bluuebunny/crossref_metadata_2025 huggingface dataset:
A proofless proof of the Barwise compactness theorem
Authors: Howard Mark
Published: May 2006

We prove a theorem (1.7) about partial orders which can be viewed as a version of the Barwise compactness theorem which does not mention logic. The Barwise compactness theorem is easily equivalent to 1.7 + “Every Henkin set has a model”. We then make the observation that 1.7 gives us the definability of forcing for quantifier-free sentences in the forcing language and use this to give a direct proof of the truth and definability lemmas of forcing.
A normalization theorem for set theory
Authors: Bailin Sidney C.
Published: May 2006

In this paper we present a normalization theorem for a natural deduction formulation of Zermelo set theory. Our result gets around M. Crabbe's counterexample to normalizability (Hallnäs [3]) by adding an inference rule of the form

and requiring that this rule be used wherever it is applicable. Alternatively, we can regard the result as pertaining to a modified notion of normalization, in which an inference

is never considered reducible if A is T Є T, even if R is an elimination rule and the major premise of R is the conclusion of an introduction rule. A third alternative is to regard (1) as a derived rule: using the general well-foundedness rule

we can derive (1). If we regard (2) as neutral with respect to the normality of derivations (i.e., (2) counts as neither an introduction nor an elimination rule), then the resulting proofs are normal.
Forcing constructions for uncountably chromatic graphs
Authors: Komjáth Péter , Shelah Saharon
Published: May 2006

In this paper we solve some of Pál Erdős's favorite problems on uncountably chromatic graphs. Generalizing a finite graph theory result of Tutte, Erdős and R. Rado showed that for every infinite cardinal κ there exists a triangle-free, κ-chromatic graph of size κ. For κ = ℵ0, Erdős established the existence of ℵ0-chromatic graphs excluding even C4, C5,…, Cn, i.e. circuits up to a given length. For κ < ℵ0 the situation is different. As shown by Erdős and A. Hajnal, a graph is necessarily countably chromatic if it omits any finite bipartite graph. We can, however, exclude any finite list of nonbipartite graphs (this obviously reduces to excluding finitely many odd circuits). They posed an even stronger conjecture, namely, that similar examples must occur in every uncountably chromatic graph. To be specific, they conjectured that for every infinite κ, every κ-chromatic graph contains a κ-chromatic triangle-free subgraph. Here we show that this may not be true for κ = ℵ1 i.e. we exhibit a model where it is false. We must emphasize that the conjecture is probably false already in ZFC, but we have been unable to show this.
Ordinal numbers and the Hilbert basis theorem
Authors: Simpson Stephen G.
Published: May 2006

In [5] and [21] we studied countable algebra in the context of “reverse mathematics”. We considered set existence axioms formulated in the language of second order arithmetic. We showed that many well-known theorems about countable fields, countable rings, countable abelian groups, etc. are equivalent to the respective set existence axioms which are needed to prove them.

One classical algebraic theorem which we did not consider in [5] and [21] is the Hilbert basis theorem. Let K be a field. For any natural number m, let K[x1,…,xm] be the ring of polynomials over K in m commuting indeterminates x1,…,xm. The Hilbert basis theorem asserts that for all K and m, every ideal in the ring K[x1,…,xm] is finitely generated. This theorem is of fundamental importance for invariant theory and for algebraic geometry. There is also a generalization, the Robson basis theorem [11], which makes a similar but more restrictive assertion about the ring K〈x1,…,xm〉 of polynomials over K in mnoncommuting indeterminates.

In this paper we study a certain formal version of the Hilbert basis theorem within the language of second order arithmetic. Our main result is that, for any or all countable fields K, our version of the Hilbert basis theorem is equivalent to the assertion that the ordinal number ωω is well ordered. (The equivalence is provable in the weak base theory RCA0.) Thus the ordinal number ωω is a measure of the “intrinsic logical strength” of the Hilbert basis theorem. Such a measure is of interest in reference to the historic controversy surrounding the Hilbert basis theorem's apparent lack of constructive or computational content.
Markov's principle, isols and Dedekind finite sets
Authors: McCarty Charles
Published: May 2006

Markov's principle is more than a convenience in constructive arithmetic and analysis; it is absolutely essential to significant areas of constructive cardinal arithmetic. In turn, logical relations among intuitively appealing principles of constructive cardinal arithmetic parallel relations between MPS and other “problematic axioms” for constructive mathematics, such as the limited principle of omniscience. Finally, simple closure properties on the Dedekind finite sets provide ready examples of statements which are strictly weaker than Markov's principle and yet are independent of extensions of IZF.

