Ramsey's theorem [5] asserts that every infinite set X has the following partition property (RP): For every partition of the set [X]2 of two-element subsets of X into two pieces, there is an infinite subset Y of X such that [Y]2 is included in one of the pieces. Ramsey explicitly indicated that his proof of this theorem used the axiom of choice. Kleinberg [3] showed that every proof of Ramsey's theorem must use the axiom of choice, although rather weak forms of this axiom suffice. J. Dawson has raised the question of the position of Ramsey's theorem in the hierarchy of weak axioms of choice.

In this paper, we prove or refute the provability of each of the possible implications between Ramsey's theorem and the weak axioms of choice mentioned in Appendix A.3 of Jech's book [2]. Our results, along with some known facts which we include for completeness, may be summarized as follows (the notation being as in [2]):

A. The following principles do not (even jointly) imply Ramsey's theorem, nor does Ramsey's theorem imply any of them:

the Boolean prime ideal theorem,

the selection principle,

the order extension principle,

the ordering principle,

choice from wellordered sets (ACW),

choice from finite sets,

choice from pairs (C2).

B. Each of the following principles implies Ramsey's theorem, but none of them follows from Ramsey's theorem:

the axiom of choice,

wellordered choice (∀kACk),

dependent choice of any infinite length k (DCk),

countable choice (ACN0),

nonexistence of infinite Dedekind-finite sets (WN0).

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.

Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e., are there κ+ stationary subsets of κ with pairwise intersections nonstationary? Our first observation is:

Theorem. NS isκ+-saturated iff for every normal ideal J on κ there is a stationary set A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ NS}.

Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define “greatly Mahlo” cardinals and obtain the following:

Theorem. If κ is greatly Mahlo then NS is notκ+-saturated.

Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-indescribable cardinal.

These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ+-saturated.

In this paper we show that certain notions of consistency currently in use in illative combinatory logic are not always necessary nor always sufficient for the acceptability of the system.

The three notions of consistency we will consider are:

(A) Not all terms of the system are assertible.

(B) Not all propositions of the system are assertible.

(C) The system is Q-consistent, i.e. if Q (for equality) is in or is added to the system as a primitive with axioms such as

Post defined a system of propositional calculus to be inconsistent if all propositions were assertible. In a more general system where there may be terms which are not propositions (although all propositions are terms), we need to distinguish notions (A) and (B) as two possible consistency notions “in the sense of Post”.

In [3] Curry and Feys assume that Q is in the system defined as λxλy.Ξ(C*x)(C*y) or as a primitive, but Seldin in [5] also talks about Q-consistency of a system (F22) to which Q has to be added. We therefore consider the definition of Q-consistency that includes all three possibilities.

In [3] Curry and Feys state that Q-consistency is essential for acceptability. This is because there the purpose was to have an assertional system of illative combinatory logic in which combinatory equality was represented by the assertion of QXY. However if Q is to represent some other form of equality, Q-consistency may become inessential for acceptability or even incompatible with it.

In the Kripke semantics for propositional modal logic, a frame W = (W, ≺) represents a set of “possible worlds” and a relation of “accessibility” between possible worlds. With respect to a fixed frame W, a proposition is represented by a subset of W (regarded as the set of worlds in which the proposition is true), and an n-ary connective (i.e. a way of forming a new proposition from an ordered n-tuple of given propositions) is represented by a function fw: (P(W))n → P(W). Finally a state of affairs (i.e. a consistent specification whether or not each proposition obtains) is represented by an ultrafilter over W. {To avoid possible confusion, the reader should forget that some people prefer the term “states of affairs” for our “possible worlds”.}

In a broader sense, an n-ary connective is represented by an n-ary operatorf = {fw∣ W ∈ Fr}, where Fr is the class of all frames and each fw: (P(W))n → P(W). A connective is modal if it corresponds to a formula of propositional modal logic. A connective C is coherent if whether C(P1,…, Pn) is true in a possible world depends only upon which modal combinations of P1,…,Pn are true in that world. (A modal combination of P1,…,Pn is the result of applying a modal connective to P1,…, Pn.) A connective C is strongly coherent if whether C(P1, …, Pn) obtains in a state of affairs depends only upon which modal combinations of P1,…, Pn obtain in that state of affairs.

