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A result concerning cardinalities of ultraproducts
Authors: Keisler H. Jerome , Prikry Karel
Published: May 2006

The cardinality problem for ultraproducts is as follows: Given an ultrafilter over a set I and cardinals αi, i ∈ I, what is the cardinality of the ultraproduct ? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter over a set I of power κ (uniform means that all elements of have power κ). It is open whether every countably incomplete has the property that, for all infinite α, the ultra-power has power ακ. However, it is shown in [4] that certain countably incomplete , namely the κ-regular , have this property.

This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the κ-good ultrafilters, do have this property. The κ-good ultrafilters are important in model theory because they are exactly the ultrafilters such that every ultraproduct modulo is κ-saturated (see [5]).

Let be an ultrafilter on a set I. Let αi, n, i ∈ I, n ∈ ω, be cardinals and αi, n, ≥ αi, m if n < m. Let

.

Then ρn are nonincreasing and therefore there is some m and ρ such that ρn = ρ if n ≥ m. We call ρ the eventual value (abbreviated ev val) of ρn.
A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property
Authors: Gabbay D. M. , De Jongh D. H. J.
Published: May 2006

The intuitionistic propositional logic I has the following (disjunction) property

.

We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + (∼ϕ → (ψ ∨ α))→ ((∼ϕ→ψ) ∨ (∼ϕ→α)) and Scott's system I + ((∼ ∼ϕ→ϕ)→(ϕ ∨ ∼ϕ))→ (∼∼ϕ ∨ ∼ϕ). It was shown in [3c] that the first system has the finite-model property.

In this note we shall construct a sequence of intermediate logics Dn with the following properties:

These systems are presented both semantically and syntactically, using the remarkable correspondence between properties of partially ordered sets and axiom schemata of intuitionistic logic. This correspondence, apart from being interesting in itself (for giving geometric meaning to intuitionistic axioms), is also useful in giving independence proofs and obtaining proof theoretic results for intuitionistic systems (see for example, C. Smorynski, Thesis, University of Illinois, 1972, for independence and proof theoretic results in Heyting arithmetic).
Models for recursion theory
Authors: Moldestad Johan , Normann Dag
Published: May 2006

Several results in the theory of recursion in higher types indicate that the effect of a higher type functional on the lower types does not reflect the high type, i.e. the same effect could be obtained by functionals of relatively low type. The two main results here are:

Plus -1 - Theorem (G. Sacks [6] for k = 1, [7] for k > 1). Let H be a normal functional of type ≥ k + 1. Then there exists a normal functional F of type k + 1 such that k-sc(F) = k-sc(H), i.e. the same subsets of tp(k − 1) are recursive in F and H.

Plus - 2 - Theorem (L. Harrington [1]). Let H be a normal functional of type ≥ k + 2. Then there exists a normal functional F of type k +2 such that k-en(H) = k-en(F), i.e. the same subsets of tp(k − 1) are semirecursive in F and H.

The results in this paper also indicate that higher types cannot have too much influence on lower types. The key is the Skolem-Löwenheim theorem. Among the results we mention:

(1) Let n < m. A ⊆ tp(n) × tp(m) be Kleene-semicomputable. Let x ∈ B ⇔ ∀y∈tp(m), ⟨x, y⟩ ∈ A. Then B is . This result may be relativized to a functional of type n + 1.

(2) Let k0 be the type-k-functional that is constant zero. Let F be a functional of type < k. Then, for i ≤ k −2, i-sc(F ,k0) = i-sc(F), i-en(F, k0) = ∀tp(i)(i-en(F)).
Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods
Authors: Pincus David
Published: May 2006

Fraenkel-Mostowski models are a particularly simple and conceptual tool for proving consistency results involving the axiom of choice, AC. These models satisfy the theory, FM, of a well founded universe of sets built from a ground set of individuals. Zermelo-Fraenkel set theory, ZF, is the extension of FM in which the set of individuals is assumed to be empty. In this paper we show that there is a large class of statements whose consistency with ZF can be proven directly by means of a Fraenkel-Mostowski model.

