In the paper we prove the following theorem:Theorem. There is a model N of open induction in which the set of primes is bounded and N is such that its field of fractions 〈N*, +, ·, <〉 is elementarily equivalent to 〈Q, +, ·, <〉 (the standard rationals).We fix an ω1-saturated model 〈M, +, ·, <〉 of PA. Let 〈M*, +, ·, <〉 denote the field of fractions of M. The model N that we are looking for will be a substructure of 〈M*, +, ·, <〉.If A ⊆ M* then let Ā denote the ring generated by A within M*, Ậ the real closure of A, and A* the field of fractions generated by A. We haveLet J ⊆ M. Then 〈M*, +, ·〉 is a linear space over J*. If x1,…,xk ∈ M*, we shall say that x1,…,xk are J-independent if 〈1, x1,…, xk〉 are J*-independent in the usual sense. As usual, we extend the notion of J-independence to the case of infinite sets.If A ⊆ M* and X ⊆ A, then we say that X is a J-basis of A if X is a maximal subset of A which is J-independent.Definition 1.1. By a J-form ρ we mean a function from (M*)k into M*, of the formwhere q0,…, qk ∈ J*If υ ∈ M, we say that ρ is a υ-form if the numerators and denominators of the qi's have absolute values ≤ υ.
Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma