Showing papers in "Journal of Symbolic Logic in 1975"

Constructive Set Theory

Abstract: This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis . The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwer's mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishop's book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Godel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/ x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣ x ∣ > 1/ n ) or, to take the Dialectica interpretation seriously, by adjoining to the Godel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.

One Hundred and Two Problems in Mathematical Logic

Abstract: This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.

Consciousness, philosophy, and mathematics

Abstract: Publisher Summary This chapter describes the importance of consciousness, philosophy, and mathematics. Consciousness in its deepest home seems to oscillate slowly, will-lessly, and reversibly between stillness and sensation. It seems that only the status of sensation allows the initial phenomenon of the said transition. This initial phenomenon is a move of time. By a move of time, a present sensation gives way to another present sensation in such a way that consciousness retains the former one as a past sensation, and moreover, through this distinction between present and past, recedes from both and from stillness, and becomes mind. Mathematics comes into being, when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common substratum of all two-ities, as basic intuition of mathematics, is left to an unlimited unfolding, creating new mathematical entities in the shape of predeterminately, or more or less freely proceeding infinite sequences of mathematical entities acquired, and in the shape of mathematical species.

Historical background, principles and methods of intuitionism

Abstract: Publisher Summary This chapter discusses the historical background, principles, and methods of intuitionism. The historical development of the mental mechanism of mathematical thought is naturally closely connected with the modifications which, in the course of history, have come about in the prevailing philosophical ideas firstly concerning the origin of mathematical certainty; secondly, concerning the delimitation of the object of mathematical science. The mental mechanism of mathematical thought during so many centuries has undergone a few fundamental changes because of the circumstance that, in spite of all revolutions undergone by philosophy in general, the belief in the existence of properties of time and space, immutable, and independent of language, and experience, remained intact until far into the 19th century. It is found that for the elementary theory of natural numbers, the principle of complete induction, and more or less considerable parts of algebra and theory of numbers, exact existence, absolute reliability, and noncontradictority were universally acknowledged independently of language and without proof.

Ernest W. Adams. Subjunctive and indicative conditionals. Foundations of language, vol. 6 (1970), pp. 89–94.

Abstract: The purpose of this note is to dispute Michael Ayers' claim that "there is no special problem of subjunctive conditionals".' Ayers argues for the following theses: (1) there is no special problem of counterfactual conditionals, (2) the subjunctive conditional should not be confused with the counter factual, (3) the subjunctive conditional does not, as many authors have argued, differ from its indicative counterpart in that the former logically entails either that its antecedent is false, or that the speaker believes it to be false, whereas the latter does not, and (4) there appear to be no special problems in attempting to characterize the conditions of verification and falsification of subjunctive conditionals which are not shared equally by indicative con ditionals. Now, I agree with theses (1)-(3), but I feel that thesis (4) is mis leadingly put, and when it is less misleadingly formulated, it is false. What I wish to argue is that subjunctive and indicative conditionals differ not so much as to their conditions of verification and falsification, as in the degrees to which they are justified or supported by evidence. There are occasions on which subjunctive conditionals are very well justified by evidence, but the corresponding indicatives are quite unjustified. Furthermore, the kind of justification here spoken of is fundamental to a characterization of 'the logic' of these sorts of statements. First I will exhibit three closely related examples of situations in which a subjunctive is justified, but its corresponding indicative is not. Then I will speculate briefly on the significance of these examples. The upshot will be to conclude that subjunctive and indicative conditionals are indeed logically distinct species, and there remains a special problem of analyzing subjunctives even after the indicatives are analyzed. A hypothetical example of the kind I have in mind is the following. Sup pose that on a given occasion three persons, A, B and V, are isolated in a room which is sealed off from the outside. During this time, the third person, V ('the victim'), is murdered by being shot through the heart. The circum stances are such that only A and B could have done the shooting, though both deny it and accuse the other, and no one else witnessed the murder. An in vestigation is therefore instituted. It establishes that A had in fact a very

On Spectra, and the Negative Solution of the Decision Problem for Identities having a Finite Nontrivial Model

Abstract: An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By “nontrivial” it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a “recursive algorithm” as the term is used today.In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (the language of groupoids), the set of identities that have no nontrivial model, is recursively inseparable from the set of identities such that the sentence has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algorithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskiǐ [14] and by Perkins [17]. The other parts of the corollary seem to be new.

Categories of Frames for Modal Logic

Abstract: §1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = { y ∣ (∃ x )( y x Є X )}; and if is a CAMA then is a frame, where is the set of atoms of and b 1 b 2 ⇔ b 1 ∩ mb 2 ≠∅. Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence. The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames. The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].

Forcing in Model Theory

Abstract: The forcing concept of Paul J. Cohen has had an immense effect on the development of Axiomatic Set Theory but it also possesses an obvious general significance. It therefore was to be expected that it would have an impact also on general Model Theory. In the present talk, I shall show that this expectation is indeed justified and that the forcing notion provides us with a new tool in Model Theory, which leads to a better understanding of concepts that have by now become classical in this area and to their further development.

First-Order Definability in Modal Logic

Abstract: In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp . Definition 1. A modal frame ℱ = 〈 W, R 〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

The Inadequacy of the Neighbourhood Semantics for Modal Logic

Abstract: We present two finitely axiomatized modal propositional logics, one between T and S 4 and the other an extension of S 4, which are incomplete with respect to the neighbourhood or Scott-Montague semantics. Throughout this paper we are referring to logics which contain all the classical connectives and only one modal connective □ (unary), no propositional constants, all classical tautologies, and which are closed under the rules of modus ponens (MP), substitution, and the rule RE (from A ↔ B infer α A ↔ □ B ). Such logics are called classical by Segerberg [6]. Classical logics which contain the formula □ p ∧ □ q → □( p ∧ q ) (denoted by K ) and its “converse,” □{ p ∧ q )→ □ p ∧ □ q (denoted by R ) are called regular; regular logics which are closed under the rule of necessitation, RN (from A infer □ A ), are called normal . The logics that we are particularly concerned with are all normal, although some of our results will be true for all regular or all classical logics. It is well known that K and R and closure under RN imply closure under RE and also that normal logics are also those logics closed under RN and containing □{ p → q ) → {□ p → □ q ).

Extending Gödel's negative interpretation to ZF

Abstract: In [5] Godel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Godel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Godel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

On Partitions into Stationary Sets

Abstract: We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L . We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω 1 . This is an extension of a result of Silver who proved it for λ = ω 2 , providing a partial answer to Problem 68 of Friedman [2]. The main results of this paper were obtained independently by both authors. If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ. Consider the statements: (J 1 ) There is a class E of limit ordinals and a sequence C λ defined on singular limit ordinals λ such that (i) E ⋂ μ is Mahlo in μ for all regular > ω; (ii) C λ is closed and unbounded in λ; (iii) if γ C λ , then γ is singular, γ ∉ E and C γ = γ ⋂ C λ . For each infinite cardinal κ: (J 2,κ ) There is a set E ⊂ κ + and a sequence C λ (Lim(λ), λ + ) such that (i) E is Mahlo in κ + ; (ii) C λ is closed and unbounded in λ; (iii) if cf(λ) C λ (iv) if γ C λ then γ ∉ E and C γ = ϣ ⋂ C λ .

A Simplification of the Bachmann Method for Generating Large Countable Ordinals

Abstract: In [2] Bachmann showed how a hierarchy of normal functions on the countable ordinals Ω can be constructed using certain uncountable ordinals together with appropriate fundamental sequences for limit ordinals, as indexing ordinals for the hierarchy. This method, explained in more detail in §1 below, has been generalized by Pfeiffer [6] and Isles [4] to produce larger hierarchies. Using the hierarchies constructed in this way it is possible, as an immediate consequence of the definitions, to assign ω -sequences 〈α n 〉 n ∈ω to limit ordinals α in an initial segment I of Ω such that . Indeed this was the motivation for Bachmann's original construction. However the initial segments I constructed in this way are of interest even without the fundamental sequences because they occur naturally as the proof-theoretic ordinals associated with particular formal theories. The fundamental sequences necessary for the Bachmann method are not intrinsically of interest proof-theoretically and only serve to obscure the definitions of the associated initial segments. Feferman and Weyhrauch [9] suggested an alternative definition for a sequence of functions on the countable ordinals. This definition was generalized by Aczel [1] who showed how the new functions corresponded to those in Bachmann's hierarchy. (Independently, Weyhrauch established the same results for an initial segment of the sequence of Bachmann functions, i.e. those functions indexed by α e Ω+1 .)

Canonical Partition Relations

Abstract: Several canonical partition theorems are obtained, including a simultaneous generalization of Neumer's lemma and the Erdos-Rado theorem. The canonical partition relation for infinite cardinals is completely determined, answering a question of Erdos and Rado. Counterexamples are given showing that in several ways these results cannot be improved.

A Note on Modal Formulae and Relational Properties

Abstract: Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators □ and ⋄. The semantic structures are frames , i.e., pairs W, R > with R ⊆ W 2 . Let F, V be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of W . Then the relation may be defined, for arbitrary formulae α, following the Kripke truth-definition. From this relation we may further define Now, to every modal formula α there corresponds some property P α of R . A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first-order predicate. For these particular P α we have for all F and w ∈ W . These formulae P α are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when P α may be taken to be some first-order formula. For example, it can be seen that for all F and w ∈ W . It is customary to talk about a related correspondence, namely when for all F we have Note that this correspondence holds whenever the first one above holds.

