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Re: How to check whether an infinite set is closed under addition?
You have 3 choices: 1) You check all possible cases, which are countably infinite. Since you don't an infinite amount of time, you have to either use the statistical approach involving a finite but sufficient sample size or the standard proof technique involving induction. 2) You use the axioms of universal algebra, but you are stuck if people ask you why and how your objects satisfy those axioms. Well, you can simply say, "Well, my objects just meet the requirements. Have faith, my friend!" 3) You define all your positive integers as sets and all your additions as set unions, and prove closure trivially in ZFC. Else, you can define all positive integers as binary sequences and addition as a CPU instruction to show closure. However, set theorists may ask you to prove that your definition of positive integers is equivalent to their definition of natural numbers. Set theorists don't add, so you don't have to worry about that one. Whenever you deal with infinities, you must count on faith as a result of understanding. Faith without understanding is blind, but without faith (in axioms, implicit or explicit) you cannot prove anything.