Upward Löwenheim-Skolem Theorem
In particular: throughout.
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Theorem
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This ambiguity relates to the following: I suspect model of a first-order theory $\LL$, which is more specific than what is linked to now
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Let $T$ be an $\LL$-theory with an infinite model.
Then for each infinite cardinal $\kappa \ge \card \LL$, there exists a model of $T$ with cardinality $\kappa$.
Proof
The idea is:
- to extend the language by adding $\kappa$ many new constants
and:
- to extend the theory by adding sentences asserting that these constants are distinct.
It is shown that this new theory is finitely satisfiable using an infinite model of $T$.
Compactness then implies that the new theory has a model.
Some care needs to be taken to ensure that we construct a model of exactly size $\kappa$.
Let $\LL^*$ be the language formed by adding new constants $\set {c_\alpha: \alpha < \kappa}$ to $\LL$.
Let $T^*$ be the $\LL^*$-theory formed by adding the sentences $\set {c_\alpha \ne c_\beta: \alpha, \beta < \kappa, \ \alpha \ne \beta}$ to $T$.
We show that $T^*$ is finitely satisfiable:
Let $\Delta$ be a finite subset of $T^*$.
Then $\Delta$ contains:
- finitely many sentences from $T$
along with:
- finitely many sentences of the form $c_\alpha \ne c_\beta$ for the new constant symbols.
Since $T$ has an infinite model, it must have a model $\MM$ of cardinality at most $\card \LL + \aleph_0$.
This model already satisfies everything in $T$.
So, since we can find arbitrarily many distinct elements in it, it can also be used as a model of $\Delta$ by interpreting the finitely many new constant symbols in $\Delta$ as distinct elements of $\MM$.
Since $T^*$ is finitely satisfiable, it follows by the Compactness Theorem that $T^*$ itself is satisfiable.
Since $T^*$ ensures the existence of $\kappa$ many distinct elements, this means it has models of size at least $\kappa$.
It can be proved separately or observed from the ultraproduct proof of the compactness theorem that $T^*$ then has a model $\MM^*$ of exactly size $\kappa$.
That proof needs to be proved, and / or a link needs to be provided to that ultraproduct proof and its implications explained.
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Since $T^*$ contains $T$, $\MM^*$ is a model of $T$ of size $\kappa$.
$\blacksquare$
Source of Name
This entry was named for Leopold Löwenheim and Thoralf Albert Skolem.
Also see
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.7$ First-Order Logic Semantics: Theorem $\mathrm{II.7.17}$