A statement, Φ, of set theory is said to be transferable if there is a metatheorem: If Φ is true in a Fraenkel-Mostowski model then Φ is consistent with ZF. Jech and Sochor introduced, in [12], the class of boundable statements and proved them to be transferable. Most existential contradictions of AC are boundable. It remains to find criteria under which Ψ ∧ Φ is transferable where Ψ is a universal consequence of AC and Φ is an existential contradiction of AC. To this end we give two classes of statements. Each class is closed under conjunction, contains the boundable statements, and contains a number of universal consequences of AC. Nearly every Fraenkel-Mostowski consistency in the literature falls into one of these two classes.

In §2 we give two generalizations of the boundable statements. In §§3 and 4 the classes of transferable statements are discussed. In §5 we discuss the transfer problem and prove a metatheorem concerning nontransferable statements.
On n-quantifier induction
Authors: Parsons Charles
Published: May 2006

In this paper we discuss subsystems of number theory based on restrictions on induction in terms of quantifiers, and we show that all the natural formulations of ‘n-quantifier induction’ are reducible to one of two (for n ≠ 0) nonequivalent normal forms: the axiom of induction restricted to (or, equivalently, ) formulae and the rule of induction restricted to formulae.

Let Z0 be classical elementary number theory with a symbol and defining equations for each Kalmar elementary function, and the rule of induction

restricted to quantifier-free formulae. Given the schema

let IAn be the restriction of IA to formulae of Z0 with ≤n nested quantifiers, IAn′ to formulae with ≤n nested quantifiers, disregarding bounded quantifiers, the restriction to formulae, the restriction to , formulae. IRn, IRn′, , are analogous.

Then, we show that, for every n, , , IAn, and IAn′, are all equivalent modulo Z0. The corresponding statement does not hold for IR. We show that, if n ≠ 0, is reducible to ; evidently IRn is reducible to . On the other hand, IRn′ is obviously equivalent to IAn′ [10, Lemma 2].
Major subspaces of recursively enumerable vector spaces
Authors: Kalantari Iraj
Published: May 2006

The main point of this paper is a further development of some aspects of the recent theory of recursively enumerable (r.e.) algebraic structures. Initial work in this area is due to Frölich and Shepherdson [4] and Rabin [10]. Here we are only concerned with vector space structure. The previous work on r.e. vector spaces is due to Dekker [2], [3], Metakides and Nerode [8], Remmel [11], Retzlaff [13], and the author [5].

Our object of study is V∞ a countably infinite dimensional fully effective vector space over a countable recursive field . By fully effective we mean that V∞. under a fixed Godel numbering has the following properties:

(i) Operations of vector addition and scalar multiplication on V∞ are presented by partial recursive functions on the Gödel numbers of elements of V∞.

(ii) V∞ has a dependence algorithm, i.e., there is a uniform effective procedure which applied to any n vectors of V∞ determines whether or not they are linearly independent.

We also study , the lattice of r.e. subspaces of V∞ (under the operations of intersection, ⋂ and (weak) sum, +). We note that if is not distributive and is merely modular (see [1]). This fact indicates the essential difference between the lattice of r.e. sets and .
Amalgamation of nonstandard models of arithmetic
Authors: Blass Andreas
Published: May 2006

Any two models of arithmetic can be jointly embedded in a third with any prescribed isomorphic submodels as intersection and any prescribed relative ordering of the skies above the intersection. Corollaries include some known and some new theorems about ultrafilters on the natural numbers, for example that every ultrafilter with the “4 to 3” weak Ramsey partition property is a P-point. We also give examples showing that ultrafilters with the “5 to 4” partition property need not be P-points and that the main theorem cannot be improved to allow a prescribed ordering of lower skies.
The Hanf number for complete Lω1,ω-sentences (without GCH)
Authors: Baumgartner James E.
Published: May 2006

The Hanf number for sentences of a language L is defined to be the least cardinal κ with the property that for any sentence φ of L, if φ has a model of power ≥ κ then φ has models of arbitrarily large cardinality. We shall be interested in the language Lω1,ω (see [3]), which is obtained by adding to the formation rules for first-order logic the rule that the conjunction of countably many formulas is also a formula.

