> Axiom of choice assumes that you can always pick the best possible option, however you need to be able to account for your worst possible options as well. All mathematical optimization falls outside of classical logic.
Not sure what you mean here. The axiom of choice just says that the Cartesian product of a set of nonempty sets is nonempty. Can you explain what you mean by "best possible option" and "worst possible options"?
Imma paraphrase Wikipedia. You are right but I find the informal definition to be more illuminating. “Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite”. Now imagine that there’s an opponent who decides the object selection of certain bins.
This is a silly example but compare playing chess single player vs multiplayer. In the single player chess, you still have two sides, you are still trying to make your side to win, however you decide all the moves (even for the other party). IRL, a lot of choices are outside of your control. So in some sense, you need to account for your opponent's best moves (which are generally the worst moves for you).
Two player games to me are all about quantifiers. If the opponent plays, it means we have to prove a “for all”. (This includes all worst case scenarios obviously.) And own plays are “exist” (just the best option at that time). No axiom of choice involved, just FO logic.
Your formulation doesn't work well if games are allowed to be infinite, because standard propositional logic doesn't handle formulae of infinite length.
There is something called the Axiom of Determinacy which states that all two-player games of perfect information have a winning strategy for one player. This axiom applies to games of any length, including all types of uncountable infinities. The Axioms of Choice and Determinancy are incompatible; if one of them is true, the other can be proven false.
I don't know if this is related to what GP was talking about, but it reminded me.
Neither propositional logic nor first-order logic (FOL) allow formulae of infinite length.
While there are logics with formulae of infinite length (see [1, 2] for an overview), I would not consider them logics on par with propositional or FOL. Why? Because as a finite human you cannot actually write down infinite formulae! Instead, you use some finite abbreviation mechanism (typically some variant of FOL, extended with set theoretic axioms like ZFC) so you can denote (in a finite way) those infinite formulae.
I'd suggest to see infinite formulae as a semantics gadget that is sometimes useful in model-theoretic investigations.
It's not surprising that we encounter questions independent from ZFC (or whatever your preferred foundation of mathematics) when we look at infinite games. It's but an instance of our inability to define infinite sets in an impredicative way (i.e. the usual inductive definition of the natural numbers as the least set closed under successor).
You can convert any terminating formula (including recursive) to a finite logic representation in second order logic with exponential number of terms. This by use of Peano numbering.
Proving whether the given (inductive or recursive) formula is actually terminating is the halting problem though.
Linear logic is closely related to martingales too.
I’m having difficulty following your comments about adversarial choice as related to the axiom of choice. What you’re saying doesn’t sound well-defined (in the mathematical sense).
It's actually very well defined (or which part are you confused by). I'm stating these things informally as a formal treatment is right now beyond me and it also doesn't really exist. Look up Chu spaces esp their relationship to linear logic. This paper is pretty informative
It's a path that unifies several very formally related. The unification itself isn't fully formalized, you need to look at the symmetries in the well formalized subparts.
With regards to (2), this is one way of solving the problem. However, your solution uses O(n) memory (where n is the size of the linked list), and the tortoise-and-hare algorithm uses only O(1) memory. Here's how it works: http://en.wikipedia.org/wiki/Cycle_detection#Tortoise_and_ha...
Complexity optimization is a generally useful skill.
"moving the goal posts" is called "every month at work, when you finish 1 task and then move on to another task, instead of going home and getting paid to rest on your laurels"
I don't think it has to be "wrong". If I were asking this question, and I was provided with "store them in a hash", I would say something along the lines of "that's a correct solution" and then add additional constraints.
"How you adapt in the face of additional constraints" is very much relevant to programming.
Or maybe he thought it wasn't going to last? It's one of his recurring themes(/fears?) that it's all going to end.
> “I’ve got maybe 10 years to ladle the butter into a jar for my kids and then die”
I think it started with him saying his career would be over in a year. Last year when I saw him he said "two, maybe three years tops" before people would forget about him, and now I guess he feels a little more confident that he'll be sticking around for a while.
