Zhen Lin
Reputation
392/400 score
 May7 comment Can “proving mathematical theorems” as a 1 player puzzle be studied in combinatorial game theory? See Wikipedia about game semantics. May7 comment Can “proving mathematical theorems” as a 1 player puzzle be studied in combinatorial game theory? Actually, it's a two-player game between the verifier and the refuter. May6 asked Is a divisor in the hyperplane class necessarily a hyperplane divisor? May6 comment Existence of not locally small categories If those authors don't admit the existence of locally small categories, why would they talk about such things? To talk about such things would require them to admit there are categories that are not locally small! But one sometimes speaks of ‘metacategories’. May6 comment formalized provability predicate and implication relation If your formal proof system is reasonable, then yes. (Here is an example of an unreasonable proof system: no axioms, no rules of inference. Nothing is provable.) May6 awarded Caucus May6 comment Existence of not locally small categories The point is that there are people who take ‘category’ to mean locally small by definition. So for them a category that is not locally small does not exist – by definition! May6 comment Existence of not locally small categories Well, the collection of all sets is certainly well-defined, yet some people say such a thing does not exist. The question is of "legitimacy" more than whether we can define it at all. May6 comment Irreducible closed subsets of projective varieties Well, it is certainly the case that every closed irreducible subset of a projective variety is a projective variety. But if your definition of "projective variety" does not include irreducibility then even every closed subset of a projective variety is a projective variety. Therefore if $Y$ has a irreducible closed subset that is not a projective variety, then $Y$ cannot be projective. May6 comment Pole of differential That's not correct. Both $x$ and $y$ evaluate to $\infty$ at $[0 : 1 : 0]$. (Otherwise $x$ and $y$ would be functions that are regular everywhere on a projective variety, hence, constant – an absurdity.) May6 comment Irreducible closed subsets of projective varieties Some people include irreducibility in their definition of "variety". May6 comment Pole of differential $x / y$ is a rational function with a simple pole at infinity. Do you understand that calculation? May6 comment Examples of epimorphisms which are not split epimorphisms? You mean coequaliser. May6 comment What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom? Yes, I am aware of Solovay's result. That does not count as a statement in ordinary mathematics to me. May6 awarded Nice Question May5 answered Determine whether two primitive recursive functions are equal May5 comment Confusion about cofinality Yes. But if $[0, \kappa)$ is the set of all ordinals of cardinality less than $\kappa$, then the answer is the same, provided the well-ordering principle is available. May4 comment What's stronger: projective or locally free? flat or locally free? The question does not apply to general abelian categories, since there is no notion of "locally free" and no (obvious) notion of "flat". May4 comment The Axiom of Choice and definability The inductive step is far too vague. Here is one problem: in the set-forming axioms, one can use parameters, so the "constant" terms must have variables. But once you allow that, then one can derive a form of meta-AC: suppose we have a proof of $\forall x . x \in X \to \exists y . y \in x$; then we can construct a definable term $c (x)$ such that $\forall x . x \in X \to c (x) \in x$; then by replacement, $\exists f . (f : X \to \bigcup X) \land \forall x . x \in X \to f (x) \in x$. May4 comment The Axiom of Choice and definability My gut feeling is that your "Theorem" is false, because the same kind of reasoning can be used to "deduce" the axiom of choice. (See, for instance, the derivation of AC in Martin-Löf type theory.)