MPS, Markov's principle with variables over sets, is equivalent to each of these elementary properties of ⊿, the class of Dedekind finite sets:

1. ∀A(A Є ⊿ ↔ CP(A)).

2. [*]A Є ⊿ ↔ ∀B (A + B is infinite ↔ B is infinite).

3. [**]A Є ⊿ ↔ ∀B((A + 1) x B is infinite ↔ B is infinite).

CP is Tarski's cancellation property. Consequently, MPS is tantamount, in constructive mathematics, to the standard classical characterizations of ⊿ in terms of cardinality.

[*] and [**] imply that ⊿ is closed under addition and multiplication. Closure under addition is, in turn, constructively equivalent to the closure of ⊿ under all combinatorial functions and to the fact that ⊿ is closed under each strict combinatorial function individually. Generally speaking, a function on P(ω) is combinatorial whenever it preserves finiteness, respects cardinality and has a “moduluslike” associate function. It is strict when it is strictly increasing relative to the subset ordering, [cf. §6 for precise definitions.] From this, we see that MPS is foundational for cardinal arithmetic on the constructive Dedekind finite sets.
On the structure of semialgebraic sets over p-adic fields
Authors: Scowcroft Philip , van den Dries Lou
Published: May 2006

In his Singular points of complex hypersurfaces Milnor proves the following “curve selection lemma” [10, p. 25]:

Let V ⊂ Rm be a real algebraic set, and let U ⊂ Rm be an open set defined by finitely many polynomial inequalities:

Lemma 3.1. If U ∩ V contains points arbitrarily close to the origin (that is if 0 ∈ Closure (U ∩ V)) then there exists a real analytic curve

with p(0) = 0 and with p(t) ∈ U ∩ V for t > 0.

Of course, this result will also apply to semialgebraic sets (finite unions of sets U ∩ V), and by Tarski's theorem such sets are exactly the sets obtained from real varieties by means of the finite Boolean operations and the projection maps Rn+1 → Rn. If, in this tiny extension of Milnor's result, we replace ‘R’ everywhere by ‘Qp’, we obtain a p-adic curve selection lemma, a version of which we will prove in this essay. Semialgebraic sets, in the p-adic context, may be defined just as they are over the reals: namely, as those sets obtained from p-adic varieties by means of the finite Boolean operations and the projection maps . Analytic maps are maps whose coordinate functions are given locally by convergent power series.
Operative vs. combinatory spaces
Authors: Ivanov Lyubomir
Published: May 2006

The algebraic systems of combinatory spaces [3] and operative spaces [1] have been designed to provide appropriate settings for the development of abstract recursion theory. As shown in [1, Chapter 27], these systems are closely related; namely, every combinatory space has a companion operative space with a storing operation St such that Skordev recursiveness in the former equals st-recursiveness in the latter. The problem of characterization of those operative spaces which have companion combinatory spaces was solved in [2] by introducing a class of operative spaces called Skordev spaces and showing that these give an alternative axiomatics of Skordev's combinatory spaces.

In this paper we study operative spaces with two special constants S1 and S2 which, in computer science terms, axiomatize the availability of stacking facility. We establish that any such space has a Skordev subspace and, moreover, all Skordev spaces can be obtained in this way. The construction ensures that st-recursiveness in the Skordev space concerned is equivalent to recursiveness in S1 and S2 in the wider operative space. Therefore, it quite unexpectedly turns out that Skordev recursiveness can be expressed via relative recursiveness in operative spaces in a straightforward manner, without employing higher spaces as done in [1].
Hereditarily finite Finsler sets
Authors: Booth David
Published: May 2006

Hereditarily finite sets are sets which are finite, whose members are finite, the members of whose members are finite, and so on. In ZF there are but countably many such sets; they constitute Vω. Were ZF to lose its axiom of regularity, however, one could not guarantee that the number of hereditarily finite sets would remain countable.

