Throughout the paper A will be a transitive set closed under finite subsets and the formulas in various classes mentioned are allowed to contain parameters from A (or from B in §2).

By use of a refinement of Moschovakis' notion of the game-quantifier [13], [14], [15] we are able to obtain a game-theoretic description of s-Π11 predicates over countable sets which then leads to a classification of positive Σ1 inductive sets.

Similar results are then proved for certain sets of cofinality ω. As a consequence we obtain the compactness results of Green [8], [11], Nyberg [16] and Makkai [12].

The use of games to classify inductive sets was initiated by Moschovakis [13], [14], [15] and has been extended to Q-inductive sets by Aczel [2]. Games were also used in a slightly different setting by Vaught [18] and Makkai [12]. In fact, Vaught's proof of the compactness theorem is very close to our proof in §1 and Makkai's extension to cofinality ω sets uses a result similar to Theorem 3 in §2.

We are indebted to the referee for many helpful suggestions, in particular, for bringing to our attention the related works of Vaught and Makkai cited above.

We lean upon the usual variant of the logic of quantum mechanics [1]. Here the propositions correspond to the results of the quantum experiments. A beautiful essay [2] may be connected with a conservation of semantics. But we try to broaden the semantics, admitting propositions about the possibility of results of experiments. Doing so, we fulfil the old wish of W. A. Fock, who attracts our attention to the importance of the modal categories for the interpretation of the quantum theory [3].

We begin with the formal description of the modal system Br+. The alphabet contains the signs ~, ∨ and Ↄ (negation, alternative and the sign of necessity), the set V of the prepositional variables and parentheses. The rules of formation are: if A ∈ V, then A is a proposition; if X and Y are propositions, then ~ X, X ∨ Y and Ↄ X are propositions also; there are no other propositions. ℬ will design the set of all propositions.

The rules of transformation are the following:

.

Here X, Y ∈ ℬ. The sign ⊨ denotes the derivability in Br+ (in one step here).

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 area of interest of this paper is recursively enumerable vector spaces; its origins lie in the works of Rabin [16], Dekker [4], [5], Crossley and Nerode [3], and Metakides and Nerode [14]. We concern ourselves here with questions about maximal vector spaces, a notion introduced by Metakides and Nerode in [14]. The domain of discourse is V∞ a fully effective, countably infinite dimensional vector space over a recursive infinite field F.

By fully effective we mean that V∞, under a fixed Gödel numbering, has the following properties:

(i) The operations of vector addition and scalar multiplication on V∞ are represented by recursive functions.

(ii) There is a uniform effective procedure which, given n vectors, determines whether or not they are linearly dependent (the procedure is called a dependence algorithm).

We denote the Gödel number of x by ⌈x⌉ By taking {εn ∣ n > 0} to be a fixed recursive basis for V∞, we may effectively represent elements of V∞ in terms of this basis. Each element of V∞ may be identified uniquely by a finitely-nonzero sequence from F Under this identification, εn corresponds to the sequence whose n th entry is 1 and all other entries are 0. A recursively enumerable (r.e.) space is a subspace of V∞ which is an r.e. set of integers, ℒ(V∞) denotes the lattice of all r.e. spaces under the operations of intersection and weak sum. For V, W ∈ ℒ(V∞), let V mod W denote the quotient space. Metakides and Nerode define an r.e. space M to be maximal if V∞ mod M is infinite dimensional and for all V ∈ ℒ(V∞), if V ⊇ M then either V mod M or V∞ mod V is finite dimensional. That is, M has a very simple lattice of r.e. superspaces.

The notion of the next admissible set has proved to be a very useful notion in definability theory and generalized recursion theory, a unifying notion that has produced further interesting results in its own right. The basic treatment of the next admissible set above a structure ℳ of urelements is to be found in Barwise's [75] book Admissible sets and structures. Also to be found there are many of the interesting characterizations of the next admissible set. For further justification of the interest of the next admissible set the reader is referred to Moschovakis [74], Nadel and Stavi [76] and Schlipf [78a, b, c].

One of the most interesting single properties of is its ordinal (ℳ). It coincides, for example, with Moschovakis' inductive closure ordinal over structures ℳ with pairing functions—and over some, such as algebraically closed fields of characteristic 0, without pairing functions (by recent work of Arthur Rubin) (although a locally famous counterexample of Kunen, a theorem of Barwise [77], and some recent results of Rubin and the author, show that the inductive closure ordinal may also be strictly smaller in suitably pathological structures). Further justification for looking at (ℳ) alone may be found in the above-listed references. Loosely, we can consider the size of to be a useful measure of the complexity of ℳ. One of the simplest measures of the size of —and yet a very useful measure—is its ordinal, (ℳ). Keisler has suggested thinking of (ℳ) as the information content of a model—the supremum of lengths of wellfounded relations characterizable in the model.

