The ambient model of arithmetic

Let Lall be the language containing symbols for every relation and function on the natural numbers N; each symbol from Lall has a canonical interpretation in N. Let M be an ℵ1-saturated model of the true arithmetic in the language Lall. 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 ak, k ∈ N, is a countable family of elements of M then there exists a non-standard t ∈ M and a sequence (bi)i<t ∈ M such that bk = ak for all k ∈ N.

We shall often denote this sequence of length t simply (ai)i<t.

For example, if all elements {ak}k∈N obey some definable property P then – by induction in M (aka overspill, see the Appendix) – also some bs with a non-standard index s < t will obey P. Such an element bs will serve well as ‘a limit’ (interpreted here informally) of the sequence {ak}k∈N.

Another property implied by the ℵ1-saturation (and equal to it if we used a countable language) is the following:

(2) If Ak, k ∈ N, is a countable family of definable subsets of M such that ∩i<kAi ≠ ∅ for all k ≥ 1, then ∩kAk ≠ ∅.