bluuebunny/crossref_metadata_2025 huggingface dataset:
The ℵ1-categoricity of strictly upper triangular matrix rings over algebraically closed fields
Authors: Rose Bruce I.
Published: May 2006

Let n ≥ 3. The following theorems are proved.

Theorem. The theory of the class of strictly upper triangular n × n matrix rings over fields is finitely axiomatizable.

Theorem. If R is a strictly upper triangular n × n matrix ring over a field K, then there is a recursive map σ from sentences in the language of rings with constants for K into sentences in the language of rings with constants for R such that K ⊨ φ if and only if R φ σ(φ).

Theorem. The theory of a strictly upper triangular n × n matrix ring over an algebraically closed field is ℵ1-categorical.
An ideal game
Authors: Galvin F. , Jech T. , Magidor M.
Published: May 2006

Let us consider the following infinite game between two players, Empty and Nonempty. We are given a large set S. Empty opens the game by choosing a large subset S0 of S; then Nonempty chooses a large set S1 ⊆ S0; then Empty chooses large S2 ⊆ S, etc. The game is over after ω moves. If ⋂n=0xSn is empty then Empty wins, and if ⋂n=0∞Sn is nonempty then Nonempty wins.

If “large” means “infinite”, then Empty can beat Nonempty rather easily: he chooses So countable, S0 = {a0, a1,…, an,…}, and then he chooses S2 such that a0 ∉ S2, S4 such that a1, ∉ S4 and so on.

Next we assume that S is a set of uncountable cardinality, and that “large” means “of cardinality ∣S∣”. Then still Empty can win, but his winning strategy is somewhat more sophisticated: Let us identify S with a cardinal number κ. Thus each subset of S of size κ is a set of ordinals below κ. For each X ⊆ κ of size κ, let fx be the unique order-preserving mapping of X onto κ, and let F(X) = {x ϵ X: f(x) is a successor ordinal}. Empty's strategy is to play S0 = F(K), and when Nonempty plays S2k − 1, let S2k = F(S2k − 1).
Minimal pairs and high recursively enumerable degrees
Authors: Cooper S. B.
Published: May 2006

A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].)

Theorem. Let a be a recursively enumerable degree for which a′ = 0″. Then there are recursively enumerable degrees b0 and b1 such that0 < bi < a for each i ≤ 1, and b0 ⋂ b1 = 0.

The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1].

By Robinson [3] we can choose a r.e. representative A of the degree a, with uniformly recursive tower {As, ∣ s ≥ 0} of finite approximations to A, such that CA dominates every recursive function where

We define, stage by stage, finite sets Bi,s, i ≤ 1, s ≥ 0, in such a way that Bi, s + 1 ⊇ Bi,s for each i, s, and {Bi,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.
Continuity and elementary logic
Authors: Tharp Leslie H.
Published: May 2006

The purpose of this paper is to investigate continuity properties arising in elementary (i.e., first-order) logic in the hope of illuminating the special status of this logic. The continuity properties turn out to be closely related to conditions which characterize elementary logic uniquely, and lead to various further questions.
The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces
Authors: Henson C. Ward
Published: May 2006

The basic setting of nonstandard analysis consists of a set-theoretical structure together with a map * from into another structure * of the same sort. The function * is taken to be an elementary embedding (in an appropriate sense) and is generally assumed to make * into an enlargement of [13]. The structures and * may be type-hierarchies as in [11] and [13] or they may be cumulative structures with ω levels as in [14]. The assumption that * is an enlargement of has been found to be the weakest hypothesis which allows for the familiar applications of nonstandard analysis in calculus, elementary topology, etc. Indeed, practice has shown that a smooth and useful theory can be achieved only by assuming also that * has some stronger properties such as the saturation properties first introduced in nonstandard analysis by Luxemburg [11].

This paper concerns an entirely new family of properties, stronger than the saturation properties. For each cardinal number κ, * satisfies the κ-isomorphism property (as an enlargement of ) if the following condition holds:

For each first order language L with fewer than κ nonlogical symbols, if and are elementarily equivalent structures for L whose domains, relations and functions are all internal (relative to * and ), then and are isomorphic.
The Boolean algebras of abelian groups and well-orders
Authors: Myers Dale
Published: May 2006