We formalise the notion of those infinite binary sequences z that admit a single program P which expresses the entire algorithmical structure of z. Such a program P minimizes the information which must be used in a relative computation for z. We propose two concepts with different strength for this notion, the learnable and the super-learnable sequences. We establish three different equivalent characterizations of learnable (super-learnable, resp.) sequences. In particular, we prove that a sequence z is learnable (super-learnable, resp.) if and only if there is a computable probability measure p such that p is Schnorr (Martin-Löf, resp.) p-random. There is a recursively enumerable sequence which is not learnable. The learnable sequences are invariant with respect to all total and effective transformations of infinite binary sequences.

In this note we give a simple recursive axiomatization for the class of structures of type (ℶω ℵ0). This solves a problem of Vaught which is Problem 13 in the book [1] of Chang and Keisler. The same technique is used to get a recursive axiomatization for the class of κ-like structures where κ is strongly ω-inaccessible.

Let us fix throughout some recursive first-order language L, and until further notice let us suppose that included in L is a distinguished unary predicate symbol U. For cardinals κ and λ with κ ≥ λ ≥ ℵ0, we say the structure has type (κ, λ) if card(A)= κ and card . Let K(κ, λ) be the class of all structures of type (κ, λ). For each ordinal α define 2ακby 20κ = κ, and 2ακ= ⋃ {2λ: λ = 2βκ for some β < α} when α > 0. Let

Vaught proved the following theorem in [7].

Theorem (Vaught). Suppose a is a sentence such that for each n < ω there are κ, λ with κ > 2λn and a model of σ of type (κ, λ). Then whenever κ ≥ λ ≥ ℵ0, the sentence σ has a model of type (κ, λ).

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.

Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures.

The function μ from P(X) to the closed interval [0, 1] is a measure on X if μ. is finitely additive on disjoint sets and μ(X) = 1. (P is the power set.) μ is nonprincipal if μ ({x}) = 0 for each x Є X. μ is an ultrafilter if Range μ= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite X follows from the axiom of choice.

Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo–Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.

Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.

Let L be a countable first-order language and L(Q) be obtained by adjoining an additional quantifier Q. Q is a generalization of the quantifier “there exists uncountably many x such that…” which was introduced by Mostowski in [4]. The logic of this latter quantifier was formalized by Keisler in [2]. Krivine and McAloon [3] considered quantifiers satisfying some but not all of Keisler's axioms. They called a formula φ(x) countable-like if

for every ψ. In Keisler's logic, φ(x) being countable-like is the same as ℳ⊨┐Qxφ(x). The main theorem of [3] states that any countable model ℳ of L[Q] has an elementary extension N, which preserves countable-like formulas but no others, such that the only sets definable in both N and M are those defined by formulas countable-like in M. Suppose C(x) in M is linearly ordered and noncountable-like but with countable-like proper segments. Then in N, C will have new elements greater than all “old” elements but no least new element — otherwise it will be definable in both models. The natural question is whether it is possible to use generalized quantifiers to extend models elementarily in such a way that a noncountable-like formula C will have a minimal new element. There are models and formulas for which it is not possible. For example let M be obtained from a minimal transitive model of ZFC by letting Qxφ(x) mean “there are arbitrarily large ordinals satisfying φ”.

We shall prove the following theorem:

Theorem. For any finite lattice there is a model of ZF in which the partial ordering of the degrees of constructibility is isomorphic with the given lattice.

Let M be a standard countable model of ZF satisfying V = L. Let K be the given finite lattice. We shall extend M by forcing.

The paper is divided into two parts. The first part concerns the definition of the set of forcing conditions and some properties of this set expressible without the use of generic filters.

We define first a representation of a lattice and then the set of conditions. In Lemmas 1, 2 we show that there are some canonical isomorphisms between some conditions and that a single condition has some canonical automorphisms.

In Lemma 3 and Definition 7 we show some methods of defining conditions. We shall use those methods in the second part to define certain conditions with special properties.

Lemma 4 gives a connection between the sets P and Pk (see Definitions 4 and 5). It is next employed in the second part in Lemma 10 in an essential way.

Indeed, Lemma 10 is necessary for Lemma 13, which is the crucial point of the whole construction. Lemma 5 is also employed in Lemma 13 (exactly in its Corollary).

