Showing papers in "Journal of Symbolic Logic in 1977"

Naming, Necessity, and Natural Kinds

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The Semantics of Entailment.

Abstract: Once upon a time, modal logics “had no semantics”. Bearing a real world G, a set of worlds K, and a relation R of relative possibility between worlds, Saul Kripke beheld this situation and saw that it was formally explicable, and made model structures. It came to pass that soon everyone was making model structures, and some were deontic, and some were temporal, and some were epistemic, according to the conditions on the binary relation R. None of the model structures that Kripke made, nor that Hintikka made, nor that Thomason made, nor that their co-workers and colleagues made, were, however, relevant. This caused great sadness in the city of Pittsburgh, where dwelt the captains of American Industry. The logic industry was there represented by Anderson, Belnap & Sons, discoverers of entailment and scourge of material impliers, strict impliers, and of all that to which their falsehoods and contradictions led. Yea, every year or so Anderson& Belnap turned out a new logic, and they did call it E, or R, or E i , or P W, and they beheld each such logic, and they were called relevant. And these logics were looked upon with favor by many, for they captureth the intuitions, but by many more they were scorned, in that they hadeth no semantics. Word that Anderson & Belnap had made a logic without semantics leaked out. Some thought it wondrous and rejoiced,’ that the One True Logic should make its appearance among us in the Form of Pure Syntax, unencumbered by all that set-theoretical garbage. Others said that relevant logics were Mere Syntax. Surveying the situation Routley, and quite independently Urquhart, found an explication of the key concept of relevant implication. Building on Routley [ 19721 , and with a little help from our friends Dunn and Urquhart

Computational complexity, speedable and levelable sets

Abstract: 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.

On Splitting Stationary Subsets of Large Cardinals

Abstract: 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.

Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces

Abstract: ?0. Introduction. 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 Godel 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 Godel number of x by rx 1. By taking {En I n > O} 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, En corresponds to the sequence whose nth 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, ST(V.) denotes the lattice of all r.e. spaces under the operations of intersection and weak sum. For V, W E ST( 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 E S( V4), if V D 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. In this paper, we present a new method of constructing desired r.e. spaces and use it to produce maximal spaces with a variety of unexpected properties. In ?1 we show that any maximal space with extendible basis has a complemented lattice of r.e. superspaces. ?2 contains the construction of an r.e. space whose lattice of r.e. superspaces contains at least one noncomplemented element. This method of construction is exploited in ?3 to produce a space which is maximal in a stronger sense, namely no nontrivial element of its lattice of r.e. superspaces is complemented. Finally, recall R. Soare's result: for any two maximal sets of integers A and B, there exists (F, an automorphism of the

Definability of measures and ultrafilters

Abstract: 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.

Ramsey's Theorem in the Hierarchy of Choice Principles

Abstract: 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).

An Intuitionistically Plausible Interpretation of Intuitionistic Logic

Abstract: 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 “⊦ IPC A ”, 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 ⊦ IPC A ”, 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 ⊦ IPC A . 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.

Fragments of first order logic, I: universal Horn logic

Abstract: 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 Consistency Problem for NF

Abstract: 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].

Transfinite Extensions of Friedberg's Completeness Criterion

Abstract: 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 >-8 then there is a b such that bn) = 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(w) there is a b such that b ') = 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') = bw) = a U b = 0(-).) In ?3 we generalize the methods of ?2 to show that if a is a recursive ordinal and a 0(a) then there is a b such that b'a) = a, i.e. the Friedberg result can be extended to all recursive ordinal levels. Thomason [6] used a forcing argument to show: If A ?hey (the Kleene set of notations for the recursive ordinals) then there is a B such that A-h6B (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 A ?TC then there is a B such that A =TJB.

Chang's conjecture and powers of singular cardinals

Abstract: In [2] Galvin and Hajnal showed, as a corollary to a more general result, that if , is a strong limit cardinal, then . They established similar bounds for powers of singular cardinals of cofinality greater than ω. Jech and Prikry in [3] showed that the Galvin-Hajnal bound can be improved if we assume that ω 1 carries an ω 2 saturated ω 1 complete, nontrivial ideal. (See [7] for definitions), namely: under the given assumption provided is a strong limit cardinal. In this paper we show that the same conclusion can be derived from Chang's Conjecture (see below) which is, at least consistencywise, a weaker assumption than the existence of an ω 2 saturated ideal on ω 1 . We do not know if assumptions like these are necessary for obtaining the result. Our notations and terminology should be understood by any reader acquainted with set theory. Chang's Conjecture is the following model theoretic assumption introduced by C. C. Chang: which is deciphered as follows: Every structure 〈 A, R ,…〉 in a countable type where ∣ A ∣ = ω 2 , R ⊆ A , ∣ R ∣ = ω 1 has an elementary substructure: 〈 A ′, R ′,…〉 where ∣ A ′∣ = ω 1 and ∣ R ′∣ = ω 0 . The consistency of Chang's Conjecture modulo the existence of Ramsey cardinals is claimed in [5].