In Mostowski set theory, in which atomic sets are permissible, each atom, in isolation, would form a hereditarily finite collection. The number of hereditarily finite sets could be anything one should choose.

Even in a world that did not permit the free adjunction of arbitrary, meaningless atoms, the number of hereditarily finite sets could remain large. In Finsler set theory, it is shown as Theorem 22, below, that there are uncountably many hereditarily finite sets.

The reader who is interested in this paradoxical sounding fact can turn directly to §4 after grasping these introductory concepts. §3 is an exhaustive list of the smallest Finsler sets; it is hoped that this list will prove useful in checking future attempts to classify the finite Finsler sets.

Finsler set theory is not a firmly axiomatized theory. It is, at its present stage, a family of theories undergoing evolution. It permits the usual mathematical operations with sets. One can employ ordinal numbers, cardinal numbers, and the usual methods of Cantorian set theory freely. But there is a somewhat different interpretation attached to the concept “set” than one is used to in Zermelo-Fraenkel set theory, ZF.
A generalization of Specker's theorem on typical ambiguity
Authors: Kaye Richard
Published: May 2006

We generalize Specker's theorem on typical ambiguity, that NF and TST + Ambiguity have the same stratified consequences, to the subschemes Amb(Γ) of ambiguity restricted to classes of sentences Γ with certain natural closure conditions.
The role of parameters in bar rule and bar induction
Authors: Rathjen Michael
Published: May 2006

For several subsystems of second order arithmeticTwe show that the proof-theoretic strength ofT+ (bar rule) can be characterized in terms ofT+ (bar induction)□, where the latter scheme arises from the scheme of bar induction by restricting it to well-orderings with no parameters. In addition, we demonstrate that,ACA0+ (bar rule) andACA0+ (bar induction)□prove the same-sentences.
Boolean valued Lie algebras
Authors: Nishimura Hirokazu
Published: May 2006

In this paper we study a certain class of Lie algebras over commutative von Neumann algebras satisfying a certain finiteness condition. By using Boolean valued methods developed by Takeuti [8]−[11], we will establish the basic structure and representation theorems.
The Σ21 theory of axioms of symmetry
Authors: Weitkamp Galen
Published: May 2006

The axiom of symmetry (Aℵ0) asserts that for every function F: ω2 → ω2 there is a pair of reals x and y in ω2 so that y is not in the countable set {(F(x))n: n< ω} coded by F(x) and x is not in the set coded by F(y). A(Γ) denotes axiom Aℵ0 with the restriction that graph(F) belongs to the pointclass Γ. In §2 we prove A(). In §3 we show A(), A() and ω2 ⊈ L are equivalent. In §4 several effective versions of A(REC) are examined.
Near-equational and equational systems of logic for partial functions. I
Authors: Craig William
Published: May 2006

Equational logic for total functions is a remarkable fragment of first-order logic. Rich enough to lend itself to many uses, it is also quite austere. The only predicate symbol is one for a notion of equality, and there are no logical connectives. Proof theory for equational logic therefore is different from proof theory for other logics and, in some respects, more transparent. The question therefore arises to what extent a logic with a similar proof theory can be constructed when expressive power is increased.

The increase mainly studied here allows one both to consider arbitrary partial functions and to express the condition that a function be total. A further increase taken into account is equivalent to a change to universal Horn sentences for partial and for total functions.

