Kleene [7] and Kreisel [8] defined independently the countable (continuous) functionals. Kleene [7] defined the countable functionals of type k to be total functionals of type k acting in a continuous way when restricted to countable arguments of type k − 1. He also defined the associates for countable functionals. They are functions α: N → N containing information about how the functional acts on countable arguments. Kleene [7] showed that the countable functionals are closed under the computations derived from S1–S9 of his paper [6], and that every computable functional has a recursive associate.

Kreisel defined the continuous functionals to be equivalence-classes of associates. By his definition it is meaningless to let a continuous functional act upon anything but continuous arguments.

One disadvantage of Kleene's approach is that two different functionals may have the same associates We will later see that there may be two functionals φ1 and φ2 with the same associates but such that the relations

are not the same.

In more recent papers on the countable functionals it is normal to regard the hierarchy 〈Ct(k)〉kϵN of countable functionals as a type-structure such that the functionals in Ct(k + 1) are maps from Ct(k) to N(Ct(0) = N), see e.g. Bergstra [1] and Gandy and Hyland [3].

We will then enjoy the streamlined formalism of a type-structure in which S1–S9 have meaning, but avoid the ambiguities of Kleene's original approach.

We will presuppose a brief familiarity with the theory of the countable functionals.

R. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that λ < α*, where α* is the Σ1-projectum of α and λ is the Σ2-cofinality of α. The theorem is also established for the metarecursive case, α = ω1, where α* = λ = ω.

In [8] and [9] Urquhart presented the semilattice semantics for a relevance logic with conjunction and disjunction. These semantics combined the simplest known semantics for relevant implication, with minimal modification of the truth conditions for conjunction and disjunction. A slightly different approach leads to the distinct positive relevance logic of [7] and [5]. Sound and complete axiomatic logics for the semilattice semantics were readily available, so long as disjunction was left out.

The purpose of this paper is to report the remedy of this lack. In [1] the author developed work of Fine ([3], announced in [4]) and Prawitz [6] to show soundness and completeness of an axiomatic logic and two natural deduction logics, with respect to Urquhart's semantics. Thus Prawitz' positive relevance logic is equivalent to the semilattice logic, rather than the logic of [5]. We present here the axiomatic logic and show its equivalence to the semilattice semantics.

In §1 the languages and logic are given, along with a few proof-theoretic remarks. In §2 the semantics are defined, and the soundness proof is sketched. In §3, the completeness proof is given.

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.

Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:

For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.

We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].

We prove, relative to suitable hypotheses, that it is consistent for there to be unboundedly many measurable cardinals each of which possesses a large degree of strong compactness, and that it is consistent to assume that the least measurable is partially strongly compact and that the second measurable is strongly compact. These results partially answer questions of Magidor on the relationship of strong compactness to measurability.

In a recent article in this Journal (see [3]), J.P. Jones states and proves a theorem which purports to give an “absolute epistemological upper bound on the complexity of mathematical proofs” for recursively axiomatizable theories. However, Jones' statement of this result is misleading, and in fact defective, as can be seen by a close analysis of it. Such an analysis is the object of the present note.

The main point is that Jones' “epistemological bound” can in no way be considered a computational bound on the complexity of proofs. Not only is the “proof-theoretic interpretation of the number 243” contained in Jones' article objectionable but, more fundamentally, there is in a strong sense no way one can hope to recover anything like the full force suggested by Jones' original statement of the theorem.

We wish to insist that our comments concern only the difficulties surrounding Jones' Corollary 1 on p. 338 of his article and not his ingenious construction of universal Diophantine representations of r.e. sets presented in the same article.

Through the ability of arithmetic to partially define truth and the ability of infinite integers to simulate limit processes, nonstandard models of arithmetic automatically have a certain amount of saturation: Any encodable partial type whose formulae all fall into the domain of applicability of a truth definition must, by finite satisfiability and Overspill, be nonstandard-finitely satisfiable—whence realized. This fact was first exploited by A. Robinson, who in Robinson [1963] cited the unrealizability in a given model of a certain encodable partial type to prove Tarski's Theorem on the Undefinability of Truth. A decade later, H. Friedman brought this phenomenon to the public's attention by using it to establish impressive embeddability criteria for countable nonstandard models of arithmetic. Subsequently, Wilkie considered models expandable to “strong theories” and, among such models, complemented Friedman's embeddability criteria with elementary embeddability and isomorphism criteria. Oddly enough, the fact that some kind of saturation property was being employed was not explicitly acknowledged in any of this work.