For any linear order type τ with first element let be the Boolean set algebra generated by the left-closed right-open (including [x, ∞)) intervals of some linear order of type τ. Let η and ω be the order types of the rationals and natural numbers respectively (when not used as an order type, ω will, as usual, be the set of nonnegative integers). We show that the Boolean algebra of the elementary theory of well-orders, i.e., the Boolean algebra of elementary classes of well-orders or, equivalently, the algebra of equivalence classes of sentences of the elementary theory of well-orders, is isomorphic to and that the Boolean algebra of the elementary theory of abelian groups is isomorphic to . Both results are obtained by applying Hanf's structure diagram technique to the work of Mostowski, Tarski, and Szmielew. One may formalize discussion of algebras of proper classes by assuming the classes are included in a universe which is a set in a larger universe.
Given a Boolean algebra, let 0 be its zero and 1 its unit, let ≤ be its associated partial ordering, and, for any elements a, b, and c of the algebra, let “a + b = c ” be the assertion that c is the disjoint sum of a and b. A subset of the algebra disjointly generates the algebra iff each of the algebra's elements is a disjoint sum of a finite number of the subset's elements.
The finite controllability of the Maslov case
Authors: Aanderaa Stål , Goldfarb Warren D.
Published: May 2006

In this paper we show the finite controllability of the Maslov class of formulas of pure quantification theory (specified immediately below). That is, we show that every formula in the class has a finite model if it has a model at all. A signed atomic formula is an atomic formula or the negation of one; a binary disjunction is a disjunction of the form A1 ⋁ A2, where A1 and A2 are signed atomic formulas; and a formula is Krom if it is a conjunction of binary disjunctions. Finally, a prenex formula is Maslov if its prefix is ∃···∃∀···∀∃···∃ and its matrix is Krom.

A number of decidability results have been obtained for formulas classified along these lines. It is a consequence of Theorems 1.7 and 2.5 of [4] that the following are reduction classes (for satisfiability): the class of Skolem formulas, that is, prenex formulas with prefixes ∀···∀∃···∃, whose matrices are conjunctions one conjunct of which is a ternary disjunction and the rest of which are binary disjunctions; and the class of Skolem formulas containing identity whose matrices are Krom. Moreover, the following results (for pure quantification theory, that is, without identity) are derived in [1] and [2]: the classes of prenex formulas with Krom matrices and prefixes ∃∀∃∀, or prefixes ∀∃∃∀, or prefixes ∀∃∀∀ are all reduction classes, while formulas with Krom matrices and prefixes ∀∃∀ comprise a decidable class. The latter class, however, is not finitely controllable, for it contains formulas satisfiable only over infinite universes. The Maslov class was shown decidable by Maslov in [11].
Completeness properties of heyting's predicate calculus with respect to re models
Authors: Gabbay Dov M.
Published: May 2006

Validity in recursive structures was investigated by several authors. Kreisel [10] has shown that there exists a consistent sentence of classical predicate calculus (CPC) that does not possess a recursive model. The sentence is a conjunction of the axioms of a variant of Bernays set theory, including the axiom of infinity. The language contains additional constants besides ϵ. Later Kreisel [2] and Mostowski [3] presented a sentence (not possessing recursive models) which was a conjunction of axioms of a variant of Bernays set theory without the axiom of infinity but still with additional constants besides ϵ. Later Mostowski [4] improved the result by giving a sentence which can be demonstrated in Heyting arithmetic to be consistent and to have no recursive models. Rabin [6] obtained a simple proof that some sentence of set theory with the single nonlogical constant ϵ does not have any recursively enumerable models.

More generally, Mostowski [5] has shown that the set of all sentences valid in all RE models is not arithmetical and Vaught [1] improved this result by showing that it holds for a language of one binary relation. In fact, Vaught gives· a way of translating n-place relations to 2-place ones that preserves the RE characteristic of the model. For further results pertaining to recursive models see Vaught [1].
Negative-existentially complete structures and definability in free extensions
Authors: Weispfenning Volker
Published: May 2006

Let R be a commutative ring with 1 and R[X1, …, Xn] the polynomial ring in n variables over R. Then for any relation f(X) = 0 in R[X] there exists a conjunction of equations φf such that f(X) = 0 holds in R[X] iff φf holds in R; φf is of course the formula saying that all the coefficients of f(X) vanish. Moreover, φf is independent of R and formed uniformly for all polynomials f up to a given formal degree. In this paper we investigate first order theories T for which a similar phenomenon holds. More precisely, we let TAH be the universal Horn part of a theory T and look at free extensions of models of T in the class of models of TAH. We ask whether an atomic relation t1(X, a) = t2(X, a) or R(t1(X, a), …, tn(X, a)) in can be equivalently expressed by a finite or infinitary formula φ(a) in , such that φ(y) depends only on ti{X, y) and not on or a1, …, am ∈ A.

