Łoś, Theorem
المؤلف:
Bell, J. L. and Slomson, A. B
المصدر:
Models and Ultraproducts: an Introduction. Amsterdam, Netherlands: North-Holland, 1971.
الجزء والصفحة:
...
13-2-2022
959
Łoś' Theorem
Let 
be a set, and let 
be an ultrafilter on 
, let 
be a formula of a given language 
, and let ![<span style=]()
{A_i:i in I}" src="https://mathworld.wolfram.com/images/equations/LosTheorem/Inline6.svg" style="height:22px; width:79px" /> be any collection of structures which is indexed by the set 
. Denote by ![[x]_U](https://mathworld.wolfram.com/images/equations/LosTheorem/Inline8.svg)
the equivalence class of 
under 
, for any element 
of the product 
. Then the ultraproduct 
satisfies 
via a valuation ![s=[(x_i)_(i in I)]_U](https://mathworld.wolfram.com/images/equations/LosTheorem/Inline15.svg)
in 
,
![<span style=]() {i in I:A_i|=_(x_i)phi} in U. " src="https://mathworld.wolfram.com/images/equations/LosTheorem/NumberedEquation1.svg" style="height:22px; width:138px" /> |
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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