George Weaver

The Fraenkel‐Carnap question for Dedekind algebras

03B15, 03C85It is shown that the second-order theory of a Dedekind algebra is categorical if it is finitely axiomatizable.This provides a partial answer to an old and neglected question of Fraenkel and Carnap: whether every finitelyaxiomatizable semantically complete second-order theory is categorical. It follows that the second-order theoryof a Dedekind algebra is finitely axiomatizable iff the algebra is finitely characterizable. It is also shown thatthe second-order theory of a Dedekind algebra is quasi-finitely axiomatizable iff the algebra is quasi-finitelycharacterizable.

Fraenkel-Carnap properties

In the late 20’s Fraenkel and Carnap raised the question of whether or not every finitely axiomatizable, seman-tically complete theory is categorical. Carnap announced a positive answer to this question for theories for-mulated in the simple theory of types. However, that proof of this result (the Gabelbarkeitssatz) was flawed(see [1, p.26]).Whileexamplesoffinitely axiomatizablefirst-ordertheoriesthatare semanticallycompletebutnotcategorical are known, it is still open whether every finitely axiomatizable, semantically complete second-ordertheory is categorical. Awodey and Reck [2] outline a proof, due to Dana Scott, that the Fraenkel-Carnap questionhas a positive solution in the case of those second-order languages whose non-logical vocabulary is empty(see [2, p. 83, Proposition 3]). This paper also provides several conditions sufficient for the categoricity of thosefinitely axiomatizable, semantically complete theories in second-order languages whose non-logical vocabularyis non-empty ([2, p. 84]). A related approach to answering the Fraenkel-Carnap question was suggested in [7]:to restrict attention to various classes of mathematical systems. Given a finite set of non-logical constants,

The First-Order Theories of Dedekind Algebras

A Dedekind Algebra is an ordered pair (B,h) where B is a non-empty set and h is an injective unary function on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called configurations of the Dedekind algebra. There are N0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on omega called its configuration signature. The configuration signature of a Dedekind algebra counts the number of configurations in the decomposition of the algebra in each isomorphism type.The configuration signature of a Dedekind algebra encodes the structure of that algebra in the sense that two Dedekind algebras are isomorphic iff their configuration signatures are identical. Configuration signatures are used to establish various results in the first-order model theory of Dedekind algebras. These include categoricity results for the first-order theories of Dedekind algebras and existence and uniqueness results for homogeneous, universal and saturated Dedekind algebras. Fundamental to these results is a condition on configuration signatures that is necessary and sufficient for elementary equivalence.

Homogeneous and Universal Dedekind Algebras

A Dedekind algebra is an order pair (B, h) where B is a non-empty set and h is a similarity transformation on B. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ℵ0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type which occur in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. It is shown that configuration signatures can be used to characterize the homogeneous, universal and homogeneous-universal Dedekind algebras. This characterization is used to prove various results about these subclasses of Dedekind algebras.

From finitary to infinitary second-order logic

A back and forth condition on interpretations for those second-order languages without functional variables whose non-logical vocabulary is finite and excludes functional constants is presented. It is shown that this condition is necessary and sufficient for the interpretations to be equivalent in the language. When applied to second-order languages with an infinite non-logical vocabulary, excluding functional constants, the back and forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second-order languages whose non-logical vocabulary is infinite for which the back and forth condition is both necessary and sufficient. It is also shown that some applications of the back and forth construction for second-order languages can be extended to the infinitary second-order languages. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Logical Consequence in Modal Logic: Alternative Semantic Systems for Normal Modal Logics

Publisher Summary This chapter discusses the alternative semantic systems of normal modal logics. The novelty of the approach discussed in the chapter is twofold: first, to treat modal logics as consequence systems rather than logistic systems; and second, to view modal logics as extensions of non-modal ones. This second feature leads to the realization that certain usually desired properties of modal logics (e.g., compactness, weak completeness, and strong completeness) are at least in part determined by properties of the semantic systems of their non-modal “base.”

König's Infinity Lemma and Beth's Tree Theorem

König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called Königs Infinity Lemma. Konig, D. [1927. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Acta Litterarum ac Scientiarum, Szeged, 3, 121–30] includes a graph theoretic formulation: an infinite, locally finite and connected (undirected) graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite (undirected) tree with a root has a infinite branch. This tree lemma can be traced to [Beth, E. W. 1955. ‘Semantic entailment and formal derivability’, Mededelingen der Kon. Ned. Akad. v. Wet., new series 18, 13, 309–42]. It is argued that Beth independently discovered the tree lemma in the early 1950s and that it was later recognized among logicians that Beths result was a consequence of the infinity lemma. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree.

A General Setting for Dedekind's Axiomatization of the Positive Integers

A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. Each Dedekind algebra can be decomposed into a family of disjoint, countable subalgebras called the configurations of the algebra. There are ℵ0 isomorphism types of configurations. Each Dedekind algebra is associated with a cardinal-valued function on ω called its configuration signature. The configuration signature counts the number of configurations in each isomorphism type that occurs in the decomposition of the algebra. Two Dedekind algebras are isomorphic iff their configuration signatures are identical. The second-order theory of any countably infinite Dedekind algebra is categorical, and there are countably infinite Dedekind algebras whose second-order theories are not finitely axiomatizable. It is shown that there is a condition on configuration signatures necessary and sufficient for the second-order theory of a Dedekind algebra to be finitely axiomatizable. It follows that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable.

Classifying [aleph]o-Categorical Theories II: The Existence of Finitely Axiomatizable Proper Class II Theories

Clark and Krauss [1977] presents a classification of complete, satisfiable and ℵo-categorical theories in first order languages with finite non-logical vocabularies. In 1988 the first author modified this classification and raised three questions about the distribution of finitely axiomatizable theories. This paper answers two of those questions.

Syntactic features and synonymy relations: a unified treatment of some proofs of the compactness and interpolation theorems

This paper introduces the notion of syntactic feature to provide a unified treatment of earlier model theoretic proofs of both the compactness and interpolation theorems for a variety of two valued logics including sentential logic, first order logic, and a family of modal sentential logic includingM,B,S4 andS5. The compactness papers focused on providing a proof of the consequence formulation which exhibited the appropriate finite subset. A unified presentation of these proofs is given by isolating their essential feature and presenting it as an abstract principle about syntactic features. The interpolation papers focused on exhibiting the interpolant. A unified presentation of these proofs is given by isolating their essential feature and presenting it as a second abstract principle about syntactic features. This second principle reduces the problem of exhibiting the interpolant to that of establishing the existence of a family of syntactic features satisfying certain conditions. The existence of such features is established for a variety of logics (including those mentioned above) by purely combinatorial arguments.