We will show that for a wide class of theories T “defining formulas” φ(y) in this sense exist and can be taken as infinite disjunctions of positive existential formulas.
Uniform inductive definability and infinitary languages
Authors: Nyberg Anders M.
Published: May 2006

Introduction. The purpose of this paper is to show how results from the theory of inductive definitions can be used to obtain new compactness theorems for uncountable admissible languages. These will include improvements of the compactness theorem by J. Green [9].

In [2] Barwise studies admissible sets satisfying the Σ1-compactness theorem. Our approach is to consider admissible sets satisfying what could be called the abstract extended completeness theorem, that is, sets where the consequence relation of the admissible fragment LA is Σ1-definable over A. We will call such sets Σ1-complete. For countable admissible sets, Σ1-completeness follows from the completeness theorem for LA.

Having restricted our attention to Σ1-complete sets we are led to a stronger notion also true on countable admissible sets, namely what we shall call uniform Σ1-completeness. We will see that this notion can be viewed as extending to uncountable admissible sets, properties related to both the “recursion theory” and “proof theory” of countable admissible sets.

By following Barwise's recent approach to admissible sets allowing “urelements,” we show that there is a natural connection between certain structures arising from the theory of inductive definability, and uniformly Σ1-complete admissible sets . The structures we have in mind are called uniform Kleene structures.
The truth about some Post numbers
Authors: Segerberg Krister
Published: May 2006

There is a mistake in [6], discovered by Robert V. Kohn: contrary to the claim on p. 714, the set {◊Xi: i ≥ 2} is not independent in S3. As a result three problems thought to be settled are open once more, viz., whether the Post numbers of S2, S3, and K4 are denumerable. In [6] it was claimed they are not. Now we shall prove that that claim is indeed correct for S2 and K4 but, surprisingly enough, not for S3.
Ramsey's theorem and recursion theory
Authors: Jockusch Carl G.
Published: May 2006

Let N be the set of natural numbers. If A ⊆ N, let [A]n denote the class of all n-element subsets of A. If P is a partition of [N]n into finitely many classes C1, …, Cp, let H(P) denote the class of those infinite sets A ⊆ N such that [A]n ⊆ Ci for some i. Ramsey's theorem [8, Theorem A] asserts that H(P) is nonempty for any such partition P. Our purpose here is to study what can be said about H(P) when P is recursive, i.e. each Ci, is recursive under a suitable coding of [N]n. We show that if P is such a recursive partition of [N]n, then H(P) contains a set which is Πn0 in the arithmetical hierarchy. In the other direction we prove that for each n ≥ 2 there is a recursive partition P of [N]n into two classes such that H(P) contains no Σn0 set. These results answer a question raised by Specker [12].

A basic partition is a partition of [N]2 into two classes. In §§2, 3, and 4 we concentrate on basic partitions and in so doing prepare the way for the general results mentioned above. These are proved in §5. Our “positive” results are obtained by effectivizing proofs of Ramsey's theorem which differ from the original proof in [8]. We present these proofs (of which one is a generalization of the other) in §§4 and 5 in order to clarify the motivation of the effective versions.
Which number theoretic problems can be solved in recursive progressions on Π11-paths through O?
Authors: Kreisel G.
Published: May 2006

This note is an addendum to an incompleteness result of Feferman and Spector [1]: for any such progression satisfying the (natural) conditions listed in [1] there is a true Π10-formula which cannot be derived in the progression. The idea of their proof yields a stronger result.

If each numerical instance A[x/n] of the formula A (with the free numerical variable x) is decided in a recursive progression along a Π11-path then the set {n: ⊦ A[x/n]} is recursive.

In other words, only recursively solvable problems are decidable in such progressions. This yields the incompleteness result of [1] when specialized to some Π10 formula A which defines a nonrecursive set, since then some instance A[x/n] must be undecided and A[x/n] is a closed Π10 formula. The stronger result is needed to extend Church's thesis for total functions reckonable in formal systems to (total) functions reckonable in recursive progressions on Π11-paths. A partially reckonable function always has a Π11 graph (and if its graph is Δ11 then it is by C. Jockusch, r.e.; so some Π11 sets are not partially reckonable). The relevance of these facts for the model of mathematical reasoning provided by (variants of) Turing's ordinal logics is analyzed in (b) (ii) of Part II, which goes into our current knowledge about Church's thesis.
Redundancies in the Hilbert-Bernays derivability conditions for Gödel's second incompleteness theorem
Authors: Jeroslow R. G.
Published: May 2006

This paper presents three generalizations of the Second Incompleteness Theorem of Gödel (see [2]) which apply to a broader class of formal systems than previous generalizations (as, e.g., the generalization in [1]).

