More Jonsson Algebras

 We prove that on many inaccessible cardinals there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal λ is λ × ω-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra of cardinality . Lastly, we give further information on guessing of clubs. Received: 10 March 1992 / First revised version: 11 August 1997 / Second revised version: 12 September 2000 / Published online: 5 November 2002     

Jonsson algebras in successor cardinals

Israel Journal of Mathematics - We shall show here that in many successor cardinals λ, there is a Jonsson algebra (in other words Jn(λ), or λ is not a Jonsson cardinal). In...     

Intuitionistic Fixed Point Theories for Strictly Positive Operators

In this paper it is shown that the intuitionistic .xed point theory (strict) for α times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and sentences over the theory for α times iterated arithmetic comprehension without set parameters.This generalizes results previously due to Buchholz [5] and Arai [2].     

On Jonsson stability and some of its generalizations

We consider Jonsson analogues of the concepts of stability and P-stability. We also consider properties and connections of a Jonsson theory and its center that concern these concepts.     

Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

On the consistency strength of ‘Accessible’ Jonsson Cardinals and of the Weak Chang Conjecture

Using the core model K we determine better lower bounds for the consistency strength of some combinatorial principles:I. Assume that λ is a Jonsson cardinal which is ‘accessible’ in the sense that at least one of (1)-(4) holds: (1) λ is a successor cardinal; (2) λ = ω_ξ and ξ<λ; (3) λ is singular of uncountable cofinality; (4) λ is a regular but not weakly hyper-Mahlo. Then 0² exists.II. For λ = ?⁺ a successor cardinal we consider the weak Chang Conjecture, wCC(λ), which is a consequence of the Chang transfer property (λ⁺, λ)?(λ, ?).III. If λ = ?⁺?ω₂, then wCC(λ) implies the existence of 0².IV. We can determine the consistency strenght of wCC(ω₁). We include a relatively simple definition of the core model which together with the results of Dodd and Jensen suffices for our proofs.     

Testing Ten Theories

Using the most comprehensive data set now available, this investigation tests the precision of all exchange theories that now contend. Beyond precision, the investigation focuses on broad issues of effectiveness including consistency, parsimony, and whether the theories can be applied to structures larger than normally studied in the lab. Seeking greater parsimony, this investigation introduces a new model by combining parts of two contending theories. We find that all ten theories have scientific merit for all can predict with some effectiveness for the exchange structures experimentally investigated. Nevertheless, the ten vary in precision. Elementary Theory is the most precise. The new Expected-value Resistance model ranks second in precision and is the simplest. Both apply to large networks as well as the best of the other theories.     

On Positive and Weakly Positive Implicative BCI-Algebras

Prove that the notion of positive implicative BCI-algebras coincides with that of weakly positive implicative BCI-algebras, thus the whole results in the latter are still true in the former, in particular, one of these results answers definitely the first half of J. Meng and X.L. Xin’s open problem: Does the class of positive implicative BCI-algebras form a variety? The second half of the same problem is: What properties will the ideals of such an algebra have? Here, some further properties are obtained.     

E *-Stable Theories

S. Shelah proved that stability of a theory is equivalent to definability of every complete type of that theory. T. Mustafin introduced the concept of being T ^*-stable, generalizing the notion of being stable. However, T ^*-stability does not necessitate definability of types. The key result of the present article is proving the definability of types for E ^*-stable theories. This concept differs from that of being T ^*-stable by adding the condition of being continuous. As a consequence we arrive at the definability of types over any P-sets in P-stable theories, which previously was established by T. Nurmagambetov and B. Poizat for types over P-models.     

On Finite-Valued Cohomology Theories

In this paper, we describe cohomology theories constructed on the category of pairs of topological spaces and continuous maps and based on finite-valued cochains with coefficients in abelian group.     

Antiadditive Primitive Connected Theories

Our main goal is to prove that an infinite group is interpreted in every primitive connected non-superstable theory. Previously, we have introduced the concept of primitive connected theories, for which the quantifier elimination theorem was proved generalizing a similar elimination result for modules due to Baur, Monk, and Garavaglia. Here, we study primitive connected theories in which an infinite group is not interpreted, that is, theories that differ radically from theories of modules, but have a similar structure theory. Such are said to be antiadditive. (Note that theories of modules, as distinct from antiadditive ones, may be non-superstable.)     

Non-Pointed Strongly Protomodular Theories

We give criterions for strong protomodularity and prove that the strong protomodularity of an algebraic theory is inherited by its models in a category with finite limits. We give examples of strongly protomodular theories with several constants: C ^*-algebras, rings, Heyting algebras and Boolean algebras.