xameer
- compactness theorem: any theory w arbitrarily large al finite models, or 1 infinite model, has models of al cardinality (this is Upward Löwenheim–#Skolem theorem). So, for instance, there are nonstandard models of #Peanoarithmetic with uncountably many 'natural numbers
infinite structures can never be discriminated in FO: Löwenheim– #Skolem theorem, => no fo theory w infinite model can have a unique model up to #isomorphism.
The most famous example is probably Skolem's theorem, that there is a countable non-standard model of arithmetic.