1 - The definition of the models

Summary

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 ℵ₁-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 ℵ₁-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 ℵ₁-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<kA_i ≠ ∅ for all k ≥ 1, then ∩_kA_k ≠ ∅.