I want to know the applications of axiom of choice in various problems of mathematics?

Question

I want to know the applications of axiom of choice in various problems of mathematics?

Answer 1

I want to know why you posted a declaration instead of a question.

Answer 2

By allowing the construction of a ("chosen") set, the axiom of choice simplifies many proofs. In many cases, a proof could have been obtained without the axiom, but with greater difficulty. In other cases, the axiom is needed for proof.



Wikipedia has a nice article about a plausible game theory fact called "Deteminacy":

http://en.wikipedia.org/wiki/Axiom_of_determinacy



The article shows how to *disprove* the determinacy "fact" using the axiom of choice.

Answer 3

you have no longer certainly study very many math texts, have you ever? I even have, and that i won't be in a position of remember any that "hold forth distinctive self obtrusive truths." "arithmetic is in keeping with a beginning of axioms that are assumed without evidence." Erm, no longer precisely. You look to think of that there exists a collection of axioms undergirding all of arithmetic. No such project exists. in certainty, diverse mathematical structures anticipate diverse instruments of axioms, and one set of axioms often contradicts yet another. (Your extreme-college algebra would not artwork in noncommutative or nonassociative contexts, case in point.) you're additionally incorrect in asserting that mathematicians regard the axioms as "actual." they do no longer anticipate that the axioms are "actual" in any purpose experience. They declare in simple terms that IF the axioms carry THEN their conclusions persist with. or maybe the regulations for drawing conclusions are no longer inevitably fixed: distinctive and conflicting fashions of excellent judgment exist. you besides might say that axioms are no longer problem to evidence -- it fairly is technically actual, yet axioms _can_ be invalidated. Any set of axioms might desire to be consistent. it fairly is obtainable to show that a collection of axioms is inconsistent, wherein case that set of axioms is an beside the point beginning for a mathematical gadget. nevertheless, even nevertheless you have mischaracterized mathematical foundations, i'm curious: what axioms could you enumerate as foundational to "Biblical certainty?"