Two ways of increasing expressive power will be considered. In both cases, the notion of equality is modified and nonlogical function symbols are interpreted as ranging over partial functions, instead of ranging only over total functions. In one case, the only further change is the addition of symbols that denote logical functions, such as the binary projection functionAethat maps each pair ‹a0,a1› of elements of a setAinto the elementa0. An addition of this kind results in a language, and also in a system of logic based on this language, which we callequationalIn the other case, instead of adding a symbol forAe, one admits those special universal Horn sentences in which the conditions expressed by the antecedent are, in a sense, pure conditions of existence. Languages and systems of logic that result from a change of this kind will be callednear-equational. According to whether the number of existence conditions that one may express in antecedents is finite or arbitrary, the resulting language and logic shall befiniteorinfinitary, respectively. Each of our finite near-equational languages turns out to be equivalent to one of our equational languages, and vice versa.
Nonfinite axiomatizability results for cylindric and relation algebras
Authors: Maddux Roger D.
Published: May 2006

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.
A new proof of Sahlqvist's theorem on modal definability and completeness
Authors: Sambin G. , Vaccaro V.
Published: May 2006

There are not many global results on modal logics. One of these is the following theorem by Sahlqvist on completeness and correspondence for a wide class of modal formulae (including many well known logics, e.g. D, T, B, S4, K4, S5, …) (see [S]).

Sahlqvist's Theorem. Let A be any modal formula which is equivalent to a conjunction of formulae of the form □m(A1 → A2), where m ≥ 0, A2 is positive and A1 is obtained from propositional variables and their negations applying ∧, ∨, ♢, and □ in such a way that no positive occurrence of a variable is in a subformula of the form B1 ∨ B2 or ♢ B1 within the scope of some □. Then A corresponds effectively to a first order formula, and L + A is canonical whenever Lis a canonical logic.

A formula A satisfying the above conditions is henceforth called a Sahlqvist formula. Unfortunately, till now, the only complete proof was the original proof of Sahlqvist (a proof of the correspondence half has also been given by van Benthem [vB]). It is so complicated and long that even in an advanced textbook of modal logic [HC] it has not found a place. Here, by considering general frames as topological spaces, an attitude which we developed in [TD], we give a proof of Sahlqvist's theorem simplified to such an extent that one can easily grasp the key idea on which it is based and apply the resulting algorithm to specific modal formulae in a straightforward manner, suitable even for implementation on a personal computer. This key idea also improves on previous preliminary work in the same direction (see [S1], [S2]).
Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion
Authors: Jockusch C. G. , Lerman M. , Soare R. I. , Solovay R. M.
Published: May 2006

Let We be the eth recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleene's recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that We = Wf(e). In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets.

Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper.

Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete (i.e. A has degree0′) iff there is a function f recursive in A with no fixed point.
Some restrictions on simple fixed points of the integers
Authors: McColm G. L.
Published: May 2006

A function is recursive (in given operations) if its values are computed explicitly and uniformly in terms of other “previously computed” values of itself and (perhaps) other “simultaneously computed” recursive functions. Here, “explicitly” includes definition by cases.

We investigate those recursive functions on the structure N = 〈ω, 0, succ, pred〉 that are computed in terms of themselves only, without other simultaneously computed recursive functions.
A general treatment of equivalent modalities
Authors: Bellissima Fabio , Mirolli Massimo
Published: May 2006

The problem of the nonequivalent modalities available in certain systems is a classical problem of modal logic. In this paper we deal with this problem without referring to particular logics, but considering the whole class of normal propositional logics. Given a logic L let P(L) (the m-partition generated by L) denote the set of the classes of L-equivalent modalities. Obviously, different logics may generate the same m-partition; the first problem arising from this general point of view is therefore to determine the cardinality of the set of all m-partitions. Since, as is well known, there exist normal logics, and since one immediately realizes that there are infinitely many m-partitions, the problem consists in choosing (assuming the continuum hypothesis) between ℵ0 and . In Theorem 1.2 we show that there are m-partitions, as many as the logics.

The next problem which naturally arises consists in determining, given an m-partition P(L), the number of logics generating P(L) (in symbols, μ(P(L))). In Theorem 2.1(ii) we show that ∣{P(L): μ(P(L)) = }∣ = . Now, the set {L′) = P(L)} has a natural minimal element; that is, the logic L* axiomatized by K ∪ {φ(p) ↔ ψ(p): φ, ψ are L-equivalent modalities}; P(L) and L* can be, in some sense, identified, thus making the set of m-partitions a subset of the set of logics.
On Ehrenfeucht-Fraïssé equivalence of linear orderings
Authors: Oikkonen Juha
Published: May 2006

C. Karp has shown that if α is an ordinal with ωα = α and A is a linear ordering with a smallest element, then α and α ⊗ A are equivalent in L∞ω up to quantifer rank α. This result can be expressed in terms of Ehrenfeucht-Fraïssé games where player ∀ has to make additional moves by choosing elements of a descending sequence in α. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraïssé games of length ω1. One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending ω1-sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O].
Pointless metric spaces
Authors: Gerla Giangiacomo
Published: May 2006

In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).