Although this problem is still open, there are a few related results concerning (i) theories equiconsistent with NF and (ii) consistent fragments of NF. Concerning (i) we will discuss the following results

(1) Specker's result about the equiconsistency of NF with an extension of the theory of types [16],

(2) the equiconsistency of NF with fragments of NF containing NFU [2], [10],

and a new one about

(3) the equiconsistency of NF with an extension of Zermelo's set theory with the axiom of comprehension restricted to bounded formulas.

Concerning (ii) we will give a simplified proof and a generalization of (4) Jensen's result about the consistency of NFU [13].

Only a few words will be devoted to

(5) Grishin's results and refinements [4], [5], [8], [9], treated with more details in [4].

The reader who is unfamiliar with NF or who has a natural repulsion for this system is first invited to read the excellent account contained in [7, Chapter III, §3].

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 main purpose of this note is to present new semi-model-theoretic proofs of some axiomatization results. Principally, we prove the result of [van Dalen and Statman] on the axiomatization of the equality fragment of the intuitionistic theory of apartness by ω-fold stability axioms. Further examples are also discussed.

The intuitionistic theories of equality and apartness are given by the following nonlogical axioms:

EQ .

AP .

In AP, one can define a notion of equality by means of the abbreviation:

.

The corresponding interpretation of E Q in AP is not faithful — since equality is a negation, we have stability of equality:

.

The equality fragment of AP is axiomatized by a suitable generalization of the stability of equality. To state it, one first defines a sequence of partial apartness relations:

.

A simple induction shows, for all n,

whence

.

Let L be any finitary language. By restricting our attention to the universal Horn sentences of L and appealing to a semantical notion of logical consequence, we can formulate the universal Horn logic of L. The present paper provides some theorems about universal Horn logic that serve to distinguish it from the full first order predicate logic. Universal Horn equivalence between structures is characterized in two ways, one resembling Kochen's ultralimit theorem. A sharp version of Beth's definability theorem is established for universal Horn logic by means of a reduced product construction. The notion of a consistency property is relativized to universal Horn logic and the corresponding model existence theorem is proven. Using the model existence theorem another proof of the definability result is presented. The relativized consistency properties also suggest a syntactical notion of proof that lies entirely within the universal Horn logic. Finally, a decision problem in universal Horn logic is discussed. It is shown that the set of universal Horn sentences preserved under the formation of homomorphic images (or direct factors) is not recursive, provided the language has at least two unary function symbols or at least one function symbol of rank more than one.

This paper begins with a discussion of how algebraic relations between structures can be used to obtain fragments of a given logic. Only two such fragments seem to be under current investigation: equational logic and universal Horn logic. Other fragments which seem interesting are pointed out.

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 most interesting aspects of the theory of computational complexity is the speed-up phenomenon such as the theorem of Blum [6, p. 326] which asserts the existence of a 0, 1-valued total recursive function with arbitrarily large speed-up. Blum and Marques [10] extended the speed-up definitions from total to partial recursive functions, or equivalently, to recursively enumerable (r.e.) sets, and introduced speedable and levelable sets. They classified the effectively speedable sets as the subcreative sets but remarked that “the characterizations we provided for speedable and levelable sets do not seem to bear a close relationship to any already well-studied class of recursively enumerable sets.” The purpose of this paper is to give an “information theoretic” characterization of speedable and levelable sets in terms of index sets resembling the jump operator. From these characterizations we derive numerous consequences about the degrees and structure of speedable and levelable sets.

Let N denote the natural numbers. If A ⊆ N, we write Ā for the complement of A in N. A set A ⊆ N is cohesive if (i) A is infinite and (ii) for any recursively enumerable set W either W ∩ A or ∩ A is finite. A r.e. set M ⊆ N is maximal if is cohesive.

