202601190158

Tags : Model Theory

Łoś’s Theorem

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Theorem

Suppose $I$ is a non-empty set, ${\mathcal{M}}_i$ are non-empty $L$ structures. Let $\mathcal{M}$ be their direct product and let $U$ be an ultrafilter on $I$. Then for every formula $\phi \in L_n$ and every $n$-tuple $\bar{a}$ from $M$.

$$M/U\models \phi (\bar{a}/U)\text{ iff }\Vert \phi (\bar{a})\Vert \in U.$$

Consider formulas $\phi ,{\phi }_1,{\phi }_2$ and $\theta$, then:

• $\mathcal{M}/U\models \neg \phi (\bar{a}/U)$ iff $\mathcal{M}/U/\models \phi (\bar{a}/U)$ iff $\Vert \phi (\bar{a})\Vert \notin U$ iff $\Vert \neg \phi (\bar{a})\Vert \in U$.
• For conjunction and existential operator, the property follows from Reduced Product.

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References

• Reduced Product
• Filters
• Finiteness Theorem