In [3] Friedberg showed that every Turing degree ≥ 0′ is the jump of some degree. Using the relativized version of this theorem it can be shown by finite induction that if a ≥ 0(n)0(n) then there is a b such that b(n) = a (our notation is defined in §1). It is natural to ask whether these results can be extended into the transfinite. Is it true for example, that whenever a ≥ 0(ω) there is a b such that = b(ω)= a? In §2 we use forcing to prove this result. (The usefulness of forcing in answering questions involving the jump operation on degrees of unsolvability has been previously demonstrated in Selman [5] where, for example, forcing was used to construct degrees a and b such that a(ω) = b(ω) = a ∪ b = 0(ω).) In §3 we generalize the methods of §2 to show that if α is a recursive ordinal and a ≥ 0(α) then there is a bsuch that b(α) = a, i.e. the Friedberg result can be extended to all recursive ordinal levels.

Thomason [6] used a forcing argument to show: If (the Kleene set of notations for the recursive ordinals) then there is a B such that (the set of notations for ordinals recursive in B). In §4 we show this result holds when hyperarithmetic reducibility is replaced by Turing reducibility: If then there is a B such that .

The sequent calculus formulation (L-formulation) of the theory of functionality without the rules allowing for conversion of subjects of [3, §14E6] fails because the (cut) elimination theorem (ET) fails. This can be most easily seen by the fact that it is easy to prove in the system

and

but not (as is obvious if α is an atomic type [an F-simple])

The error in the “proof” of ET in [14, §3E6], [3, §14E6], and [7, §9C] occurs in Stage 3, where it is implicitly assumed that if [x]X ≡ [x] Y then X ≡ Y. In the above counterexample, we have [x]x ≡ ∣ ≡ [x](∣x) but x ≢ ∣x. Since the formulation of [2, §9F] is not really satisfactory (for reasons stated in [3, §14E]), a new seguent calculus formulation is needed for the case in which the rules for subject conversions are not present. The main part of this paper is devoted to presenting such a formulation and proving it equivalent to the natural deduction formulation (T-formulation). The paper will conclude in §6 with some remarks on the result that every ob (term) with a type (functional character) has a normal form.

The conventions and definitions of [3], especially of §12D and Chapter 14, will be used throughout the paper.

The purpose of this paper is to introduce the notion of “omitting models” and to derive a very natural theorem concerning it (Theorem 1). A corollary of this theorem is the remarkable theorem of Vaught [3] which states that a countable complete theory cannot have precisely two nonisomorphic countable models. In fact, we show that our theorem implies Rosenstein's theorem [2] which, in turn, implies Vaught's theorem.

T stands for a countable complete theory whose (countable) language is denoted by L. Following [1], a countably homogeneous model of T is a countable model of T with the property that, for any two n-tuples a1, …, an and b1,…,bn of the universe of whose types are the same, there is an automorphism of which maps ai, on bi, for i = 1, …, n [1, p. 129 and Proposition 3.2.9, p. 131]. “Homogeneous model” always means “countably homogeneous model.” “Type of T” always stands for “n-type of T” where n ≥ s 0, i.e., for the type of some n-tuple of individuals of the universe of some model of T. We often use that two homogeneous models which realize the same types are isomorphic [1, Proposition 3.2.9, p. 131].

It is well known that every type of T is realized by at least one countable model of T. The main definition of this paper is:

Definition 1. A set of countable models of T is omissible or “may be omitted” if every type of T is realized by at least one countable model of T which is not isomorphic to a model in the set.

The main theorem of the paper is:

Theorem 1. If a countable complete theory is not ω-categorical, every finite set of its homogeneous models may be omitted.

The theorem is proved in §1 and in §2 it is shown how Vaught's and Rosenstein's theorems follow from it. §3 discusses some general aspects of omitting models.

Let ℳ be a structure for a language ℒ on a set M of urelements. HYP(ℳ) is the least admissible set above ℳ. In §1 we show that pp(HYP(ℳ)) [= the collection of pure sets in HYP(ℳ)] is determined in a simple way by the ordinal α = ° (HYP(ℳ)) and the ℒxω theory of ℳ up to quantifier rank α. In §2 we consider the question of which pure countable admissible sets are of the form pp(HYP(ℳ)) for some ℳ and show that all sets Lα (α admissible) are of this form. Other positive and negative results on this question are obtained.

