At the end I'm glad that you brought up the relationship between collection and choice; I thought they looked similar based off all the times I've used it in my analysis classes, and I was happy to see it was sensible that I had that reaction!
0:00 Separation: "Every subset of a set A is also a set." That is not a good description, since it lacks the definition of a subset. A better definition: "Given any First Order predicate φ(x), the elements of A for which φ holds, also make a set." Notice, that this is a form of General Comprehension, restricted to set A. 4:30 Replacement: "Given a set A and a function f, the image of f restricted to A is a set." "So what is a function?" Zermelo said it is any rule that assigns exactly one set to another set. Skolem-Fraenkel said it is imprecise, and defined a function as follows: "f is a First Order formula of 2 variables (x,y) that is true iff y is the value corresponding to x." Only now has Replacement become a well-defined axiom. 5:45 Although actually, both Separation and Replacement are not axioms, they are infinite collections of axioms, one for each formula. It is not a problem, since we can computably generate every of these axioms (they are countable, because there is only a countable number of formulas). Moreover, any finite number of axioms in ZFC can prove themselves Consistent, therefore proving themselves Inconsistent, so having these infinitary axioms is necessary. 7:00 But Borcherds also says there are extensions of ZF which allow Classes (like Gödel's or Bernay's) and these do have a finite number of axioms. "The theorems about Set Theory they prove turn out to be the same as those ZF proves." 7:45 Uses of Separation and Replacement. "Separation is obvious, you often form subsets of sets." Replacement is needed when forming the Von Neumann hierarchy. To create V2ω you need Replacement+Union, and Borcherds says you need that duo to reach every Limit Ordinal. BUT WHAT ABOUT Vω?? It is a Limit Ordinal! Even at 3:00 of his Infinity axiom video, Borcherds said you need apply Replacement as soon as Vω! (At this point I gave up on trying to understand what a godforsaken "model" actually is. To hell with "models", they must be just 'stronger versions' of set theory!) 10:45 "Is Separation necessary?" "It almost follows from Replacement. - Given a predicate P, just take the function φ(x) which returns "x" when Px holds, and returns "[one designated element of that set]" whenever Px does not hold. This way we can find every nonempty subset of a set." That was more or less how Borcherds described it, and it seems weird, I would do it such that φ returns a pre-chosen value that we know will be removed from our resulting subset. If we could have φ be a partial function, and not give a result upon false inputs, then it would be easy. But if it must give a function, we can have it return the powerset of our parent set, because its powerset will be obviously different from any of its subsets. After that, just remove from our "output image" every set of cardinality > than our parent set, and boom, I just achieved Separation. 12:45 Oh, Borcherds says that we cannot prove the existence of ∅ without Separation, which would, in fact, destroy my above proposition. "You could just replace it (Separation) with an axiom that ∅ exists."
Well spotted! He overused phi in this as a predicate name and function name. From the example given the predicate phi(a) could be True or False = True. Phi (b) is False and phi(c) is True, but phi(a) wasn't determined.
At the end I'm glad that you brought up the relationship between collection and choice; I thought they looked similar based off all the times I've used it in my analysis classes, and I was happy to see it was sensible that I had that reaction!
很喜欢关于集合论的讲解，思路清晰，内容全面，感谢！
0:00 Separation: "Every subset of a set A is also a set." That is not a good description, since it lacks the definition of a subset.
A better definition: "Given any First Order predicate φ(x), the elements of A for which φ holds, also make a set."
Notice, that this is a form of General Comprehension, restricted to set A.
4:30 Replacement: "Given a set A and a function f, the image of f restricted to A is a set."
"So what is a function?" Zermelo said it is any rule that assigns exactly one set to another set.
Skolem-Fraenkel said it is imprecise, and defined a function as follows:
"f is a First Order formula of 2 variables (x,y) that is true iff y is the value corresponding to x."
Only now has Replacement become a well-defined axiom.
5:45 Although actually, both Separation and Replacement are not axioms, they are infinite collections of axioms, one for each formula. It is not a problem, since we can computably generate every of these axioms (they are countable, because there is only a countable number of formulas).
Moreover, any finite number of axioms in ZFC can prove themselves Consistent, therefore proving themselves Inconsistent, so having these infinitary axioms is necessary.
7:00 But Borcherds also says there are extensions of ZF which allow Classes (like Gödel's or Bernay's) and these do have a finite number of axioms. "The theorems about Set Theory they prove turn out to be the same as those ZF proves."
7:45 Uses of Separation and Replacement.
"Separation is obvious, you often form subsets of sets."
Replacement is needed when forming the Von Neumann hierarchy. To create V2ω you need Replacement+Union, and Borcherds says you need that duo to reach every Limit Ordinal.
BUT WHAT ABOUT Vω?? It is a Limit Ordinal! Even at 3:00 of his Infinity axiom video, Borcherds said you need apply Replacement as soon as Vω!
(At this point I gave up on trying to understand what a godforsaken "model" actually is. To hell with "models", they must be just 'stronger versions' of set theory!)
10:45 "Is Separation necessary?"
"It almost follows from Replacement. - Given a predicate P, just take the function φ(x) which returns "x" when Px holds, and returns "[one designated element of that set]" whenever Px does not hold. This way we can find every nonempty subset of a set."
That was more or less how Borcherds described it, and it seems weird, I would do it such that φ returns a pre-chosen value that we know will be removed from our resulting subset. If we could have φ be a partial function, and not give a result upon false inputs, then it would be easy.
But if it must give a function, we can have it return the powerset of our parent set, because its powerset will be obviously different from any of its subsets. After that, just remove from our "output image" every set of cardinality > than our parent set, and boom, I just achieved Separation.
12:45 Oh, Borcherds says that we cannot prove the existence of ∅ without Separation, which would, in fact, destroy my above proposition.
"You could just replace it (Separation) with an axiom that ∅ exists."
11:53 if phi(x) is true => ph(x) = x and phi(x) false => phi(x) = a, then does this mean phi(a) is both true and false?
Well spotted! He overused phi in this as a predicate name and function name. From the example given the predicate phi(a) could be True or False = True. Phi (b) is False and phi(c) is True, but phi(a) wasn't determined.
yeeeee
we missed algebraic topology class thanks a lot