SetTheoryTimeless

SetTheoryTimeless - Econ 204 Set Formation and the Axiom of...

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Econ 204 Set Formation and the Axiom of Choice In this supplement, we discuss the rules underlying set formation and the Axiom of Choice. We generally begin with a set of elements, such as the natural numbers N , the rational numbers Q ,therea lnumbers R , or an abstract set like the set X of all points of an unspeciFed metric space. Given any set X ,w ecanfo rm2 X , often called the power set of X ;2 X is the set of all subsets of X . Thus, we can form the set N of all natural numbers, 2 N , the set of all subsets of N ; , { 1 , 2 } , { 2 , 4 , 6 ,... } are elements of 2 N . We can also form 2 2 N =2 ( 2 N ) , the set of all subsets of the set of all subsets of the natural numbers. An element of 2 2 N is a set of subsets of the natural numbers; for example, {∅} , {∅ , N } , {{ 1 } , { 2 } , { 2 , 4 , 6 ,... }} and {{ 2 } , { 4 } , { 6 } ,... } are elements of 2 2 N . Let X be any set, and P ( x ) a mathematical statement about a variable x .Then { x X :

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SetTheoryTimeless - Econ 204 Set Formation and the Axiom of...

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