an isomorphism between A and B and f^-1 denote the inverse of f. Then a1 = a2 f(a1) = f(a2) and b1 = b2 f^-1(b1) = f^-1(b2). Set-theoretically, this can be easily proven, knowing that every function is a left-total and univalent (deterministic) relation and that an isomorphism must be both injective and surjective (meaning it has an inverse which is also both injective and surjective). The left-to right direction of both sides of the conjunction is guaranteed by the univalence criterion, while the right-to-left direction is guaranteed by f (and f^-1) being injective.
@jdelouche
11 ай бұрын
This is really cooooool
@mzg147
3 жыл бұрын
The discord link at 5:00 is dead, is there another way to join?
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