In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].

In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.
Undecidability of modal and intermediate first-order logics with two individual variables
Authors: Gabbay D. M. , Shehtman V. B.
Published: May 2006

The interest in fragments of predicate logics is motivated by the well-known fact that full classical predicate calculus is undecidable (cf. Church [1936]). So it is desirable to find decidable fragments which are in some sense “maximal”, i.e., which become undecidable if they are “slightly” extended. Or, alternatively, we can look for “minimal” undecidable fragments and try to identify the vague boundary between decidability and undecidability. A great deal of work in this area concerning mainly classical logic has been done since the thirties. We will not give a complete review of decidability and undecidability results in classical logic, referring the reader to existing monographs (cf. Suranyi [1959], Lewis [1979], and Dreben, Goldfarb [1979]). A short summary can also be found in the well-known book Church [1956]. Let us recall only several facts. Herein we will consider only logics without functional symbols, constants, and equality.

(C1) The fragment of the classical logic with only monadic predicate letters is decidable (cf. Behmann [1922]).

(C2) The fragment of the classical logic with a single binary predicate letter is undecidable. (This is a consequence of Gödel [1933].)

(C3) The fragment of the classical logic with a single individual variable is decidable; in fact it is equivalent to Lewis S5 (cf. Wajsberg [1933]).

(C4) The fragment of the classical logic with two individual variables is decidable (Segerberg [1973] contains a proof using modal logic; Scott [1962] and Mortimer [1975] give traditional proofs.)

(C5) The fragment of the classical logic with three individual variables and binary predicate letters is undecidable (cf. Surańyi [1943]). In fact this paper considers formulas of the following type

φ,ψ being quantifier-free and the set of binary predicate letters which can appear in φ or ψ being fixed and finite.
Random models and solvable Skolem classes
Authors: Goldfarb Warren
Published: May 2006

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.

The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.
Derivation rules as anti-axioms in modal logic
Authors: Venema Yde
Published: May 2006

We discuss a ‘negative’ way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized byderivation rules, the ‘non-ξ rules’, styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If⋀is a derivation system having a set of axioms that are special Sahlqvist formulas and⋀+is the extension of⋀with a set of non-ξ rules, then⋀+is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.
Additive reducts of real closed fields
Authors: Marker David , Peterzil Ya'acov , Pillay Anand
Published: May 2006

In [MP] Marker and Pillay showed that if X ⊂ Cn is constructible but (C, +, X) is not locally modular, then multiplication is definable in the structure (C, +,X). That result extended earlier results of Martin [M] and Rabinovich and Zil'ber [RZ]. Here we will examine additive reducts of R and Qp.

Definition. A subset X of Rn is called semialgebraic if it is definable in the structure (R, +,·). A subset X of Rn is called semilinear if it is definable in the structure (R, +, <,λr)r∈b, where λr is the function x ↦ rx [scalar multiplication by r].

Every semilinear set is a Boolean combination of sets of the form {: p () = 0} and {: q() > 0}, where p() and q() are linear polynomials.

Van den Dries asked the following question: if X is semialgebraic but not semilinear, can we define multiplication in (R, +, <,X)? This was answered negatively by Pillay, Scowcroft and Steinhorn.

Theorem 1.1 [PSS]. Suppose X ⊂ Rnis semialgebraic and X ⊂ Infor some bounded interval I. Then multiplication is not definable in (R, +, <,X,λr)r∈R.

In particular if X = · ∣ [0, l ]2, the graph of multiplication restricted to the unit interval, then X is not semilinear so we have a negative answer to van den Dries' question. Peterzil showed that this is the only restriction.
Undecidability of the identity problem for finite semigroups
Authors: Albert Douglas , Baldinger Robert , Rhodes John
Published: May 2006

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups.

