Let be a set, and let
be an ultrafilter on
, let
be a formula of a given language
, and let
be any collection of structures
which is indexed by the set
.
Denote by
the equivalence class of
under
, for any element
of the product
. Then the ultraproduct
satisfies
via a valuation
in
,
Łoś' Theorem
See also
Nonstandard Analysis, Structure, Transfer PrincipleThis 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.Referenced on Wolfram|Alpha
Łoś' TheoremCite this as:
Insall, Matt. "Łoś' Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LosTheorem.html