Imbedding of the Quantum Logic in the Modal System of Brower

Abstract: 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] ?1 Modal system We begin with the formal description of the modal system Br' The alphabet contains the signs - , v and D1 (negation, alternative and the sign of necessity), the set V of the propositional variables and parentheses The rules of formation are: if A E V, then A is a proposition; if X and Y are propositions, then - X, X v Y and DX are propositions also; there are no other propositions X3 will design the set of all propositions

Maximal and Cohesive vector spaces

Abstract: 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 ν 0 , ν 1 , …, ν n−1 of elements of U determines whether or not ν 0 , ν 1 …, ν n−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 V 1 and V 2 are subspaces of U such that V 1 ∩ V 2 ={ } (where is the zero vector of U ), then we write V 1 ⊕ V 2 for ( V 1 ∪ V 2 )*. If V 1 ⊆ V 2 ⊆ U are subspaces, we write V 2 / V 1 for the quotient space.

A Note on the Number of Zeros of Polynomials and Exponential Polynomials

Abstract: ?0. The negative solution of Hilbert's Tenth Problem brought with it a number of unsolvable Diophantine problems. Moreover, by actually providing a Diophantine characterization of recursive enumerability, the proof of the negative solution opened the door to the techniques of recursion theory. In this note, we wish to apply several recursion-theoretic facts and an improvement on the exponential Diophantine representation to refine the exponential case of a result of Davis [1972] regarding the difficulty of determining the number of zeros of a polynomial. P, Q, etc. will denote polynomials or exponential polynomials-exactly which will be clear from the context. Let # (P) denote the number of distinct nonnegative zeros of P. Further, let C = {0, 1, * * *, N4o} be the set of possible values of # (P). For A C C, we define A * to be

General random sequences and learnable sequences

Abstract: ABSTRACr. 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-Lf, 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.

The Use of Kripke's Schema as a Reduction Principle

Abstract: 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 readsThere is a weaker versionThe 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 givesOf 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.

A model theoretic approach to Malcev conditions

Abstract: A variety V (equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …, fn, x1, …, xm) where θ is a conjunction of equations in the function variables f1, …, fn and the individual variables x1, …, xm, if there are polynomial symbols p1, …, pn in the language of V such that ∀x1, …, xmθ(p1 …, pn, x1, …, xm) is a law of V. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Gratzer [1]. Since then and especially since Jonsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.

A Complete $L_{\omega 1\omega}$-Sentence Characterizing $\mathbf{\aleph}_1$

Abstract: 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.

Experimental Logics and $\Pi^0_3$ Theories

Abstract: 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].)

A sequent calculus for type assignment

Abstract: ' The first version of this paper was written in March, 1973. The subsequent revisions concern the exposition and not the mathematical results. An abstract of this paper appeared in Notices of the American Mathematical Society, vol. 20 (1973), p. A-502, no. 73T-E85. This research was supported by Illinois State appropriated funds administered by the Mathematics Department of Southern Illinois University at Carbondale. 2 The system also appears in [7, ?9C], and was originally proposed in [14, ?3E6]. 3 Thus, the equivalence with the natural deduction formulation (T-formulation) also fails unless Rule Cut is postulated for the system. Furthermore, [7, Theorems 9.19 and 9.20 and Corollary 9.20.1] are false and the proof of [7, Theorem 9.21] is invalid although the theorem can be proved another way. A correction to this part of [7, ?9C] has been indicated (without the details of the proofs, as is the basic policy of this work) in the recently published Italian edition, and will appear in any future printings of the English edition. [14, ?3E6] and [3, ?14E6] should be deleted. 4 It is true in A-calculus that the corresponding property holds; however, the possibility of using A-calculus for this system was already excluded for the reason stated in [3, ?14E6]. This defect in the proof was pointed out to me by Roger Hindley in a letter dated October 27, 1972; I found the counterexample in the process of trying to patch up the proof. 5 This is essentially the same as the system of Girard in [5, Chapter 1, ?2]; Girard's system is an extension of ordinary functionality in that it has additional primitive formation rules for types, but the sequent calculus formulation for it is constructed on the same principles as the system of [2].

Amalgamation of nonstandard models of arithmetic

Abstract: 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. Let A and B be two elementarily equivalent models. We consider models M having (isomorphic copies of) A and B as elementary submodels, and we ask about the possible relative positions of A and B inside M. "Relative position" is a vague expression; it refers, at least, to the way A and B intersect in M, but if, for example, the models are ordered then we are also interested in the order relation in M between members of A and members of B. Although we prove one theorem in this general model-theoretic context, our main concern is with models of arithmetic, i.e. elementary extensions of the standard model, which consists of the set N of natural numbers together with all relations and functions on N. We show that, for any two such models, any reasonable intersection and any reasonable ordering of elements above the intersection actually occur in some joint extension of the two models. ("Reasonable" is another vague expression; its meaning here will be explained below.) We describe some situations in which the intersection of two models, or the intersection and the relative ordering, completely determine the joint extension generated by the two models. By applying these results to ultrapowers of the standard model of arithmetic, we obtain several results about ultrafilters on N. We find situations in which the fiber product of two ultrafilters over a third is again an ultrafilter, and we clarify the connection between P-points and partition properties.

Embedding Theorems for Boolean Algebras and Consistency Results on Ordinal Definable Sets

Abstract: 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.