The content of all three of our results is that it is versions of the third derivability condition of Hilbert and Bernays (see p. 286 of [3]) which are crucial to Gödel's theorem. The second derivability condition plays some role in certain logics, but even there, only in the far weaker form of a definability condition (see Theorem 2).

The elimination of the first derivability condition allows the application of the Consistency Theorem to cut-free logics which cannot prove that they are closed under cut.

It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. The derivation of this result requires a “new twist” on old arguments. Specifically, we use a new variation on the standard self-referential lemma (this is Lemma 5.1 of [l]) to obtain a somewhat different self-referential construction than has previously been employed (this is our lemma in §2 below). With this new construction, we consider the sentence φ which “expresses” the fact: “My negation is provable.” Then, using hypotheses associated with Gödel's First Incompleteness Theorem, one shows that ⊢¬φ is impossible in a consistent logic. Finally, using only a version of the third Hilbert-Bernays derivability condition, one shows that the provability of the consistency statement implies ⊢¬φ, and hence that consistency is unprovable.
Post's problem and his hypersimple set
Authors: Jockusch Carl G. , Soare Robert I.
Published: May 2006

A standard enumeration of the recursively enumerable (r.e.) sets is an acceptable numbering {Wn}n∈N of the r.e. sets in the sense of Rogers [5, p. 41], together with a 1:1 recursive function f with range In his quest for nonrecursive incomplete r.e. sets Post [4] constructed a hypersimple set Hf, relative to a fixed but unspecified standard enumeration f. Although it was later shown that hyper-simplicity does not guarantee incompleteness, the ironic possibility remained that Post's own particular hypersimple set might be incomplete. We settle the question by proving that H, may be either complete or incomplete depending upon which standard enumeration f is used. In contrast, D. A. Martin has shown [3] that Post's simple set S [4, p. 298] is complete for any standard enumeration. Furthermore, what most modern recursion theorists would regard as the “natural” construction of a hypersimple set (which we give in §1) is also complete for any standard enumeration.

There are two conclusions to be drawn from these results. First, they substantiate the often repeated remark among recursion theorists that Post's hypersimple set construction is a precursor of priority constructions because priorities play a strong role, and because there is a great deal of “restraint” which tends to keep elements out of the set. Secondly, the results warn recursion theorists that more properties than might have been supposed depend upon which standard enumeration is chosen at the beginning of the construction of some r.e. set.
Reflecting stationary sets
Authors: Magidor Menachem
Published: May 2006

We prove that the statement “For every pair A, B, stationary subsets of ω2, composed of points of cofinality ω, there exists an ordinal α such that both A ∩ α and B ∩ α are stationary subsets of α is equiconsistent with the existence of weakly compact cardinal. (This completes results of Baumgartner and Harrington and Shelah.)

We also prove, assuming the existence of infinitely many supercompact cardinals, the statement “Every stationary subset of ωω+1 has a stationary initial segment.”
A transfer principle in harmonic analysis
Authors: Takeuti Gaisi
Published: May 2006

Let G be a locally compact abelian group, Γ its dual group, and μ its Haar measure. For a function f: G → C, the Fourier transform of f is defined by

for every γ ∈ Γ.

We extend this definition for a function f on G, whose values are pairwise commutable normal operators in a Hilbert space. Then we study harmonic analysis for this extended Fourier transform.

Our method is Boolean valued analysis as introduced in [11] and [12]. Instead of developing the theory in a step-by-step manner, we shall develop a general machinery showing how to transform classical theorems to theorems in our situation.

In Chapter 1, we summarize the basic knowledge on Hilbert space, on Boolean valued model of set theory, and on Boolean valued analysis. In Chapter 2, we develop the theory of integration. Since we deal with unbounded operators as well as bounded operators, we need a new theory of integration. For a separable Hilbert space, the value of our integration coincides with the value of the usual integration with an adequate generalization for unbounded operators.

In Chapter 3, we establish machinery for our transfer principle and carry out the transformation of many classical theorems to theorems for our case.
The Baire category theorem and cardinals of countable cofinality
Authors: Miller Arnold W.
Published: May 2006

Let κB be the least cardinal for which the Baire category theorem fails for the real line R. Thus κB is the least κ such that the real line can be covered by κ many nowhere dense sets. It is shown that κB cannot have countable cofinality. On the other hand it is consistent that the corresponding cardinal for 2ω1 be ℵω. Similar questions are considered for the ideal of measure zero sets, other ω1, saturated ideals, and the ideal of zero-dimensional subsets of Rω1.
Abstract logic and set theory. II. Large cardinals
Authors: Väänänen Jouko
Published: May 2006