If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety.

We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.
Kripke models for linear logic
Authors: Allwein Gerard , Dunn J. Michael
Published: May 2006

We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutativity and associativity are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, similar to Curry's subtraction operator, which is residuated with Linear Logic's cotensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic.
Decidability and undecidability of theories with a predicate for the primes
Authors: Bateman P. T. , Jockusch C. G. , Woods A. R.
Published: May 2006

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.
Finitely constrained classes of homogeneous directed graphs
Authors: Latka Brenda J.
Published: May 2006

Given a finite relational language L is there an algorithm that, given two finite sets and of structures in the language, determines how many homogeneous L structures there are omitting every structure in and embedding every structure in ?

For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether Γ, the class of finite tournaments omitting every tournament in Γ. is well-quasi-order?

First, we give a nonconstructive proof of the existence of an algorithm for the case in which Γ consists of one tournament. Then we determine explicitly the set of tournaments each of which does not have an antichain omitting it. Two antichains are exhibited and a summary is given of two structure theorems which allow the application of Kruskal's Tree Theorem. Detailed proofs of these structure theorems will be given elsewhere.

The case in which Γ consists of two tournaments is also discussed.
Semilinear cell decomposition
Authors: Liu Nianzheng
Published: May 2006

We obtain a p-adic semilinear cell decomposition theorem using methods developed by Denef in [Journal für die Reine und Angewandte Mathematik, vol. 369 (1986), pp. 154–166]. We also prove that any set definable with quantifiers in (0,1, +, —, λq, Pn){n∈ℕ,q∈ℚp} may be defined without quantifiers, where λq is scalar multiplication by q and Pn is a unary predicate which denotes the nonzero nth powers in the p-adic field ℚp. Such a set is called a p-adic semilinear set in this paper. Some further considerations are discussed in the last section.
Embeddability of ptykes
Authors: Girard Jean-Yves , Normann Dag
Published: May 2006

The notion of a ptyx (plural, ptykes) is obtained by generalising the ordinals On and the dilators Dil introduced by Girard [1] into a typed hierarchy. The notion is due to Girard, and a detailed treatment will appear in Girard [2].

In this Introduction we will give a summary of the theory of ptykes. It will be an advantage to be familiar with the theory of dilators as introduced in Girard [1] or as presented in Girard and Normann [3].

The class On of ordinals is organised into a category ON by using strictly increasing maps f: x → y as morphisms. A dilator will be a functor F: ON → ON commuting with pullbacks and direct limits. Associated with each dilator F there is a denotation system DF, obtained as follows: For each x ∈ On and y ∈ F(x), y can be given a unique denotation (c; x0,…, xn−1;x)F, where c∈F(n) (n = {0,…, n − 1}), y = F(ϕ)(c) (where ϕ is defined by ϕ(i) = xi for i < n), and for no m < n and ψ: m → n do we have c ∈ im(ψ).

We let the trace Tr(F) be the set of pairs (c, n) occurring in a denotation.
Frequency computations and the cardinality theorem
Authors: Harizanov Valentina , Kummer Martin , Owings Jim
Published: May 2006

In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers,

J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-computable. Trakhtenbrot's work was extended by E. B. Kinber [Ki1], using similar techniques. In 1986 R. Beigel [B] made a powerful conjecture, much more general than the above results. Partial verification, falling short of a full proof, appeared in [O]. Using new techniques, M. Kummer has recently established the conjecture, which will henceforth be referred to as the cardinality theorem (CT). It is the goal of this paper to show the connections between these various theorems, to review the methods used by Trakhtenbrot, and to use them to prove a special case of CT strong enough to imply Kinber's theorem (see §3). We thus have a hierarchy of results, with CT at the top. We will also include a discussion of Kummer's methods, but not a proof of CT.
Does V equal L?
Authors: Maddy Penelope
Published: May 2006

Does V = L? Is the Axiom of Constructibility true? Most people with an opinion would answer no. But on what grounds? Despite the near unanimity with which V = L is declared false, the literature reveals no clear consensus on what counts as evidence against the hypothesis and no detailed analysis of why the facts of the sort cited constitute evidence one way or another. Unable to produce a well-developed argument one way or the other, some observers despair, retreating to unattractive fall-back positions, e.g., that the decision on whether or not V = L is a matter of personal aesthetics. I would prefer to avoid such conclusions, if possible. If we are to believe that L is not V, as so many would urge, then there ought to be good reasons for this belief, reasons that can be stated clearly and subjected to rational evaluation. Though no complete argument has been presented, the literature does contain a number of varied argument fragments, and it is worth asking whether some of these might be developed into a persuasive case.