It is in the unpublished dissertation of Wilmers that these submerged saturation properties first surfaced. [As I have only seen accounts of it (most notably Murawski [1976/1977]) and not the dissertation itself, what I have to say about it will not quite be accurate. (Indeed, the referee has refuted, without providing alternate information, every conjecture I have made about the contents of this thesis.)

Roughly speaking, a pattern is a finite sequence coding the set of natural numbers n for which the Σn+1 projectum is less than the Σn projectum for a given admissible ordinal. We prove that for each pattern there exists an ordinal realizing it. Several results on the orderings of patterns are given. We conclude the paper with remarks on Δn projecta. The main technique, used throughout the paper, is Jensen's Uniformisation Theorem.

T. Jech and K. Prikry introduced the concept of precipitous ideals as a counterpart of measurable cardinals for small cardinals (Jech and Prikry [1]). To get a precipitous ideal on ℵ1( W. Mitchell used the Cohen extension by the standard Levy collapse that makes κ = ℵ1, where κ is a measurable cardinal in the ground model (Jech, Magidor, Mitchell and Prikry [2]). His proof essentially used the fact that , where is the notion of forcing of Levy collapse and j is the elementary embedding obtained by a normal ultrafilter on κ.

On the other hand, we know that has the κ-chain condition. In this paper, we show that the κ-chain condition for notions of forcing plays an essential role for preserving normal precipitous ideals. Namely,

Theorem 1. Let κ be a regular uncountable cardinal and I a normal ideal on κ. Let be a notion of forcing with the κ-chain condition. Then, I is precipitous iff ⊩“the ideal on κ generated by I is precipitous”.

As an application of Theorem 1, we have the following theorem.

Theorem 2. Let κ be a regular uncountable cardinal, and Γ be a κ-saturated normal ideal on κ. Then, {a < κ; the ideal of thin sets on a is precipitous} has either Γ-measure one or Γ-measure zero, and the ideal of thin sets on κ is precipitous iff {α< κ; the ideal of thin sets on α is precipitous) has Γ-measure one.

Unless otherwise stated, every structure in this paper is countable in a countable, actually recursive language and every formula is one of .

The definition of the so-called canonical Scott-sentence of a structure M, CSS(M) (compare Nadel [8]), is based on the ordinal invariant called the Scott-height of M, denoted SH(M) (compare Makkai [5]). To describe SH(M), let “b ≡αα”, for finite tuples of equal lengths b and a of elements of M and any ordinal α, stand for “b and a satisfy (in M) the same formulas of quantifier-rank ≤ α” (for the quantifier rank of a formula, see e.g., Barwise [1]); also, let “b ∼ a” mean that there is an automorphism of M mapping b to a. With a fixed M, and a in M, sh(a) (or shM(a)) denotes the least ordinal α such that for all b in M, b ≡αa implies b ∼ a. SH(M) is the least ordinal α such that for all a and b in M, b ≡αa implies b ∼ a; hence SH(M) = sup{sh(a): a in M}.

The literature on modal logic includes a number of general completeness and decidability results. The work of Schiller Joe Scroggs [5], R.A. Bull [1], Kit Fine [2], and Krister Segerberg [6] provide examples.

Scroggs showed that the proper extensions of S5 have the finite model property and are axiomatizable. (Harrop [3] then argued that logics having these properties are decidable.) Bull extended Scroggs' result by showing that the normal extensions of S4.3 have the finite model property. Fine subsequently provided a model-theoretic proof of Bull's result and also proved the axiomatizability of these logics. In a different direction Segerberg proved that every normal logic containing the characteristic axioms of Lewis' systems S4 and S5 is decidable.

The present paper is in this tradition. We extend the results of Scroggs and Segerberg by showing that every normal modal logic containing the S5 axiom has the finite model property, is axiomatizable, and thus is decidable.