A recursively presented vector space (r.p.v.s.) U over a recursive field F consists of a recursive set U ⊆ N and operations of vector addition and scalar multiplication which are partial recursive and under which U becomes a vector space. A r.p.v.s. U has a dependence algorithm if there is a uniform effective procedure which applied to any n-tuple ν0, ν1, …, νn−1 of elements of U determines whether or not ν0, ν1 …, νn−1 are linearly dependent. Throughout this paper we assume that if U is a r.p.v.s. over a recursive field F then U is infinite dimensional and U = N. If W ⊆ U, then we say W is recursive (r.e., etc.) iff W is a recursive (r.e., etc.) subset of N. If S ⊆ U, we write (S)* for the subspace generated by S. If V1 and V2 are subspaces of U such that V1 ∩ V2 ={} (where is the zero vector of U), then we write V1 ⊕ V2 for (V1 ∪ V2)*. If V1 ⊆ V2⊆U are subspaces, we write V2/V1 for the quotient space.

The comprehension principle of second order arithmetic asserts the existence of certain species (sets) corresponding to properties of natural numbers. In the intuitionistic theory of sequences of natural numbers there is an analoguous principle, implicit in Brouwer's writing and explicitly stated by Kripke, which asserts the existence of certain sequences corresponding to statements. The justification of this principle, Kripke's Schema, makes use of the concept of the so-called creative subject. For more information the reader is referred to Troelstra [5].

Kripke's Schema reads

There is a weaker version

The idea to reduce species to sequences via Kripke's schema occurred several years ago (cf. [2, p. 128], [5, p. 104]). In [1] the reduction technique was applied in the construction of a model for HAS.

On second thought, however, I realized that there is a straightforward, simpler way to exploit Kripke's schema, avoiding models altogether. The idea to present this material separately was forced on the author by C. Smorynski.

Consider a second order arithmetic with both species and sequence variables. By KS we have

(for convenience we restrict ourselves in KS to 0-1-sequences). An application of AC-NF gives

Of course ξ is not uniquely determined. This is the key to the reduction of full second order arithmetic, or HAS, to a theory of sequences.

We now introduce a translation τ to eliminate species variables. It is no restriction to suppose that the formulae contain only the sequence variables ξ1, ξ3, ξ5, …

Note that by virtue of the definition of τ the axiom of extensionality is automatically verified after translation. The translation τ eliminates the species variables and leaves formulae without species variables invariant.

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)).

Let IPC be the intuitionistic first-order predicate calculus. From the definition of derivability in IPC the following is clear:

(1) If A is derivable in IPC, denoted by “⊦IPCA”, then A is intuitively true, that means, true according to the intuitionistic interpretation of the logical symbols. To be able to settle the converse question: “if A is intuitively true, then ⊦IPCA”, one should make the notion of intuitionistic truth more easily amenable to mathematical treatment. So we have to look then for a definition of “A is valid”, denoted by “⊨A”, such that the following holds:

(2) If A is intuitively true, then ⊨ A.

Then one might hope to be able to prove

(3) If ⊨ A, then ⊦IPCA.

If one would succeed in finding a notion of “⊨ A”, such that all the conditions (1), (2) and (3) are satisfied, then the chain would be closed, i.e. all the arrows in the scheme below would hold.

Several suggestions for ⊨ A have been made in the past: Topological and algebraic interpretations, see Rasiowa and Sikorski [1]; the intuitionistic models of Beth, see [2] and [3]; the interpretation of Grzegorczyk, see [4] and [5]; the models of Kripke, see [6] and [7]. In Thirty years of foundational studies, A. Mostowski [8] gives a review of the interpretations, proposed for intuitionistic logic, on pp. 90–98.

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.

Ellerman, Comer, and Macintyre have all observed that sheaves are an interesting generalization of models and are deserving of model theoretic attention. Scott has pointed out that sheaves are Heyting algebra valued models. The reverse does not hold however since almost no genuine Boolean valued model is a sheaf.

In §1 we shall review the definition of a sheaf and prove a theorem about Boolean valued models using the sheaf construction. In §2 we shall be concerned with the set of sentences preserved by global sections. Our principal result is that global section sentences are also normal submodel sentences. (We define as a normal submodel of if is a submodel of and every point of B − A can be moved by an automorphism of which fixes each point of A.) In §3 we prove that every normal submodel sentence is the negation of a disjunction of Horn sentences and that the set of normal submodel sentences is r.e. but not recursive. §3 involves only traditional model theory and can be read independently of the first two sections.