If one regards an ordinal number as a generalization of a counting number, then it is natural to begin thinking in terms of computations on sets of ordinal numbers. This is precisely what Takeuti [22] had in mind when he initiated the study of recursive functions on ordinals. Kreisel and Sacks [9] too developed an ordinal recursion theory, called metarecursion theory, which specialized to the initial segment of the ordinals bounded by (the first nonconstructive ordinal).

The notion of admissibility was introduced by Kripke [11] and Platek [14] and served to generalize metarecursion theory. Kripke called ordinal α admissible if it satisfied certain closure properties of infinitary computations. It was shown that admissibility could be equivalently formulated in terms of the replacement schema of ZF set theory and that α = is an admissible ordinal. The study of a recursion theory on an initial segment of the ordinals bounded by some arbitrary admissible α became known as α-recursion theory.

Kripke [10] employed a Gödel numbering scheme to perform an arithmetiza-tion of α -recursion theory and created an analogue to Kleene's T-predicate (cf. [8]) of ordinary recursion theory (o.r.t.). The T-predicate then served as the basis for showing that analogues of the major results of unrelativized o.r.t. held in α-recursion theory; namely, the α-Enumeration Theorem, T Theorem, α-Recursion Theorem, and α-Universal Function Theorem.

Here an example will be given of a complete Lω1ω-sentence with a model of power ℵ1 but with no model of higher power. The continuum hypothesis is not assumed. The question of whether such an example exists was brought to the author's attention by Professor M. Makkai.

An Lω1ω-sentence is said to be complete if its models all satisfy the same Lω1ω-sentences, or, equivalently, if all of the countable L-structures satisfying the sentence are isomorphic. Scott [5] showed that any countable L -structure (where L is countable) must satisfy a complete Lω1ω-sentence. Such a sentence is called a Scott sentence for the structure. An uncountable L-structure need not satisfy any complete Lω1ω -sentence.

A complete Lω1ω-sentence σ is said to characterize the infinite cardinal k if σ has a model of power k but not of any higher power. The set of cardinals characterized by complete Lω1ω -sentences will be denoted by CC. By a result of Lopez-Escobar [3], if k ∈ CC, k <⊐ω1.

Assuming GCH (so that ⊐α = ℵα, ), Malitz [4] showed that CC = {ℵα: α < ω1}.

Without assuming GCH, Baumgartner [1] showed that ⊐α ∈ CC for all α < ω1.

Without GCH, it is unknown whether ℵn ∈ CC for n ≥ 2. Now it will be shown that ℵ1, ∈ CC.

In [1], H. Africk proved that Scott's interpolation theorem does not hold in the infinitary logic Lω1ω. In this paper we shall show that there is an interpolation theorem in Lω1ω which can be considered as an extension of Scott's interpolation theorem in Lω1ω by using a technique developed in Motohashi [2] and [3]. We use the terminology in [1]. Therefore {Ri; i ∈ J} is the set of predicate symbols in our language. Now let us divide the set of all the free variables into mutually disjoint infinite sets {VI; I ⊆ J}. Suppose that ℱ ⊆ (J). Then a formula in Lω1ω is said to be an ℱ′-formula if it is obtained from atomic formula of the form Ri(X1, …, Xn) for some I ∈ i ∈ I and X1, …, Xn in V1,, by applying ¬ (negation), ∧ (countable conjunction), ∨ (countable disjunction), → (implication), ∀ (universal quantification), and ∃ (existential quantification). Notice that every ℱ-sentence in [1] is an ℱ′. sentence (ℱ′-closed formula) in our sense.

Then we have the following theorem which is an immediate consequence of the interpolation theorem in [2].

Theorem. Let A and ? be sentences. There is an ℱ′-sentence C such that A→C and C→B are provable iff whenever and are ℱ-isomorphic structures and satisfies A, then satisfies B.

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems.

Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C.

Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.

One of the basic differences between the primitive recursive functions on the natural numbers and the primitive recursive ordinal functions (PR) is the nearly complete absence of constant functions in PR. Since ω is closed under all of the functions in PR, if α is any infinite ordinal, then λξ·α is not in PR. It is easily seen, however, that if one adds to the initial functions of PR the constant function λξ·ω, then all of the ordinals up to ω#, the next largest PR-closed ordinal, have their constant functions in this class. Since, however, such collections of functions are always countable, it is also the case that if one adds to the initial functions of PR the function λξ. α for uncountable α, then there are ordinals β < α whose constant functions are not in this collection. Because of this, the following objects are of interest:

Definition. For all α,

(i)PR(α) is the collection of functions obtained by adding to the initial primitive recursive ordinal functions, the function λξ· α;

(ii) PRsp(α), the primitive recursive spectrum of α, is the set {β < α ∣ λξβ ∈ PR(α);

(iii) Λ (α)= μρ(ρ∉ PRsp(α)).

