Upward Löwenheim–Skolem theorem
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The Upward Löwenheim–Skolem theorem states that for any first-order language L and L-theory T with an infinite model or arbitrary large finite models, T has models of arbitrarily large cardinalities.
[edit] Examples
Let L be the empty language (i.e. language of equality). Then there is no theory that axiomatizes all finite sets because any such theory would have an infinite model by the Upward Löwenheim–Skolem theorem.
[edit] Proof
Let L be a first-order language, T an L-theory with models of arbitrarily large finite cardinalities or an infinite model, and κ an infinite cardinal. Then let C be an infinite set of constants not in L of size κ and let 
. Then 
is a collection of L(C)-sentences. Now it is clear that every finite subset of is satisfiable since T has models of arbitrarily large finite cardinalities or one of infinite cardinality. Consequently, it follows from the Compactness theorem that is consistent. Therefore, by Gödel's completeness theorem, has a model. It's L-reduct is then a model of T whose cardinality is greater than or equal to | C | = κ.
[edit] References
- Rothmaler, Philipp (2000), "Introduction To Model Theory", CRC