One particularly simple approach would be to note that the existence of a measurable cardinal (MC) implies that V ≠ L,1and to argue that there is a measurable cardinal. The drawback to this approach is that its implying V ≠ L cannot then be counted as evidence in favor of MC, as it often is. Indeed, there seems to have been considerable sentiment against V = L even before the proof of its negation from MC,2and this sentiment must either be accounted for as reasonable or explained away as an aberration of some kind.
Combinatorial properties of the ideal ℬ2
Authors: Cichoń J. , Rosłanowski A. , Steprans J. , Wȩglorz B.
Published: May 2006

By ℬ2 we denote the σ-ideal of all subsets A of the Cantor set {0, 1}ω such that for every infinite subset T of ω the restriction A∣{0, 1}T is a proper subset of {0, 1}T. In this paper we investigate set theoretical properties of this and similar ideals.
μ-definable sets of integers
Authors: Lubarsky Robert S.
Published: May 2006

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.

The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.

There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.
Natural language, sortal reducibility and generalized quantifiers
Authors: Keenan Edward L.
Published: May 2006

Recent work in natural language semantics leads to some new observations on generalized quantifiers. In §1 we show that English quantifiers of type 〈1, 1〉 are booleanly generated by theirgeneralized universalandgeneralized existentialmembers. These two classes also constitute thesortally reduciblemembers of this type.

Section 2 presents our main result — the Generalized Prefix Theorem (GPT). This theorem characterizes the conditions under which formulas of the form (Q1x1…QnxnRx1…xnandq1x1…qnxnRx1…xnare logically equivalent for arbitrary generalized quantifiersQi,qi. GPT generalizes, perhaps in an unexpectedly strong form, the Linear Prefix Theorem (appropriately modified) of Keisler & Walkoe (1973).
Games and full completeness for multiplicative linear logic
Authors: Abramsky Samson , Jagadeesan Radha
Published: May 2006

We present a game semantics for Linear Logic, in which formulas denote games and proofs denote winning strategies. We show that our semantics yields a categorical model of Linear Logic and prove full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net. A key role is played by the notion of history-free strategy; strong connections are made between history-free strategies and the Geometry of Interaction. Our semantics incorporates a natural notion of polarity, leading to a refined treatment of the additives. We make comparisons with related work by Joyal, Blass, et al.
A constructive proof of McNaughton's theorem in infinite-valued logic
Authors: Mundici Daniele
Published: May 2006

We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.
Ideals of nowhere Ramsey sets are isomorphic
Authors: Plewik Szymon
Published: May 2006

We introduce a notion of ideal type such that any two ideals with the same ideal type are isomorphic. From this we infer, under the axiom t = h, that each ideal which consists of all nowhere Ramsey sets contained in some family of infinite subsets of natural numbers is isomorphic with the ideal of all nowhere Ramsey sets.
Recursion theory in a lower semilattice
Authors: Feldman Alex
Published: May 2006

In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.

The idea of looking at the lattice of recursively enumerable substructures of some recursive algebraic structure was introduced by Metakides and Nerode in [5], and since then many different kinds of algebraic structures have been studied in this way, including vector spaces, Boolean algebras, groups, algebraically closed fields, and equivalence relations. Since different algebraic structures have different recursion theoretic properties, one natural question is whether an algebraic structure with relatively little structure (such as a partial order or an equivalence relation) exhibits behavior more like classical recursion theory than one with more structure (such as vector spaces or algebraically closed fields).