The second part of the paper is devoted to the examination of the structure of degrees of constructibility in a generic model. First, we show that degrees of some “sections” of a generic real (Definition 9) form a lattice isomorphic with K. Secondly, we show that there are no other degrees in the generic model; this is the most difficult property to obtain by forcing. We prove, in two stages, that it holds in our generic models. We first show, by using special properties of the forcing conditions, that sets of ordinal numbers have no other degrees. Then we show that the degrees of sets of ordinals already determine the degrees of other sets.

Terminology. PA is Peano Arithmetic, classical first-order arithmetic with induction. ⌈A⌉ is the formal numeral in PA for the Gödel number of A. – A is the negation of A, (A&B) is the conjunction of A and B, and Bew(x) is the usual provability predicate for PA. neg(x), conj(x, y), bicond(x, y), and bew(x) are terms of PA such that for all sentences A and B of PA ⊢PA, neg(˹A˺) = ˹−A˺ ⊢PA Conj(˹A˺, ˹B˺)= ˹(A&B)˺ ⊢PA bicond(˹A˺, ˹B˺)= ˹(A ↔ B)˺, and ⊢PA bew(˹A˺) = ˹Bew(˹A˺)˺. T is the sentence ‘0 = 0’ and Con is the usual sentence expressing the consistency of PA. If A (x) is any formula of PA, then a fixed point of A(x) is a sentence S such that ⊢PAS ↔ A(˹S˺). (It is well known that every formula of PA with one free variable has a fixed point.) The P-terms are defined inductively by: the variable x is a P-term; if t(x) and u(x) are P-terms, so are neg(t(x)), conj(t(x), u(x)), and bew(t(x)). A basic P-formula is a formula Bew(t(x)), where t(x) is a P-term; and a P-formula is a truth-functional combination of basic P-formulas. An F-sentence is a member of the smallest class that contains Con and contains −A, (A&B), and −Bew(˹−A˺) whenever it contains A and B. In [B] we gave a decision procedure for the class of true F-sentences.

−Bew(x), Bew(x), and Bew(neg(x)) are examples of P-formulas, and fixed points of these particular P-formulas have been studied by Gödel, Henkin [H] and Löb [L], and Jeroslow [J], respectively. In this note we show how to decide whether or not a fixed point of any given P-formula is provable in PA.

In this paper we are going to consider experimental logics introduced by Jeroslow [4] as models of human reasoning proceeding by trial and error, i.e. admitting changes of axioms in time (some axioms are deleted, some new ones accepted). Jeroslow's notion is based on the idea that events which may cause changes in axioms and rules of reasoning are mechanical. Suppose a finite alphabet Γ to be fixed and let Γ* be the set of words in the alphabet Γ. N denotes the set of natural numbers.

0.1. Definition. An experimental logic is a recursive relation H ⊆ N × Γ*; H(t, φ) is read “the expression is accepted at the point of time t”. φ is recurring w.r.t. H (notation: RecH(φ)) if H(t, φ) holds for infinitely many t; is stable w.r.t. H (notation: StblH(φ)) if H(t,φ) holds for all but finitely many t. In symbols:

H is convergent if every recurring expression is stable.

0.2. We have the following facts: Let X ∈ Γ*. (1) X ∈ iff there is an experimental logic H such that X = {φ; RecH(φ)}- (2) X ∈ iff there is an experimental logic H such that X = {φ; StblH(φ)}. (3) X ∈ iff there is a convergent experimental logic H such that X is the set of all expressions recurring ( = stable) w.r.t. H. (See [4], [3], [7]; cf. also [5].)

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.

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).

That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).

Our basic theorem will be the following: If k > ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].

The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional (HAω + MUC). The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+ as defined there cannot be shown to have the required properties in EL + QF-AC, the reason being that a change in the definition of W12 alone does not suffice—if one wishes to establish closure under the operations of HAω the definitions of W1σ for other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.

For a proper understanding, we should perhaps note already here that on the assumption of the fan theorem, ECF+ as defined in [T4] and the new model of this note coincide (since then the definition of W12 [T4, p. 594] is equivalent to the definition for W12 in the case of ECF); but in EL it is impossible to prove this (and under assumption of Church's thesis the two models differ).

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.

Any two models of arithmetic can be jointly embedded in a third with any prescribed isomorphic submodels as intersection and any prescribed relative ordering of the skies above the intersection. Corollaries include some known and some new theorems about ultrafilters on the natural numbers, for example that every ultrafilter with the “4 to 3” weak Ramsey partition property is a P-point. We also give examples showing that ultrafilters with the “5 to 4” partition property need not be P-points and that the main theorem cannot be improved to allow a prescribed ordering of lower skies.