The definition of the models
Jan Krajı́ček · Cambridge University Press eBooks · 2010
The ambient model of arithmetic Let L all be the language containing symbols for every relation and function on the natural numbers N ; each symbol from L all has a canonical interpretation in N . Let M be an ℵ 1 -saturated model of the true arithmetic in the language L all . Such a model exists by general model-theoretic constructions; see Hodges [43]. Definable sets mean definable with parameters, unless specified otherwise. The ℵ 1 -saturation implies the following: (1) If a k , k ∈ N , is a countable family of elements of M then there exists a non-standard t ∈ M and a sequence ( b i ) i < t ∈ M such that b k = a k for all k ∈ N . We shall often denote this sequence of length t simply ( a i ) i < t . For example, if all elements { a k } k ∈ N obey some definable property P then – by induction in M (aka overspill, see the Appendix) – also some b s with a non-standard index s < t will obey P . Such an element b s will serve well as ‘a limit’ (interpreted here informally) of the sequence { a k } k ∈ N . Another property implied by the ℵ 1 -saturation (and equal to it if we used a countable language) is the following: (2) If A k , k ∈ N , is a countable family of definable subsets of M such that ∩ i < k A i ≠ ∅ for all k ≥ 1, then ∩ k A k ≠ ∅.