In this paper, we shall define the “partially ordered interpretation” of a first order theory in another first order theory and state some recent results. Although an exact definition will be given in §4 below, we now give a brief outline. First of all, let us recall the “interpretations” defined by A. Tarski et al. in [17] and the “parametrical interpretations” defined by P. Hájek in [6], [7] and U. Felgner in [3]. Since “interpretations” can be considered as a special case of “parametrical interpretations”, we consider only the latter type of “interpretations”. A parametrical interpretation I of a first order language L in a consistent theory T′ (formulated in another first order language L′) consists of the following formulas:

(i) a unary formula C(p) (i.e. a formula with one designated free variable p), which is used to denote the range of parameters,

(ii) a binary formula U(p, x), which is intended to denote the pth universe for each parameter p,

(iii) an (n + 1)-ary formula Fp(p, x1 …, xn) for each n-ary predicate symbol P in L,

such that the formulas (∃p)C(p) and (∀p)(C(p)→(∃x)U(p, x)) are provable in T". Then, given a formula A in L and a parameter p, we define the interpretation Ip (A ) of A by I at p to be the formula which is obtained from A by replacing every atomic subformula P(*, …, *) in A by Fp(p, *,…,*), and relativizing every occurrence of quantifiers in A by U(p, * ). A sentence A in L is said to be I-provable in T′ if the sentence (∀p) (C(p)→ Ip(A)) is provable in T′. Then, it is obvious that every provable sentence in L is I-provable in T′. This is a basic result of “parametrical interpretations” and is used to prove the “consistency” of a theory T in L by showing that every axiom of T is I-provable in T′ when I is said to be a parametrical interpretation of T in T′. As is shown above, the word “interpretation” is used in the following three senses: interpretations of languages, interpretations of formulas and interpretations of theories. So, in this introduction we let the word “interpretation” denote “interpretation of languages”, for short.

Let L be an elementary first order language. Let be an L-structure, and let φ be an L-formula with free variables u1, …, un, and υ. A Skolem function for φ on is an n-ary operation f on such that for all . If is an elementary substructure of , then an n-ary operation f on is said to preserve the elementary embedding of into if f(x)∈ for all x ∈ n, and (, f ∣n) ≺ (, f). Keisler asked the following question:

Problem 1. If and are L-structures such that ≺ , and if φ (u, υ) is an L-formula (with appropriate free variables), must there be a Skolem function for φ on which preserves the elementary embedding?

Payne [6] gave a counterexample in which the language L is uncountable. In [3], [5], the author announced the existence of an example in which L is countable but the structures and are uncountable. The construction of the example will be given in this paper. Keisler's problem is still open in case both the language and the structures are required to be countable. Positive results for some special cases are given in [4].

The following variant of Keisler's question was brought to the author's attention by Peter Winkler:

Problem 2. If L is a countable language, a countable L-structure, and φ(u, υ) an L-formula, must there be a Skolem function f for φ on such that for every countable elementary extension of , there is an extension of f which preserves the elementary embedding of into ?

The negative solution of Hilbert's Tenth Problem brought with it a number of unsolvable Diophantine problems. Moreover, by actually providing a Diophantine characterization of recursive enumerability, the proof of the negative solution opened the door to the techniques of recursion theory. In this note, we wish to apply several recursion-theoretic facts and an improvement on the exponential Diophantine representation to refine the exponential case of a result of Davis [1972] regarding the difficulty of determining the number of zeros of a polynomial.

P, Q, etc. will denote polynomials or exponential polynomials–exactly which will be clear from the context. Let # (P) denote the number of distinct nonnegative zeros of P. Further, let C = {0,1, …,ℵ0} be the set of possible values of # (P). For A ⊆ C, we define A * to be

With this notation, Davis' result reads:

Theorem 0.1 (DAVIS). A* is recursive iff A = ∅ or A = C.

The refinement we will prove is:

Theorem 0.2 (Exponential Case). A * is r.e. iff A = ∅ or A = {c: c ≥ a: m} for some finite m.