In [6] and [7], Metakides and Remmel studied recursion theory on orderings, and, as they point out in [6], orderings differ from most other algebraic structures in that the algebraic closure operation on orderings is trivial; but this does not present a problem for them, given the questions they explore. Moreover, they take an approach of proving general theorems which can then be applied to specific orderings. Our tack is different, although also well-established (see, for example, [3]), in which a “largest” structure is defined (in §3) which corresponds to the natural numbers in classical recursion theory. In order to distinguish substructures from subsets, a function symbol is added, namely greatest lower bound. The greatest lower bound function is fundamental to the study of orderings and occurs naturally in many of them, and thus is an appropriate addition to the theory of orderings. In §4 we redefine the concepts of simple and maximal in a manner appropriate to this structure, and prove several existence theorems.
Baire numbers, uncountable Cohen sets and perfect-set forcing
Authors: Landver Avner
Published: May 2006

The Baire number of the real line (see 1.1(c)) has uncountable cofinality [Mi]. This number is equal to the Baire number of the space 2ω and also to the Baire number (1.1(d)) of every countable partial order. Our research is motivated by the following open question (1.13) (see also [BPS] and [Mi]): can the cofinality of nκ, the Baire number of the space (2κ)κ (1.1(b)), be less than or equal to κ? This question is nontrivial when κ is regular and 2κ = κ (1.3). A. Miller proved that the answer is “no” if κ is strongly inaccessible (see [Mi] and 1.11). Assuming CH, is equal to the Baire number of the Stone space of ℘(ω)/Fin.

In §1 we discuss , the Baire number of the space (2λ)κ. If κ is regular, then , and using 1.3 and 1.5, we may rule out the easy cases and restrict ourselves to cases where κ is regular with 2<κ = κ, and κ ≤ λ 2κ. If λ > κ, then it is consistent to have cof()=κ ([Mi] and 1.7). We do not know whether is consistent (1.9). Next, the cardinal number eκ is defined (1.10). It is easy to see that nκ ≤ eκ and that cof(eκ) > κ. If κ is strongly inaccessible (s.i.), then nκ = eκ ([Mil], [Bar], [MF] and 1.11). For κ accessible, the question of whether nκ = eκ is open.

Another open question (Fremlin) is stated as follows: assume W ⊇ V and f ∈ ωω ∩ W is such that, for every g ∈ ω ω ∩ V, ∣(f = g)∣ = ℵ0, does this imply that W contains sets which are ω-Cohen (1.1(a)) over V? If κ is accessible regular with 2<κ = κ, then the answer to the analog of Fremlin's question for κ is “no”. In fact, any κ-closed forcing that adds new subsets of κ, also adds a new function f ∈ κκ such that, for every g ∈ κκ ∩V, (f = g) is stationary (4.1(a)). On the other hand, perfect-set forcing on κ (see [Ka] and §5) adds a new subset of κ but produces a minimal extension (5.4(b)), and hence adds no sets which are κ-Cohen over V.
Decomposing Baire functions
Authors: Cichoń J. , Morayne M. , Pawlikowski J. , Solecki S.
Published: May 2006

We discuss in the paper the following problem: Given a function in a given Baire class, into “how many” (in terms of cardinal numbers) functions of lower classes can it be decomposed? The decomposition is understood here in the sense of the set-theoretical union.
Continuity and nondiscontinuity in constructive mathematics
Authors: Ishihara Hajime
Published: May 2006

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of the Kreisel-Lacombe-Shoenfield-Tseĭtin theorem.
On the weak Kleene scheme in Kripke's theory of truth
Authors: Cain James , Damnjanovic Zlatan
Published: May 2006

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered:

1. There exist sentences that are neither paradoxical nor grounded.

2. There are fixed points.

3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n ∣ A(n) is true in the minimal fixed point}, where A(x) is a formula of AR + T) are precisely the sets.

4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the sets.

5. The closure ordinal for Kripke's construction of the minimal fixed point is .

In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen:

1. There may or may not exist nonparadoxical, ungrounded sentences.

2. The number of fixed points may be any positive finite number, ℵ0, or .

3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the sets, including intermediate cases.

4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the sets.

5. The closure ordinal for the construction of the minimal fixed point may be ω, , or any successor limit ordinal in between.

In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Gödel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language.