The following problem is studied: How large and how small can the Löwenheim and Hanf numbers of unbounded logics be in relation to the most common large cardinals? The main result is that the Löwenheim number of the logic with the Härtig-quantifier can be consistently put in between any two of the first weakly inaccessible, the first weakly Mahlo, the first weakly compact, the first Ramsey, the first measurable and the first supercompact cardinals.
A strengthening of Jensen's □ principles
Authors: Beller Aaron , Litman Ami
Published: May 2006

The aim of this paper is to prove strengthenings of three theorems appearing in Jensen [1].
Beth's property fails in L<ω
Authors: Badger Lee
Published: May 2006

In this paper we prove that Beth's property does not hold in L<ω. This answers a question posed by Magidor and Malitz in [8]. Beth's property is a natural closure condition on a language which says that everything implicitly definable in the language is also explicitly definable in the language. That the first-order predicate calculus (L0) has the property was first shown by Beth [4]. Lopez-Escobar proved that also has Beth's property [7]. Malitz [9] showed the Beth's property fails in Lκλ where κ ≥ λ ≥ ω1. Friedman and Silver showed that the property fails in Lκλ for κ > ω1. Also Friedman [5] showed that extensions of elementary logic using cardinality quantifiers (L)1κ do not have Beth's property.

The undefined notation used here is standard. If further clarification is needed, we refer the reader to [3]. κ and λ denote infinite cardinals. cX and ȣX∣ denote the cardinality of X. cfμ will denote the cofinality of the order μ. All languages discussed are assumed to have no function or constant symbols. All structures are relational. The type of a formula is the set of relation symbols appearing in the formula (excluding equality). The type of a set of formulas is the set of all relation symbols appearing in some formula of the set of formulas. For purpose of this paper we will assume that to each n-ary relation in a structure there is associated an n-ary relation symbol and that in any given structure this association is one-to-one. Using this convention we can define the type of a structure to be the set of those relation symbols to which there is associated a relation in the structure. Sometimes we use the term predicate instead of relation.
On the bounded monadic theory of well-ordered structures
Authors: Thomas Wolfgang
Published: May 2006

Monadic (second-order) theories of well-orderings were first studied by Büchi [1], [2], [3] using concepts of automata theory. There it was shown that the monadic theory of ω, the monadic theory of any countable ordinal, and the monadic theory of ω1 are decidable. Expansions of the well-ordering (ω, <) by further relations were considered in [4], [5], [8] and [9], for example. Concerning such expansions, Buchi and Landweber [4] asked whether there is a set P ⊂ ω such that the weak monadic theory of (ω, <, P) is decidable and the (strong) monadic theory of (ω, <, P) is undecidable. In this note we give a negative answer by proving the following general theorem: If α is an ordinal and an n-tuple of subsets of α, then the monadic theory of (α, <, ) is decidable provided the monadic theory of (cf (α), <), i.e. of the cofinality of α, and the bounded monadic theory of (or, <, ) are decidable. (In the bounded monadic theory the second-order variables range only over bounded subsets of α.) Also we show that in the bounded and the (strong) monadic theory of a structure (α, <, ) the same classes of subsets of α are definable. For the proofs we use a result of Shelah [7] and a suitable version of a combinatorial argument which was introduced by Büchi [1] and McNaughton [6] into the study of monadic theories.
Forcing with Δ perfect trees and minimal Δ-degrees
Authors: Kechris Alexander S.
Published: May 2006

This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper.

In §1, we use the general method of [3] together with reflection arguments to study the properties of forcing with Δ perfect trees, for certain Spector pointclasses Γ, obtaining as a main result the existence of a continuum of minimal Δ-degrees for such Γ's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal Δ½n+1-degrees, for all n.

Following an idea of Leo Harrington, we extend these results in §2 to show the existence of minimal strict upper bounds for sequences of Δ-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of Δ½n+1-degrees is equal to ω when n ≥ 1. (By results of Sacks and Richter the length of the natural hierarchy of -degrees is known to be equal to the first recursively inaccessible ordinal.)

We will follow in this paper standard notation and terminology, as in Moschovakis' book [7]. Letters e, i, j, k, l, m, n vary over the set of natural numbers ω, a, b, c over the Cantor space 2ω and α, β, γ, δ, … over the set of reals ωω. Finally ξ, η, κ, λ always denote ordinals.
A proof of the associated sheaf theorem by means of categorical logic
Authors: Veit Barbara
Published: May 2006

The double nature, both logical and geometrical, of topos theory is one of the most fascinating aspects of this discipline. Consequently, it might be of some interest that an essentially “geometric” fact such as the associated sheaf theorem admits a proof entirely based on methods of categorical logic.

