ultrapower
Quick Summary
An ultrapower is a construction in model theory used to create new mathematical structures from existing ones. It involves an ultrafilter, which is a special collection of subsets of a set I. This ultrafilter defines a notion of "almost all" subsets.
When applied to a family of identical structures, an ultrafilter helps form an ultraproduct, which is then called an ultrapower. A key property is that a sentence is true in the ultrapower if and only if it is true in "almost all" of the original structures. This technique is foundational for nonstandard analysis, providing a rigorous basis for concepts like infinitesimals.12
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metalogicElementary logic…(see below Ultrafilters, ultraproducts, and ultrapowers)—in particular, the ultrapower when the structures are all copies of the same structure (just as the product of a1, . . ., an is the same as the power an, if ai = a for each i). The intuitive idea in this method is…
Read More…(see below Ultrafilters, ultraproducts, and ultrapowers)—in particular, the ultrapower when the structures are all copies of the same structure (just as the product of a1, . . ., an is the same as the power an, if ai = a for each i). The intuitive idea in this method is…
Read More- metalogicUltrafilters, ultraproducts, and ultrapowers
An ultrafilter on a nonempty set I is defined as a set D of subsets of I such that
Read MoreAn ultrafilter on a nonempty set I is defined as a set D of subsets of I such that
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