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Łoś' Theorem

Let I be a set, and let U be an ultrafilter on I, let phi be a formula of a given language L, and let {A_i:i in I} be any collection of structures which is indexed by the set I. Denote by [x]_U the equivalence class of x under U, for any element x of the product product_(i in I)A_i. Then the ultraproduct (product_(i in I)A_i)/U satisfies phi via a valuation s=[(x_i)_(i in I)]_U in (product_(i in I)A_i)/U,

{i in I:A_i|=_(x_i)phi} in U.

See also

Nonstandard Analysis, Structure, Transfer Principle

This entry contributed by Matt Insall (author's link)

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References

Bell, J. L. and Slomson, A. B. Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.Hurd, A. E. and Loeb, P. A. An Introduction to Nonstandard Real Analysis. Orlando, FL: Academic Press, 1985.

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Łoś' Theorem

Cite this as:

Insall, Matt. "Łoś' Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LosTheorem.html

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