The idea of this proof comes from a previous paper [V], where the same technique was used in the context of Grothendieck topoi. That paper used a generalized notion of forcing which leads directly from classical Tarski semantics in sets to Kripke-Joyal semantics in an arbitrary Grothendieck topos and gives a precise description of the links between the two. On account of this very close relationship, we thus could establish various basic facts on Grothendieck topoi without an extensive use of categorical methods, simply by viewing given subobjects as interpretations of appropriate formulas written in the formal language that had been used all along.
A Gentzen- or Beth-type system, a practical decision procedure and a constructive completeness proof for the counterfactual logics VC and VCS
Authors: de Swart H. C. M.
Published: May 2006

In [1] and [2] D. Lewis formulates his counterfactual logic VC as follows. The language contains the connectives ∧, ∨, ⊃, ¬ and the binary connective ≤. A ≤ B is read as “A is at least as possible as B”. The following connectives are defined in terms of ≤.

A < B: = ¬(B ≤ A) (it is more possible that A than that B).

◊ A ≔ ¬(⊥ ≤ A) (⊥ is the false formula; A is possible).

□ A ≔ ⊥ ≤ ¬A (A is necessary).

(if A were the case, then B would be the case).

(if A were the case, then B might be the case).

and are two counterfactual conditional operators. (AB) iff ¬(A ¬B).

The following axiom system VC is presented by D. Lewis in [1] and [2]: V: (1) Truthfunctional classical propositional calculus.
A note on cardinal exponentiation
Authors: Shelah Saharon
Published: May 2006

Silver and subsequently Galvin and Hajnal, got bounds on , for ℵα strong limit cardinal of cofinality > ℵ0. We somewhat improve those results.
Recursion theory and the lambda-calculus
Authors: Byerly Robert E.
Published: May 2006

A semantics for the lambda-calculus due to Friedman is used to describe a large and natural class of categorical recursion-theoretic notions. It is shown that if e1 and e2 are gödel numbers for partial recursive functions in two standard ω-URS's which both act like the same closed lambda-term, then there is an isomorphism of the two ω-URS's which carries e1 to e2.
Intermediate β-r.e. degrees and the half-jump
Authors: Homer Steven
Published: May 2006

In his thesis [2] Sy Friedman showed that for any inadmissible ordinal β there is an easily definable β-r.e. degree, 01/2, with β-degree strictly between 0 and 0′. In this paper we define, for some weakly admissible β, a β-r.e. degree 03/4 intermediate between 01/2 and 0′. This definition is given using the machinery developed by Maass [6], [7] to study the β-r.e. degrees.

A weak jump for β-recursion theory, the ½-jump, was first defined by Friedman [4] and has been studied by Sacks and Homer [5]. The degree 03/4 is of interest because of the way it interacts with the half-jump. In particular 03/4 forms the boundary between the half-jump of “hyperregular” and “nonhyperregular” β-r.e. degrees less than 01/2. One consequence of these results is that there are weakly inadmissible β for which either the jump theorem or the density theorem fails.

Much of this paper is based on earlier work in this area by Sy Friedman and Wolfgang Maass. I would like to thank Gerald Sacks for many helpful conversations and comments concerning this work.
Analytic determinacy and 0#
Authors: Harrington Leo
Published: May 2006

Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this:

Theorem. If analytic games are determined, then x2 exists for all reals x.

This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π11-determinacy (where α − Π11 is the αth level of the difference hierarchy based on − Π11 see [1]). Martin has also shown that the existence of sharps implies < ω2 − Π11-determinacy.

Our method also produces the following:

Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x.

The converse to this theorem had been previously proven by Steel [7], [18].

We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results.

For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16].

Throughout this paper we will concern ourselves only with methods for obtaining 0# (rather than x# for all reals x). By relativizing our arguments to each real x, one can produce x2.
A r-maximal vector space not contained in any maximal vector space
Authors: Remmel J.
Published: May 2006

In [4], Metakides and Nerode define a recursively presented vector space V∞. over a (finite or infinite) recursive field F to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V∞ becomes a vector space. Throughout this paper, we will identify V∞ with N, say via some fixed Gödel numbering, and assume V∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Given a subspace W of V∞, we write dim(W) for the dimension of W. Given subspaces V and W of V∞, V + W will denote the weak sum of V and W and if V ∩ W = {0) (where 0 is the zero vector of V∞), we write V ⊕ W instead of V + W. If W ⊇ V, we write W mod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is a r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case that A is extendible.
Interpolation theorems for Lk,k2+
Authors: Ferro Ruggero
Published: May 2006

Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and ω1-saturated models.

In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.

In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to Lα+,ω languages with interpolant in Lα+,α where α is a strong limit cardinal of cofinality ω.