Lopez-Escobar proved [4] that the Hanf number for sentences of Lω1,ω is ⊐ω1, where the cardinals ⊐α are defined recursively by ⊐0 = ℵ0 and ⊐α = Σ{2⊐β: β < α} for all cardinals α > 0. Here ω1 denotes the least uncountable ordinal.

A sentence of Lω1,ω is complete if all its models satisfy the same Lω1,ω-sentences. In [5], Malitz proved that the Hanf number for complete sentences of Lω1,ω is also ⊐ω1, but his proof required the generalized continuum hypothesis (GCH). The purpose of this paper is to give a proof that does not require GCH.

More precisely, we will prove the following:

Theorem 1. For any countable ordinal α, there is a complete Lω1,ω-sentence σαwhich has models of power ⊐α but no models of higher cardinality.

Our basic approach is identical with Malitz's. We simply use a different combinatorial fact at the crucial point.
Models of set theory with more real numbers than ordinals
Authors: Cohen Paul E.
Published: May 2006

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.

If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.

If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).

Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.

We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].

Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.
Omitting types in set theory and arithmetic
Authors: Knight Julia F.
Published: May 2006

In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).

The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.

If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.
A reduction class containing formulas with one monadic predicate and one binary function symbol
Authors: Hughes Charles E.
Published: May 2006

A new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.
Approximation theorems and model theoretic forcing
Authors: Harnik Victor
Published: May 2006

We suggest the name “approximation theorems” for a new kind of theorems which are strong versions of preservation theorems. A typical preservation theorem has the form:

A sentence φ is preserved by the relation R (i.e. and imply ) iff there exists φ* ϵ Φ such that ⊧φ↔ φ*

where R is a relation between structures and Φ is a class of sentences (depending, of course, on R). Usually, Φ is described in syntactical terms and it is easy to see that every element of it is, indeed, preserved by R. A typical approximation theorem has the form:

For every sentence Φ there is a sentence Φ* ϵ Φ (the “approximation” of Φ) such that, for all sentences δ which are preserved under R,

(a) if ⊧δ → φ then ⊧δ → φ* and

(b) if ⊧φ → δ then ⊧φ* → δ.

An approximation theorem obviously implies the corresponding preservation theorem.

The first approximation theorem was proved by Vaught in [14] (see Corollary 2.3 below). That paper inspired the present one. Vaught's result is stated in topological terms. It says that for each Borel set B there is an invariant Borel set B* explicitly defined by an Lω1ω sentence, such that B− ⊆ B* ⊆ B+ where B− (B+) is the largest (smallest) invariant set included in (containing) B.
Ed-regressive sets of order n
Authors: Catlin Seth
Published: May 2006

A number is a nonnegative integer, and E is the set of numbers. In [3], J. C. E. Dekker introduced the concept of a regressive set of order n as the range of a one-one function f of n arguments such that (i) domain f ⊆ En and, if (x1, …, xn) ∈ domain f, and yi ≤ ≤ xi for 1 ≤ i ≤ n, then (y1, …, yn) ∈ domain f, and (ii) if 1 ≤ i ≤ n and (x1, …, xn) ∈ domain f, then f(x1 … xi−1, xi ∸ 1, xi+1 … xn) can be found effectively from f(x1 … xn). (0 ∸ 1 = 0 and, for m ≥ 1, m ∸ 1 = m − 1.) Since one can take the view, as Dekker did when first introducing regressive functions in [1], that a regressive set of order one is the range of a function of the above type which is of order one and everywhere defined, it seems natural to study the n-dimensional analogue in which (i) is replaced by “domain f = En.” It is the purpose of this paper to study such sets.
Selection in abstract recursion theory
Authors: Harrington L. A. , Macqueen D. B.
Published: May 2006

In [1] Gandy established the following selection theorem for recursion in type-2 objects.

