Model Theory for Modal Logic pp 32-42 | Cite as

# Löwenheim-Skolem Theorems

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## Abstract

A *cardinal constellation* is a map *c* defined on a set *X* such that for each *x ∈ X, c(x)* is a non-zero cardinal number. The *constellation of a modal structure* 𝕬=<*A* _{k}, **K, R, O, N**>, *c* ^{𝕬}, is that constellation *c* with domain **K** such that for *k* ∈ **K**, *c*(*k*) = card (|*𝒜* _{ k }|). Let *c* and *c′* be constellations with domains *X* and *X′*, respectively. We write *c⊆c′* iff *X ⊆ X′* and *c′| X = c′*, and we write *c≤c′* iff *X = X′* and for all *x ∈ X*, *c*(*x*) ≤ *c′*(*x*). We define card (**ML**) to be the card inality of the set of all formulas of **ML**. A cardinal constellation *c* with domain including **K** is said to be *appropriate* for 𝔄 if for all *k, k′* ∈ **K**, *k* **R** *k′* implies *c(k) ≤ c(k′)* and if 𝔄 is a structure for **ML**, then card(**ML**) ≤ *c(k)* for all *k* ∈ **K**.

## Keywords

Modal Structure Free Variable Atomic Formula Individual Constant Elementary Embedding## Preview

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