This paper contains some observations on models for A2-second-order arithmetics. It arose from unsuccessful attempts to prove the Conjecture 1 (see at the end of the paper). Two partial solutions are given in this paper. We also give a sketch of a new proof of a classical result of descriptive set-theory referred to in [3]. Unless otherwise specified the word “model” means “ω-model for second-order arithmetics with the axiom scheme of choice,” i.e., “ω-model of A2” in the terminology of [3]. We say that a model M1 is shorter than M2 iff there is in M2 a relation A which is a well-ordering in M2 such that every relation B which is a well-ordering in M1 is similar to an initial segment of A. We denote by M0 the principal model of A2, i.e., a model which has only standard integers and contains all sets of integers.

An important program in the study of the structure the lattice of r.e. sets modulo finite sets, is the classification of the orbits under Aut(), the automorphism group of , If Φ ∈ Aut() and Φ(We) =* Wh(e) for all e ∈ ω, then h is called a presentation of Φ. Define Autχ() to be the class of those elements of Aut() that have a presentation in the class of functions X, where for instance X might be the class of Δn functions for n ∈ ω. If Φ ∈ Autx() then we will say that Φ is an X-automorphism. Note that we only need to consider Δn- and Πn-orbits and automorphisms since if f is a Σn-presentation, then f is a total function and therefore Δn.

Definition. If X is a class of functions and ⊆ {Wi: i < ω}, then is an X-orbit iff is an orbit under Aut() and for all A, B ∈ there is a Φ ∈ Autx() satisfying Φ(A) =* B. (Here Φ: → .)

Harrington proved that the creative sets form a Δ0-orbit and Soare proved that the maximal sets form Δ3-orbit. We will show that there is no Boolean algebra such that {A: A is r.e. and ℒ*(A) ≈ ] forms a Δ2-orbit, where ℒ*(A) is the principal filter of A in ; ℒ*(A) = {B: B ⊆* A & B ∈ }. The idea behind the proof of this theorem is very similar to the proof by Soare that maximal sets do not form a Δ2-orbit (see Soare [1974] or Soare [1987]).

In attempting to reconstruct Rose's proof of Lemma 3.2 of [1], the present authors found what is apparently a different and simpler method, which moreover leads to a far stronger conclusion.

We are operating in the Heyting prepositional calculus as formulated on p. 3 of [1] or on pp. 82 and 101 of [2], and shall make use of relevant theorems on pp. 90, 113–119 of [2]. We shall use a, b, c, w, x, y, z as propositional variables.

We say that a conjunction is simple if each factor has one of the forms: (i) a, (ii) ¬a, (iii) a⊃b, (iv) a⊃(b∨c), (v) (a&b)⊃c, (vi) (a⊃b)⊃c.

0# can be invisibly class generic.

The notions of sets of reals being κ-Souslin (κ a cardinal) and admitting a λ-scale (λ an ordinal) are due respectively to D. A. Martin and Y.N. Moschovakis. A set is ω-Souslin if and only if it is Σ1 1 (analytic). We show that a set is ω-Souslin if and only if it admits an (ω + l)-scale. Jointly with Martin and Solovay we show that if κ is uncountable and has cofinality ω, then being κ-Souslin is equivalent to admitting a κ-scale. Our results together with those of Kechris give a new simultaneous characterization of Σ1 1 and Δ1 1 (Borel) sets (a set is Σ1 1 if it admits an (ω + 1)-scale and Δ1 1 if it admits an ω-scale) and determine completely the relation between the κ-Souslin sets and the sets admitting λ-scales.

We examine two-cardinal problems for the class of O-minimal theories. We prove that an O-minimal theory which admits some (κ, λ) must admit every (κ′, λ′). We also prove that every “reasonable” variant of Chang's Conjecture is true for O-minimal structures. Finally, we generalize these results from the two-cardinal case to the δ-cardinal case for arbitrary ordinals δ.

§1. A counterexample and preservation of “proper + X”.

Theorem. Suppose V satisfies, , and for some A ⊆ ω1, every B ⊆ ω1, belongs to L[A].

Then we can define a countable support iterationsuch that the following conditions hold:

a) EachQiis proper and ⊩Pi “Qi, has power ℵ1”.

b) Each Qi is -complete for some simple ℵ1-completeness system.

c) Forcing with Pα = Lim adds reals.

Proof. We shall define Qi by induction on i so that conditions a) and b) are satisfied, and Ci, is a Qi-name of a closed unbounded subset of ω1. Let : ξ < ω1› ∈ L[A] be a list of all functions f which are from δ to δ for some δ < ω1 and let h: ω1 → ω1, h ∈ L[A], be defined by h(α) = Min{β: β > α and Lβ[A]⊨ “∣α∣ = ℵ0”}.

Suppose we have defined Qj for every j < i; then Pi is defined, is proper (as each Qj, j < i, is proper, and by III 3.2) and has a dense subset of power ℵ (by III 4.1). Let Gi ⊆ Pi be generic so clearly there is B ⊆ ω1, such that in V[Gi] every subset of ω1 belongs to L[A, B], The following now follows:

Fact. In V[Gi], every countableN ⥽(H(ℵ2), ∈, A, B) is isomorphic toLβ[A ∩ δ, B ∩ δ] for some β < h(δ), where δ = δ(N) = ω1, ∩ N.