Theorem. Let F be a normal type-2 object. Then it is possible to select (uniformly and effectively in F) an integer from each nonempty set of integers semirecursive in F.

Notice that this really asserts that the predicates semirecursive in F are closed under existential quantification over type-0. Moschovakis [6] has essentially proven this theorem for F of arbitrary type.

In [2] Grilliot stated a powerful generalization of Gandy's result, namely:

Grilliot's Selection Theorem. Let F be a normal type-(n + 2) object (n an arbitrary integer). Then it is possible to select (uniformly and effectively in F) a nonempty recursive in F subset of each nonempty semirecursive in F set oftype-(n − 1) objects.

Notice again that this actually says that predicates semirecursive in F are closed under quantification over type-(n − 1) objects.

Despite the similarity of these two results, Gandy and Grilliot proposed rather different methods of proof. Furthermore, the proof that Grilliot presented in [2] contains an error which cannot easily be corrected. (We will comment on the nature of this error at the end of §1.) Fortunately, however, Grilliot's theorem is valid. We will present a proof of Grilliot's selection theorem which is a direct generalization of the proof of Gandy's theorem given in [6]. In fact, we will prove a general result (the theorem stated in §2) which subsumes both Gandy's and Grilliot's results.
Diversity of speed-ups and embeddability in computational complexity
Authors: Alton Donald A.
Published: May 2006

The theory of computational complexity deals with those functions which can be computed subject to certain restrictions on the resources (for instance, time or memory) available for computation. Blum [5] gave an axiomatic characterization of some of the properties which should be possessed by a measure of computational complexity and established the existence of speed-upable functions—computable functions which fail to possess optimal programs in a particularly strong sense. Recursion theorists tend to like such functions, and people concerned with the specifics of real computing tend to consider such functions somewhat pathological. In Theorem 2 we show that such pathology is rampant: there is a great diversity of behavior among the collections of “run-times” of different functions which do not possess optimal programs, where such diversity is gauged by certain algebraic criteria which have computational significance. Roughly speaking, these algebraic criteria concern the ways in which various functions can be intermixed to satisfy requirements that certain functions can or cannot be computed more easily than certain other functions. (More detailed motivation for the relevance of these algebraic notions is given later.) Specifically, in Theorem 2 we generalize the embeddability theorem of McCreight and Meyer [12] (discussed below) by making speed-upable functions responsible for the embedding.
An abstraction algorithm for combinatory logic
Authors: Abdali S. Kamal
Published: May 2006

This note presents a practical algorithm for carrying out abstraction on combinatory terms. The well-known abstraction algorithms [1, pp. 188ff.] defining abstracts in terms of the combinator sets {S, K}, {B, C, K, W}, etc. operate on one variable at a time, and result in rather long abstracts when several variables are involved. These algorithms are not practical for the applications of the combinatory logic to the theory of computing which make much use of multi-variable abstraction (e.g., [2], [3]). The present algorithm performs the abstraction with respect to all specified variables in a single step, and yields abstracts in a concise form (with sizes proportional to those of given combinatory terms).
An axiomatisation of quantum logic
Authors: Clark Ian D.
Published: May 2006

The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset.

In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa.

Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following:

(1) the greatest lower bound of a and Na exists and is 0,

(2) a ≤ b implies Nb ≤ Na,

(3) NNa = a.

Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying

where a ≤ b means a ∧ b = a, as usual.

Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following:

An orthomodular lattice gives a Finch implication algebra by defining → by

A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by

The orthocomplementation is unchanged in both cases.
A characterization of terms of the λI-calculus having a normal form
Authors: Barendregt Henk
Published: May 2006

The theorem proved in this paper answers some transitivity questions (in the geometric sense) for the type free λ-calculus: Which objects can be mapped on all other objects? How much can an object do by applying it to other objects (see footnote 2)?

The main result is that, for closed terms of the λI-calculus, the following conditions are equivalent:

(a) M has a normal form.

(b) FM = I for some λI-term F.

(c) MN1 … Nn = I for some λI-terms N1 …, Nn.