In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].

Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].
Forcing and reducibilities
Authors: Odifreddi Piergiorgio
Published: May 2006

We see far away, Newton said, if we stand on giants' shoulders. We take him seriously here and moreover (as appropriate to recursion-theorists) we will jump from one giant to another, since this paper is mostly an exegesis of two fundamental works: Feferman's Some applications of the notions of forcing and generic sets [4] and Sacks' Forcing with perfect closed sets [19]. We hope the reader is not afraid of heights: our exercises are risky ones, since the two giants are in turn on the shoulders of others! Feferman [4] rests on the basic works of Cohen [2], who introduced forcing with finite conditions in the context of set theory; Sacks [19] relies on Spector [24], who realized—in recursion theory—the necessity of more powerful approximations than the finite ones.

To minimize the risk we will try to keep technicalities to a minimum, choosing to give priority to the methodology of forcing. We do not suppose any previous knowledge of forcing in the reader, but we do require some acquaintance with recursion theory. After all, our interest lies in the applications of the forcing method to the study of various recursion-theoretic notions of degrees. The farther we go, the deeper we plunge into recursion theory.

In Part I only very basic notions and results are used, like the definitions of the arithmetical hierarchy and of the jump operator and their relationships.
A certain class of models of Peano arithmetic
Authors: Kossak Roman
Published: May 2006

This paper is devoted to the study of recursively and short recursively saturated models of PA by means of so-called nonstandard satisfaction methods. The paper is intended to be self-contained. In particular, no knowledge of nonstandard satisfaction classes is assumed. In fact we shall not use this notion explicitly.

We define a certain property of models of PA which we call the S-property and prove that properly short recursively saturated models (see Definition 2.1. below) are exactly short models with the S-property. The main result is that all properly short recursively saturated models are elementary cuts of recursively saturated models. This is a generalization to the uncountable case of the theorem of C. Smorynski [9] and is an easy application of some general results concerning cofinal extensions of models of PA which we discuss in §3.

On the way we obtain another proof of the result of Smorynski and Stavi [10] which says that short recursive and recursive saturation is preserved under cofinal extensions.

The author wants to thank H. Kotlarski and W. Marek for valuable suggestions concerning the subject of the paper.

Special thanks must also go to J. Paris for the lemma used in the proof of Theorem 3.5.
Boolean sentence algebras: Isomorphism constructions
Authors: Hanf William P. , Myers Dale
Published: May 2006

Associated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or propositions.
On sets of relations definable by addition
Authors: Lynch James F.
Published: May 2006

For every κ ∈ ω, there is an infinite set Aκ ⊆ ω and a d(κ) ∈ ω such that for all Q0, Q1, ⊆ Aκ where ∣Q0∣ = ∣ Q1∣, or d(κ) < ∣Q0∣, Q1∣ < ℵ0, the structures ‹ω, +, Q0› and ‹ω, +, Q1› are indistinguishable by first-order sentences of quantifier depth κ whose atomic formulas are of the form u = v, u + v = w, and Q(u), where u, v, and w are variables.
Two further combinatorial theorems equivalent to the 1-consistency of peano arithmetic
Authors: Clote Peter , Mcaloon Kenneth
Published: May 2006

We give two new finite combinatorial statements which are independent of Peano arithmetic, using the methods of Kirby and Paris [6] and Paris [12]. Both are in fact equivalent over Peano arithmetic (denoted by P) to its 1-consistency. The first involves trees and the second linear orderings. Both were “motivated” by anti-basis theorems of Clote (cf. [1], [2]). The one involving trees, however, is not unrelated to the Kirby-Paris characterization of strong cuts in terms of the tree property [6], but, in fact, comes directly from König's lemma, of which it is a miniaturization. (See the remark preceding Theorem 3 below.) The resulting combinatorial statement is easily seen to imply the independent statement discovered by Mills [11], but it is not clear how to show their equivalence over Peano arithmetic without going through 1-consistency. The one involving linear orderings miniaturizes the property of infinite sets X that any linear ordering of X is isomorphic to ω or ω* on some infinite subset of X. Both statements are analogous to Example 2 of [12] and involve the notion of dense [12] or relatively large [14] finite set.