A survey of partial degrees

Abstract: Partial degrees are equivalence classes of partial natural number functions under some suitable extension of relative recursiveness to partial functions. The usual definitions of relative recursiveness, equivalent in the context of total functions, are distinct when extended to partial functions. The purpose of this paper is to compare the upper semilattice structures of the resulting degrees.Relative partial recursiveness of partial functions was first introduced in Kleene [2] as an extension of the definition by means of systems of equations of relative recursiveness of total functions. Kleene's relative partial recursiveness is equivalent to the relation between the graphs of partial functions induced by Rogers' [10] relation of relative enumerability (called enumeration reducibility) between sets. The resulting degrees are hence called enumeration degrees. In [2] Davis introduces completely computable or compact functionals of partial functions and uses these to define relative partial recursiveness of partial functions. Davis' functionals are equivalent to the recursive operators introduced in Rogers [10] where a theorem of Myhill and Shepherdson is used to show that the resulting reducibility, here called weak Turing reducibility, is stronger than (i.e., implies, but is not implied by) enumeration reducibility. As in Davis [2], relative recursiveness of total functions with range ⊆{0, 1} may be defined by means of Turing machines with oracles or equivalently as the closure of initial functions under composition, primitive re-cursion, and minimalization (i.e., relative μ-recursiveness). Extending either of these definitions yields a relation between partial functions, here called Turing reducibility, which is stronger still.

$\aleph_0$-Categorical Modules

Abstract: It is shown that the first-order theory Th R ( A ) of a countable module over an arbitrary countable ring R is ℵ 0 -categorical if and only if A i finite, n ∈ ω , κ i ≤ ω . Furthermore, Th R (A) is ℵ 0 -categorical for all R -modules A if and only if R is finite and there exist only finitely many isomorphism classes of indecomposable R -modules.

Descending Sequences of Degrees

Abstract: Our unexplained notation is that of Rogers [4]. Let P ⊆ 2 N × 2 N . We call a sequence A n : n ∈ N > of subsets of N a P-sequence iff ∀n(A n +1 = the unique B such that P(A n , B)) . Theorem. Let P ⊆ 2 N × 2 N be arithmetical. Then there is no P-sequence n : n ∈ N> such that ∀n(A′ n +1 ≤ T A n ) . This theorem improves a result of Friedman [2] who showed that for no arithmetical P is there a P -sequence A n : n ∈ N > such that A n + 1 is a code for an ω-model of the relative arithmetic comprehension schema, and A n + 1 is present in the model coded by A n , for all n . Other related results are those of Harrison [3], who showed there is a sequence A n : n ∈ N > such that ∀n n + 1 ≤ T A n >, and of Enderton and Putnam [1], who showed there is no sequence A n : n ∈ N > with ∀n(A′ n + 1 ≤ T A n ) and A 0 hyperarithmetic. Our theorem is closely connected to Godel's second incompleteness theorem. Its proof is a recursion theoretic parallel to the proof of Godel's theorem. In §2 we draw a version of Godel's theorem as a corollary to ours.

Bounds for the closure ordinals of replete monotonic increasing functions

Abstract: Whenever particular ordinals are used as tools in a proof or a definition, it is necessary to find a way of representing them. If the ordinals are sufficiently small, there is a standard way (e.g. Cantor normal forms for ordinals less than e 0 ); in general, representations are often found by using functions on initial segments of the ordinals: Each term which can be obtained from by applications of a function symbol is regarded as a notation for the ordinal obtained by the same applications of the function f to the ordinal 0. In this way, f provides representations for all the ordinals in Cl f (0), the closure set of f (se e §1). (For an introduction to and development of this principle, see Feferman [F1]; and for a discussion of the significance of such representations in proof theory, see Kreisel [K1, pp. 22–34].) Thus it is natural to ask whether there are connections between frequently encountered properties of ordinal functions and the size of the ordinals for which they can provide representations. The purpose of this paper is to show that, for any integer n , the ordinal (see §2) is a bound for the closure ordinals of replete monotonic increasing n -place functions. This result is optimal for n > 2 (the bound is attained by where θα = 1 + α ) but not for n n = 2 is e 0 . (Trivially, that for n = 1 is φ .)

Uniform Enumeration Operations

Abstract: Sacks [2] has asked whether there exists a uniform solution to Post's problem, i.e. an enumeration operation W such that d W(d) d ′ for every degree d . It is shown here that if such an operation W exists it cannot itself in a particular technical sense be uniform. In fact, the jump operation is characterized amongst such uniform enumeration operations by the condition: d W(d) for all d . In addition, it is proved that the only other uniform enumeration operations such that d ≤ W(d) for all d are those which equal the identity operation above some fixed degree.

The word problem for free fields: a correction and an addendum

Abstract: In [1] it was claimed that the word problem for free fields with infinite centre can be solved. In fact it was asserted that if K is a skew field with infinite central subfield C , then the word problem in the free field on a set X over K can be solved, relative to the word problem in K . As G. M. Bergman has pointed out (in a letter to the author), it is necessary to specify rather more precisely what type of problem we assume to be soluble for K : We must be able to decide whether or not a given finite set in K is linearly dependent over its centre. This makes it desirable to prove that the free field has a corresponding property (and not merely a soluble word problem). This is done in §2; interestingly enough it depends only on the solubility of the word problem in the free field (cf. Lemma 2 and Theorem 1′ below). Bergman also notes that the proof given in [1] does not apply when K is finite-dimensional over its centre; this oversight is rectified in §4, while §3 lifts the restriction on C (to be infinite). However, we have to assume C to be the precise centre of K , and not merely a central subfield, as claimed in [1]. I am grateful to G. M. Bergman for pointing out the various inaccuracies as well as suggesting remedies.