By the same method it follows that if M is a closed term of the λK-calculus having a normal form, then for some λI-terms (sic) N1, …, Nn, MN1… Nn = I is provable in the λK-calculus.

The theorem of Böhm [2] states that if M1, M2 are terms of the λK-calculus having different βη-normal forms, then ∀A1, A2 ∃N1, …, NnMiN1 … Nn = Ai is provable in the λK-βη-calculus for i = 1, 2. As a consequence of this it was shown (implicitly) in [1, 3.2.20 1/2 (1)] that if M has a normal form, then for some λK-terms N1, …, Nn, MN1 … Nn =I is provable in the λK-calculus.

It was not clear that this also could be proved for the λK-calculus since the proof of the theorem of Böhm essentially made use of λK-terms.

We conjecture that, using the results of this paper, the full theorem of Böhm can be proved for the λI-calculus.
First-order logic based on inclusion and abstraction
Authors: Bacon John
Published: May 2006

Quine has shown that set theory may be based on inclusion and abstraction [1937], [1953]. He quantifies over (or abstracts upon) sets of all kinds, of course, including sets of sets. Here I confine Quine's approach to quantification over (abstraction upon) individuals alone, or at any rate their unit classes. Forsaking quantification over sets undercuts Quine's definition of negation, however. Smullyan sketches a first-order restriction of Quine's approach with no bound class variables for which inclusion and abstraction alone are adequate logical primitives [1957, n. 10]. However, the definition of negation requires more than one element in the universe of discourse. This requirement is met for Smullyan because he is doing arithmetic. Here, on the other hand, I presuppose only that the universe is nonempty. Accordingly, I assume a third primitive notion, the empty class. I will show that this threefold basis suffices both for classical first-order logic and for a version of “free” many-sorted logic. The monadic fragment, which I call Boolean logic with abstraction, is intermediate in strength between Boolean class logic and Lesniewski's Ontology. It affords a novel perspective on descriptions, particularly in their generic use.
Double jumps of minimal degrees over cardinals
Authors: Chong C. T.
Published: May 2006

Jockusch and Posner [4] showed that every minimal ω-degree is GL2. This is achieved by exhibiting a function f recursive in 0′ which dominates every function of minimal ω-degree. The function f has the peculiar property that for every s, f(s) is defined after a search (using 0′) over the power set of Ls (Gödel's constructible hierarchy at the level s). It can be seen that a function defined in a similar manner over an infinite successor cardinal k will not be a total function, since for example if k = ρ+, then f(ρ) will not be defined until after all the subsets of ρ have been examined, and this will take at least k steps. The following questions then naturally arise: (i) For successor cardinals k, is there a function dominating every set of minimal k-degree? (ii) For arbitrary cardinals k, is every minimal k-degree GL2 (i.e. b″ = (b ∨ 0′) for b of minimal k-degree)? In this paper we answer (i) in the negative and provide a positive answer to (ii), assuming V = L. We show in fact that if k is a successor cardinal and h ≤k 0′, then there is a function of minimal k-degree below 0′ not dominated by h (Theorem 1). This implies that any refinement of the function f described above will not be able to remove the difficulties encountered. On the other hand, we introduce the notion of ‘strong domination’ to provide a positive answer to (ii) (Theorem 2 and Corollary 1). We end this paper by indicating that for limit cardinals k, there is a function below 0′ dominating every function of minimal k-degree.
On ℵ0-categorical nilrings. II
Authors: Cherlin Gregory
Published: May 2006

Theorem. The Jacobson radical of an ℵ0-categorical associative ring is nilpotent.
Degrees joining to 0′
Authors: Posner David B. , Robinson Robert W.
Published: May 2006

It is shown that if and are sets of degrees uniformly recursive in 0′ with 0 ∈ then there is a degree b with b′ = 0′, b ∪ c = 0′ for every c ∈ , and a ≰ b for every a ∈ ˜ {0}. The proof is given as an oracle construction recursive in 0′. It follows that any nonrecursive degree below 0′ can be joined to 0′ by a degree strictly below 0′. Also, if a < 0′ and a″ = 0″ then there is a degree b such that a ∪ b = 0′ and a ∩ b = 0.
Effective aspects of profinite groups
Authors: Smith Rick L.
Published: May 2006

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized (effectively!) by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach.