Beginning in 1960, and continuing for about a decade and a half, Shepherdson's self-referential formulae dominated the applications of diagonalization in metamathematics. In 1976, it was doubly toppled from its position of supremacy by two demonstrably more powerful such sentences introduced by D. Guaspari. The goal of the present note is to understand and explain the first (and, for the time being, more important) of Guaspari's self-referential sentences, which we dub “Guaspari sentences of the first kind”, or, less poetically, “Guaspari fixed points”.

Both Shepherdson's and Guaspari's fixed points are generalizations of the Rosser sentence. But, where Shepherdson merely tacks on side-formulae, Guaspari takes a more revolutionary step: He views the basic components, PrT(⌈¬φ⌉) and PrT(⌈φ⌉), of the Rosser sentence as attempts to refute something and replaces them by refutations of something else. Ignoring their Shepherdsonesque side-formulae (tacked on for the sake of more esoteric applications), our analysis of Guaspari fixed points can be viewed as merely the isolation of those properties whose refutations can be used in this context.

In §1 we offer the outcome of this analysis—i.e. a delineation of properties whose refutations can so be used. Some simple examples are cited and the Guaspari fixed points are defined. The main theorem—a master Fixed Point Calculation—is proven in §2.

We improve the canonization theorems generalizing the Erdös-Rado theorem, and as a result complete the answer to “When does a Hausdorff space of cardinality χ necessarily have a discrete subspace of cardinality k” We also improve the results on existence of free subsets.

The Gödel Class is the class of prenex formulas of pure quantification theory whose prefixes have the form ∀y1∀y2∃x1 … ∃xn. The Gödel Class with Identity, or GCI, is the corresponding class of formulas of quantification theory extended by inclusion of the identity-sign “ = ”. Although the Gödel Class has long been kndwn to be solvable, the decision problem for the Gödel Class with Identity is open. In this paper we prove that there is no primitive recursive decision procedure for the GCI, or, indeed, for the subclass of the GCI containing just those formulas with prefixes ∀y1∀y2∃x.

Throughout this paper we take quantification theory to include, aside from logical signs, infinitely many k-place predicate letters for each k > 0, but no function signs or constants. Moreover, by “prenex formula” we include only those without free variables. A decision procedure for a class of formulas is a recursive function that carries a formula in the class to 0 if the formula is satisfiable and to 1 if not. A class is solvable iff there exists a decision procedure for it. A class is finitely controllable iff every satisfiable formula in the class has a finite model. Since we speak only of effectively specified classes, finite controllability implies solvability (but not conversely).

The GCI has a curious history. Gödel showed the Gödel Class (without identity) solvable in 1932 [4] and finitely controllable in 1933 [5].

It is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system *N has the form ω + (ω* + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined *N more closely and investigated the nonstandard real number system *R, as an ordered set, as an additive group and as a uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe *U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2].

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set.

All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example).

It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensions K ⊂ F of the rationals such that F is abelian, F is of infinite degree over K, and the Galois group of F over K, although of cardinality c, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory of finite extensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).

It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.

Assume (∃κ) [κ → (κ)<ω & Rκ ≺ V]. Then a new inner model H exists and has the following properties: (1) H ≠ HOD; (2) Th(H) = Th(HOD); (3) there is j: H → H; (4) there is a c.u.b. class of indiscernibles for H.

In this paper we investigate the relationship between the number of countable and decidable models of a complete theory. The number of decidable models will be determined in two ways, in §1 with respect to abstract isomorphism type, and in §2 with respect to recursive isomorphism type.

Definition 1. A complete theory is (α, β) if the number of countable models of T, up to abstract isomorphism, is β, and similarly the number of decidable models of T is α.

Definition 2. A model is ω-decidable if ∣∣= ω and for an effective listing {θn∣n < ω} of all sentences in the language of Th() augmented by new constant symbols i*, i < ω, {n ∣〈, i〉i<ω ⊨ θn} is recursive, where i interprets i* (in these terms, is decidable if is abstractly isomorphic to an ω-decidable model).

Definition 3. A complete theory is (α, β)r if it is (γ, β) for some γ and it has exactly αω-decidable models up to recursive isomorphism.

Specifically we will show in §1 that there is a (2, ω) theory, and in §2 that although there is a (2, 2ω) theory, there is no (2, β)r theory for any β, β < 2ω.