Use
As an example of an application of ACω, here is a proof (from ZF+ACω) that every infinite set is Dedekind-infinite:
- Let X be infinite. For each natural number n, let An be the set of all 2n-element subsets of X. Since X is infinite, each An is nonempty. A first application of ACω yields a sequence (Bn : n=0,1,2,3,...) where each Bn is a subset of X with 2n elements.
- The sets Bn are not necessarily disjoint, but we can define
- C0 = B0
- Cn= the difference of Bn and the union of all Cj, j<n.
- Clearly each set Cn has at least 1 and at most 2n elements, and the sets Cn are pairwise disjoint. A second application of ACω yields a sequence (cn: n=0,1,2,...) with cn∈Cn.
- So all the cn are distinct, and X contains a countable set. The function that maps each cn to cn+1 (and leaves all other elements of X fixed) is a 1-1 map from X into X which is not onto, proving that X is Dedekind-infinite.
Read more about this topic: Axiom Of Countable Choice
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