The fundamental notions of a co-r.e. profinite group, recursively profinite group, and the degree of a co-r.e. profinite group are defined in §1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper.

The commutator subgroup, the Frattini subgroup, thep-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3,4 and 5 are all applications of this theorem.
Rings which admit elimination of quantifiers
Authors: Berline Chantal
Published: May 2006

The aim of this paper is to provide an addendum to a paper by Rose with the same title which has appeared in an earlier issue of this Journal [2]. Our new result is:

Theorem. A ring of characteristic zero which admits elimination of quantifiers in the language (0, 1, +, ·) is an algebraically closed field.
The theory of all substructures of a structure: Characterisation and decision problems
Authors: Manders Kenneth L.
Published: May 2006

An infinitary characterisation of the first-order sentences true in all substructures of a structure M is used to obtain partial reduction of the decision problem for such sentences to that for Th(M). For the relational structure 〈R, ≤, + 〉 this gives a decision procedure for the ∃x∀y-part of the theory of all substructures, yet we show that the ∃x1x2∀y-part, and the entire theory, is Π11-complete. The theory of all ordered subsemigroups of 〈R, ≤, + 〉 is also shown Π11-complete. Applications in the philosophy of science are mentioned.
A note on finitely generated models
Authors: Pillay Anand
Published: May 2006

A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple ā in M such that (M, ā) is a prime model of its own theory. Similarly, if A ⊂ M, then M is said to be finitely generated overA if there is finite ā in M such that M is prime over A ⋃ ā. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is ā model of an ω-stable theory, M ≺ N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an ω-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.

The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite αT. However the proof in [3] for the case αT finite actually shows the following which does not assume ω-stability): Let A be atomic over a finite set, tp(ā / A) have finite Cantor-Bendixson rank, and B be atomic over A ⋃ ā. Then B is atomic over a finite set.
Finite forcing, existential types and complete types
Authors: Hirschfeld Joram
Published: May 2006

We use the spaces and n of complete types and of existential types to investigate various notions which appear in the theory of the algebraic structure of models.
Models and reality
Authors: Putnam Hilary
Published: May 2006

In 1922 Skolem delivered an address before the Fifth Congress of Scandinavian Mathematicians in which he pointed out what he called a “relativity of set-theoretic notions”. This “relativity” has frequently been regarded as paradoxical; but today, although one hears the expression “the Löwenheim-Skolem Paradox”, it seems to be thought of as only an apparent paradox, something the cognoscenti enjoy but are not seriously troubled by. Thus van Heijenoort writes, “The existence of such a ‘relativity’ is sometimes referred to as the Löwenheim-Skolem Paradox. But, of course, it is not a paradox in the sense of an antinomy; it is a novel and unexpected feature of formal systems.” In this address I want to take up Skolem's arguments, not with the aim of refuting them but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the “Löwenheim-Skolem Paradox” is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.

The structure of my argument will be as follows: I shall point out that in many different areas there are three main positions on reference and truth: there is the extreme Platonist position, which posits nonnatural mental powers of directly “grasping” forms (it is characteristic of this position that “understanding” or “grasping” is itself an irreducible and unexplicated notion); there is the verificationist position which replaces the classical notion of truth with the notion of verification or proof, at least when it comes to describing how the language is understood; and there is the moderate realist position which seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural mental powers.
Independence results
Authors: Shelah Saharon
Published: May 2006

We prove independence results concerning the number of nonisomorphic models (using the S-chain condition and S-properness) and the consistency of “ there is a universal linear order of power ℵ1”. Most of these results were announced in [Sh 4], [Sh 5].

