In the course of ordinary communication, people transmit messages (i.e., say things) which may involve the application of a truth predicate. The receiver of such a message needs to have a method which allows the extraction of non-truth-theoretic information from uses of the truth predicate; such a method can be modeled with an axiomatic system. On close examination, the choice of which axiomatic system to employ can be seen to depend on whether or not the source of the message is considered trustworthy-that is, whether the information in the message can simply be accepted, or if it must first be examined for consistency with previously known information and, on the basis of that determination, possibly be rejected. This paper explores some of the consequences involved in this framework.

Volker Halbach Axiomatic Theories of Truth. Cambridge and New York: Cambridge University Press, 2011. ix+364 pp. $85.00. ISBN 978-0-521-11581-0. Reviewed by Michael Sheard, Department of Mathematic...

A subtheory of the theory of self-referential truth known as FS is shown to be weak as a theory of truth but equivalent to full FS in its proof-theoretic strength.

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsMichael SheardMICHAEL SHEARD received his B.A. from Yale and his doctorate from the University of California, Berkeley. He taught at the University of Alaska, Fairbanks, and at Ohio State before coming in 1986 to St. Lawrence University, where he is now Professor and Chair of the Department of Mathematics. His research interests include logic, set theory, and graph theory, although he is most frequently consulted by his colleagues on questions of bird identification.

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We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

A reader coming anew to the recent work on languages which contain their own truth predicates may be perplexed by the simple question of where to begin. A first approach to the literature suggests a field which is alive and busy with investigations heading in many different directions, but there is much less indication of how various pieces fit together. There are at least two sources of this confusion. First, the literature is large and diffuse (as befits a subject which goes back over 2000 years); Visser's survey [33] aptly describes the literature as “vast but scattered, repetitive, and disconnected.” Moreover, recent interest in the field has led to a proliferation of research and publication; it seems that almost any issue of any philosophical logic journal from the mid-1980s contains some article on the topic. The second reason, in part a consequence of the first, is that while a typical article in print usually presents a good internal motivation, with clear reference to its immediate intellectual antecedents, its place in the broader picture may not be so easily discerned. The problem can be especially acute in presentations of axiomatic approaches, because decisions on certain basic questions can lie hidden in the formal and notational details which abound in any axiomatization. In fact, though, the recent research on methods for handling self-referential truth can be seen as a body of work which is very well structured, one in which a few fundamental decisions suffice to locate any particular approach in its appropriate place on the landscape. My goal is to describe this structure and in particular to stress a few critical forks in the road, which will be the recurring metaphor throughout this paper. I will also pay particular attention to pointing out where the interesting technical and mathematical questions lie.

A graph G is called uniquely hamiltonian-connected from a vertex v if there is a unique v − x hamiltonian path for every vertex x ≠ v . Vertex b is said to be penultimate to vertex c if b is the second to last vertex in the v − c hamiltonian path. An edge is called a forced edge if it appears on every hamiltonian path from v . We show that an edge xy is a forced edge if and only if x is penultimate to y and y is penultimate to x . We then show that if x is penultimate to y , then edge xy lies on the unique hamiltonian cycle of G − v and on every hamiltonian cycle in G . Using this result, some progress is made on the question of whether a graph can be uniquely hamiltonian-connected from more than one vertex.

AbstractIn a modal system of arithmetic, a theory S has the modal disjunction property if whenever S ⊢ □φ ∨ □ψ, either S ⊢ □φ or S ⊢ □ψ. S has the modal numerical existence property if whenever S ⊢ ∃x □φ(x), there is some natural number n such that S ⊢ □φ(n). Under certain broadly applicable assumptions, these two properties are equivalent.

In a language for arithmetic with a predicate T(x), intended to mean “x is the Gödel number of a true sentence”, a set S of axioms and rules of inference has the truth disjunction property if whenever S ⊢ T(\s#A) ∨ T(\s#B), either S ⊢ T(\s#A) or S ⊢ T(\s#B). Similarly, S has the truth existence property if whenever S ⊢ ∃χ T(\s#A(χ)), there is some n such that S ⊢ T(\s#A(n)). Continuing previous work, we establish whether these properties hold or fail for a large collection of possible axiomatic systems.

We add a new predicate T to the language of Peano Arithmetic, with T(x) intended to mean ‘x is the Gödel number of a true sentence of the augmented language’. We create a list of plausible axioms and rules of inference concerning this predicate T, each of which embodies some aspect of its intended interpretation as truth. We classify all subsets of the list as either consistent or inconsistent, and we measure the proof-theoretic strength of several subsets by comparing them with familiar systems of arithmetic and analysis.

For ƒ in L1[0, 1], let μ1(ƒ|M) be the set of all best L1-approximations to ƒ by nondecreasing functions, let ƒ = inf μ1(ƒ|M) and let f̂ = sup μ1(ƒ|M). We show that μ1(ƒ|M) is compact in L1 and provide a description of the elements of μ1(ƒ|M) and of its extreme points in terms ƒ and f̄f̂

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universe L into itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model” (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: if is a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinal κ, then the ultraproduct of the ground model via collapses κ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model. Definition 1.1. Let M and N be inner models (transitive, proper class models) of ZFC, and let j: M → N be an elementary embedding. The co-critical point of j is the least ordinal λ, if any exist, such that there is X ⊆ λ, X ∈ N but X ∉ M. Such an X is called a new subset of λ. It is easy to see that the co-critical point of j: M → N is a cardinal in N.

Indecomposability of ultrafilters was introduced by Keisler as a natural weakening of the concept of measurability. The property was first studied in depth by Chudnovsky and Chudnovsky, Prikry, and Silver (see [4] and [5]). One intriguing result of this early work was the following theorem: if κ is an inaccessible cardinal and there is an indecomposable ultrafilter over κ, then κ is in fact ω-Mahlo. Silver asked whether this result could be strengthened to say that an inaccessible cardinal carrying an indecomposable ultrafilter must be measurable. We prove in this paper that this is not the case; we construct a model where κ is inaccessible and carries an indecomposable ultrafilter but κ is not even weakly compact.The results in this paper form part of the research for my doctoral dissertation at the University of California, Berkeley. I would like to thank my thesis advisor, Robert Solovay, for his patient guidance during this endeavor.