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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.
KeywordsModal Structure Free Variable Atomic Formula Individual Constant Elementary Embedding
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