chapter
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U

Any filter can be extended to an ultrafilter, and, using a weak form of the Axiom of Choice, any subset of a Boolean algebra with the finite intersection property can be extended to an ultrafilter.

ultrapower An ultrapower of an L-structure A is a reduced product

∏ U A, where U is

an ultrafilter over the index set I . The reduced product is formed by declaring, for x and y in the Cartesian product

∏ I A, that x ≡U y if and

only if the set of coordinates where x and y agree is in the ultrafilter U :

{i ∈ I : x(i) = y(i)} ∈ U . The reduced product

∏ U A is then the set of all

equivalence classes under ≡U . The fundamental property of ultrapowers is

that, for any L-sentence φ, ∏ U A |= φ if and

only if {i ∈ I : A |= φ} ∈ U . But because U is an ultrafilter, ∅ /∈ U and I ∈ U , and so, the ultrapower models φ if and only if the original structure A models φ. Thus,

∏ U A ≡ A.