In subsequent papers we shall prove an analog f MA for forcing which does not destroy stationary subsets of ω1 investigate -properness for various filters and prove the consistency with G.C.H. of an axiom implying SH (for ℵ1), and connected results.
A sequent calculus formulation of type assignment with equality rules for the λβ-calculus
Authors: Seldin Jonathan P.
Published: May 2006

In [1, §14E], a sequent calculus formulation (L.-formulation) of type assignment (theory of functionality) is given for a system based either on a system of combinators with strong reduction or on a system of λη-calculus provided that the rule for subject conversion (which says that if X has type α and X cnv Y then Y has type α) is postulated for the system. This sequent calculus formulation does not work for a system based on the λβ-calculus. In [2] I introduced a sequent calculus formulation for a system without the rule of subject conversion based on any of the three systems mentioned above. Further, in [2, §5] I pointed out that if proper inclusions of the form of the statement that λx·x is a function from α to β are postulated, then functions are identified with their restrictions in the λη-calculus but not in the λβ-calculus, and that therefore there is some interest in having a sequent calculus formulation of type assignment with the rule of subject conversion for systems based on the λβ-calculus. In this paper, such a system is presented, the elimination theorem (Gentzen's Hauptsatz) is proved for it, and it is proved equivalent to the natural deduction formulation of [1, §14D].

I shall assume familiarity with the λβ-calculus, and shall use (with minor modifications) the notational conventions of [1]. Hence, the theory of type assignment (theory of functionality) will be based on an atomic constant F such that if α and β are types then Fαβ represents roughly the type of functions from α to β (more exactly it represents the type of functions whose domain includes α and under which the image of α is included in β).
Minimal upper bounds for sequences of -degrees
Authors: Kechris Alexander S.
Published: May 2006

It is proved here, assuming Projective Determinacy, that every ascending sequence of -degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1.

Our general notation and terminology will be that of [Ke1] and [Mo1]. Letters i,j, k,… denote members of ω and α, β, ϒ,… members of ωω i.e. reals. Projective Determinacy (PD) is the hypothesis that every projective set of reals in determined, while in general for a collection of sets of reals Γ, Determinacy (Γ) abbreviates the statement that every set in Γ is determined.

§1. -degrees. For each m ≥ 1 and α, β Є ωω let α ≤mβ ⇔ α Є (β), α <mβ ⇔ α ≤mβ ∧ β ≰mα, and α ≡mβ ⇔ α ≤mβ ∧ β ≤mα.Clearly ≡m is an equivalence relation on ωω. The ≡m -equivalence class of α Є ωω is called its -degree, in symbols

If d = [α]m, e = [β]m then we define
An addendum to “The work of Kurt Gödel”
Authors: Kleene Stephen C.
Published: May 2006

Gödel has called to my attention that p. 773 is misleading in regard to the discovery of the finite axiomatization and its place in his proof of the consistency of GCH. For the version in [1940], as he says on p. 1, “The system Σ of axioms for set theory which we adopt [a finite one] … is essentially due to P. Bernays …”. However, it is not at all necessary to use a finite axiom system. Gödel considers the more suggestive proof to be the one in [1939], which uses infinitely many axioms.

His main achievement regarding the consistency of GCH, he says, really is that he first introduced the concept of constructible sets into set theory defining it as in [1939], proved that the axioms of set theory (including the axiom of choice) hold for it, and conjectured that the continuum hypothesis also will hold. He told these things to von Neumann during his stay at Princeton in 1935. The discovery of the proof of this conjecture On the basis of his definition is not too difficult. Gödel gave the proof (also for GCH) not until three years later because he had fallen ill in the meantime. This proof was using a submodel of the constructible sets in the lowest case countable, similar to the one commonly given today.
The slow-growing and the Graegorczyk hierarchies
Authors: Cichon E.A. , Wainer S.S.
Published: May 2006

We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α < β ∈ I then the βth function dominates the αth one at all but finitely-many positive integers x.

Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)

The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 < λ1 < λ2 < … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.

Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.
Completeness in the theory of properties, relations, and propositions
Authors: Bealer George
Published: May 2006

Higher-order theories of properties, relations, and propositions (PRPs) are known to be essentially incomplete relative to their standard notions of validity. There is, however, a first-order theory of PRPs that results when standard first-order logic is supplemented with an operation of intensional abstraction. It turns out that this first-order theory of PRPs is provably complete with respect to its standard notions of validity. The construction involves the development of a new algebraic semantic method. Unlike most other methods used in contemporary intensional logic, this method does not appeal to possible worlds as a heuristic; the heuristic used is that of PRPs taken as primitive entities. This is important, for even though the possible-worlds approach is useful in treating modal logic, it seems to be of little help in treating the logic for psychological matters. The present approach, by contrast, appears to make a step in the direction of a satisfactory treatment of both modal and intentional logic. For, by taking PRPs as primitive entities, we remain free to tailor the statement of their identity conditions so that it agrees with the logical data—modal, psychological, etc. In this way, the present approach suggests a strategy for developing a comprehensive treatment of intensional logic.

In [1] and [2] I explore this prospect philosophically. The purpose of the present paper is to lay out the technical details of the approach and to present the completeness results.
The decidability of normal K5 logics
Authors: Nagle Michael C.
Published: May 2006

The literature on modal logic includes a number of general completeness and decidability results. The work of Schiller Joe Scroggs [5], R.A. Bull [1], Kit Fine [2], and Krister Segerberg [6] provide examples.

Scroggs showed that the proper extensions of S5 have the finite model property and are axiomatizable. (Harrop [3] then argued that logics having these properties are decidable.) Bull extended Scroggs' result by showing that the normal extensions of S4.3 have the finite model property. Fine subsequently provided a model-theoretic proof of Bull's result and also proved the axiomatizability of these logics. In a different direction Segerberg proved that every normal logic containing the characteristic axioms of Lewis' systems S4 and S5 is decidable.

The present paper is in this tradition. We extend the results of Scroggs and Segerberg by showing that every normal modal logic containing the S5 axiom has the finite model property, is axiomatizable, and thus is decidable.
Effective Galois theory
Authors: Roche Peter La
Published: May 2006

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory offiniteextensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).

It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.
Model-complete theories of e-free Ax fields
Authors: Jarden Moshe , Wheeler William H.
Published: May 2006

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem is

A complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.

The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.

James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.
Producing measurable cardinals beyond κ
Authors: Kleinberg E. M.
Published: May 2006

In this paper we prove, under the assumption of a strong partition property for an uncountable cardinal κ, the existence of more than κ-many measurable cardinals greater than κ. Our proof involves so-called seminormal measures, and, along the way, we establish several key facts about such measures.
Predicate functors revisited
Authors: Quine W. V.
Published: May 2006

Quantification theory, or first-order predicate logic, can be formulated in terms purely of predicate letters and a few predicate functors which attach to predicates to form further predicates. Apart from the predicate letters, which are schematic, there are no variables. On this score the plan is reminiscent of the combinatory logic of Schönfinkel and Curry. Theirs, however, had the whole of higher set theory as its domain; the present scheme stays within the bounds of predicate logic.

In 1960 I published an apparatus to this effect, and an improved version in 1971. In both versions I assumed two inversion functors, major and minor; also a cropping functor and the obvious complement functor. The effects of these functors, when applied to an n-place predicate, are as follows:

The variables here are explanatory only and no part of the final notation. Ultimately the predicate letters need exponents showing the number of places, but I omit them in these pages.

A further functor-to continue now with the 1971 version-was padding:

Finally there was a zero-place predicate functor, which is to say simply a constant predicate, namely the predicate ‘I’ of identity, and there was a two-place predicate functor ‘∩’ of intersection. The intersection ‘F ∩ G’ received a generalized interpretation, allowing ‘F’ and ‘G’ to be predicates with unlike numbers of places. However, Steven T. Kuhn has lately shown me that the generalization is unnecessary and reducible to the homogeneous case.
The extensions of the modal logic K5
Authors: Nagle Michael C. , Thomason S. K.
Published: May 2006

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.

This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.

By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).