We construct a model in which the tree property holds in ${\aleph _{\omega + 1}}$ and it is destructible under $Col\left( {\omega ,{\omega _1}} \right)$. On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.

Givant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.

Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

We wish to make a correction to our paper On definability of ordinals in logic with infinitely long expressions (this journal, vol. 31 (1966), pp. 365–375). Let Ω be an infinite singular cardinal and Ω+ the smallest cardinal >Ω. For the proof of the second half of Theorem 3, we incorrectly assumed that an ordinal γ is not definable in LΩ, if γ is not cofinal with any ordinal ≤Ω. However, Ω+ is indeed definable in LΩ and Theorem 3 should read:

If Ω singular, then an ordinal α is definable in Lα, if and only if σ ≤ exp(Ω+, Ω+), where exp (β,γ) denotes ordinal exponentiation βγ.

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.

We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.

If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

We study the partition relation that is a weakening of the usual partition relation . Our main result asserts that if κ is an uncountable strongly compact cardinal and , then does not hold.

Assuming every real has a sharp, we prove that for any equivalence relation either Borel reduces E0 or in a manner allows the assignment of bounded subsets of ω1 as complete invariants.

Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model has a d-basis if the types realized in are all computable and the Turing degree d can list -indices for all types realized in . We say has a d-decidable copy if there exists a model ≅ such that the elementary diagram of is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous with a 0-basis but no decidable copy.

We prove that any homogeneous with a 0′-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and is a homogeneous model with a 0-basis, we show has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous with a 0-basis has a d-decidable copy. We show that the nonlow2 degrees are 0-homogeneous bounding.

In [1, pp. 82–84] L. Åqvist considers a modal system which he calls S3.5 obtained by adding to S3 the axiom ∼□p⊃□∼□p. This system becomes S5 when the rule ├A→├□A is added to it. S3.5 is put forward to stand to S5 as S3 stands to S4 and S2 to T. In this note we show how a natural extension of the modelling for S3 in [2] can give a suitable semantics for S3.5.1

An S3 model2 is an ordered quadruple (GKRφ) where Κ is a set, G ε K and R is transitive and quasi-reflexive over K.

Let κ, λ be regular uncountable cardinals such that λ > κ+ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with s(κ) = λ starting from a ground model in which o(κ) = λ and prove that assuming ¬0¶, s(κ) = λ implies that o(κ) ≥ λ in the core model.

We study the number of normal measures on a tall cardinal. Our main results are that:

• • The least tall cardinal may coincide with the least measurable cardinal and carry as many normal measures as desired.

• • The least measurable limit of tall cardinals may carry as many normal measures as desired.

This paper is an exposition of the author's [2] investigations of a class of infinitary first order logics. These logics are sometimes characterized by saying that the syntactical role played by the positive integers in first order predicate logic is taken over by a nonstandard model of arithmetic. The result is then called a nonstandard logic.1 In particular the operations of conjunction, disjunction and quantification may be iterated infinitely (nonstandard finitely) often.

When the nonstandard model of arithmetic is an ultra power, the natural class of structures in which to interpret the nonstandard wffs is a class of ultra products. Within this framework, many model-theoretic properties of ultra products become manifest in the logical structure of the nonstandard language.

Remark 1. Finite sums and products of recursive equivalence types have been defined (see [1, p. 73 for sums and p. 130 for products]) and many of their algebraic properties revealed. It seems natural to ask if infinite sums and products can be defined in a natural manner and an algebra of such sums and products developed. The initial work in this direction is to be found in [2], [3] and [4]. In this paper we extend the above studies, concentrating for the most part on products.

We assume familiarity with the canonical languages of Post (cf. [1], [2]).

A set S of strings is said to be representable in a canonical system (F) if there is a string π such that for every string X, X ∈ S if and only if πX is provable in (F).

Suppose that K is a finite alphabet containing at least 2 symbols, and that W is a set of strings in (the symbols of) K. If W is representable in some canonical system, is it necessarily representable in a canonical system which uses only the symbols of K ? We answer this question affirmatively. Thus, e.g., it is possible to construct a 2-sign universal system over its own alphabet.2

Using the Friedman-Koepke Hyperfine Structure Theory of [2]. we provide a short construction of a gap 1 morass in the constructible universe.