We adopt the notations and definitions of [6] and [12]. We shall in particular have need of the notions of semiregular, regular and strong initial segments and of indicators.
An application of invariant sets to global definability
Authors: Miller Douglas E.
Published: May 2006

Vaught's “*-transform method” is applied to derive a global definability theorem of M. Makkai from a classical theorem of Lusin.
The measure quantifier
Authors: Morgenstern Carl F.
Published: May 2006

Originally generalized quantifiers were introduced to specify that a given formula was true for “many x's” e.g. ⊨ Qxφ(x) iff card{x ∈ ∣∣ ∣ ⊨ φ[x]} ≥ ℵ0, ℵ1, or some fixed cardinal κ. In this paper we formalize the notion that φ{x) is true “for almost all x”. This is accomplished by referring to structures = (′, μ) where ′ is a first-order structure and μ is a measure of a suitable type on the universe of ′. We will prove that the language Lμ obtained from first-order logic by adjoining a quantifier Qμ, which ranges over the measure μ, is fully compact if we assume the existence of a proper class of measurable cardinals. As a corollary to the compactness theorem we obtain the recursive enumerability of the validities of Lμ. Finally, the Magidor-Malitz quantifiers Qkn (n ∈ ω) will be added to Lμ together with analogous quantifiers Qμm (m ∈ ω) to form Lκμ<ω,<ω, which is compact for sets of sentences of cardinality < κ, where κ is a measurable cardinal > ℵ0.

An alternate approach to formalizing “for almost all” has been recently developed by Barwise, Kaufmann and Makkai [1] who follow a suggestion of Shelah [5].
Reducibilities in two models for combinatory logic
Authors: Sanchis Luis E.
Published: May 2006

This paper proposes a generalization of several reducibilities in the sense of recursive function theory. For this purpose two structures are introduced as models of combinatory logic and reducibilities are defined in a rather natural way by means of the application operation in each model. The first model we consider is called the graph model and was introduced by Dana Scott in [11]. Reducibilities in this model are generalizations of enumeration and Turing reducibilities. The second model is called the hypergraph model and induces reducibilities which are generalizations of hyperenumeration and hyperarithmetical reducibilities.

The possibility of formulating reducibilities in this way is not new. For the graph model it is implicit in [7] and for the hypergraph model in [9]. On the other hand it is possible to look at the present method as a variant of the well-known technique of relativization in recursive function theory. We think that this does not exhaust the power of the method, which is conceptually elegant and provides a natural frame for the results of this paper.

In the first part of the paper we discuss the definition and general properties of the models. Then we introduce the reducibilities in the graph model and prove several theorems which are generalizations of properties already Known for enmeration and Turing reducibilities. Next we define reducibilities in the hypergraph model and try to extend the preceding results. For this purpose we prove two theorems showing significant relations between the operators in both models. In fact we prove that each operator in the hypergraph model can be simulated on a comeager set by an operator of the graph model.
On the proof theory of the modal logic for arithmetic provability
Authors: Leivant Daniel
Published: May 2006

The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. §1 below). We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL.

Some of our results have been proved model theoretically, and similar proofs are fairly straightforward for several of the remaining ones (G. Boolos and the referee have provided such proofs for 4.1, 4.3 and 5.1 below). However, our syntactic techniques often yield more concise and obviously constructive proofs, they offer additional insight into the nature of the systems considered, and are easily adaptable to systems for which semantical analysis is problematic.

I am indebted to G. Boolos and to the referee for their valuable advice. The referee has suggested the rule GL of §3 below as an axiomatization of GL; the resulting sequential calculus has allowed a definite improvement of our original presentation.
Stability theory and set existence axioms
Authors: Harnik Victor
Published: May 2006

A series of investigations started by H. Friedman ([5], [6]) and pursued by him, by S. Simpson, and by others (see [7], [16] and the references there) had the objective of determining what are precisely the set existence axioms needed for proving theorems of “ordinary” mathematics. The study centered around -CA0 (for “-Comprehension Axiom”), a fragment of second order arithmetic in which the main body of “ordinary” mathematics can be comfortably developed (of course, after a suitable encoding of the various concepts into numbers and sets of numbers). The main question is always this: given a theorem τ provable in -CA0, do we need all of -CA0, or, maybe, is τ provable in a weaker subsystem? A surprisingly simple pattern emerged: there seem to be just five important systems, denoted (in order of increasing strength) RCA0, WKL0, ACA0, ATR0 and -CA0, such that in most cases, whenever an ordinary mathematical theorem τ is provable in -CA0, it is either provable in RCA0 or it is equivalent to one, call it S, of the other four systems, the proof of the equivalence being done in one of the systems which is weaker than S (most typically, RCA0 or ACA0). This very interesting phenomenon—or “theme” as it was called by Friedman—has been verified for many instances in the realm of analysis and algebra (cf. the references above and the forthcoming book [17]). It is the purpose of this paper to do the same for a particular branch of model theory, namely stability theory.
Sequent-systems for modal logic
Authors: Došen Kosta
Published: May 2006

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else).

This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1.

These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical.