This paper concerns the theory of morasses. In the early 1970s Jensen defined (k, α)-morasses for uncountable regular cardinals k and ordinals α < k. In the early 1980s Velleman defined (k, 1)-simplified morasses for all regular cardinals k. He showed that there is a (k, 1)-simplified morass if and only if there is (k, 1)-morass. More recently he defined (k, 2)-simplified morasses and Jensen was able to show that if there is a (k, 2)-morass then there is a (k, 2)-simplified morass.

In this paper we prove the converse of Jensen's result, i.e., that if there is a (k, 2)-simplified morass then there is a (k, 2)-morass.

Let L = 〈<, +, hq, 1〉q∈ℚ where ℚ is the set of rational numbers and hq is a one-place function symbol corresponding to multiplication by q. Then the L-theory of Scott's model for intuitionistic analysis is decidable.

In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.

We follow [8] in asking when a set of ordinals X ⊆ α is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical Σ1 Skolem function for Kα can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

Let X be an infinite but separable metric space. An open cover of X is said to be large if for each x ϵ X the set {U ϵ : x ϵ U} is infinite. The symbol Λ denotes the collection of large open covers of X. An open cover of X is said to be an ω-cover if for each finite subset F of X there is a U ϵ such that F ⊆ U, and X is not a member of , X is said to have Rothberger's property if there is for every sequence (n : n = 1,2,3,…) of open covers of X a sequence (Un : n = 1,2,3,…) such that:

(1) for each n, Un is a member of n, and

(2) {Un: n = 1,2,3,…} is a cover of X.

Rothberger introduced this property in his paper [2]. For convenience we let denote the collection of all open covers of X.

In [3] it was shown that X has Rothberger's property if, and only if, the following partition relation is true for large open covers of X:

This partition relation means:

for every large cover of X, for every coloring

such that for each U ϵ and each large cover there is an i with a large cover of X,

either there is a large cover such that f({A, B}) = 0 whenever {A,B} ϵ ,

or else there is a which is not point–finite such that f{{A, B}) = 1 whenever {A, B} ϵ .

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.

Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.

Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.

The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

We prove that if ℵα is uncountable and regular, then the Beth-closure of ℒωω(Qα) is not a sublogic of ℒαω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(ℒωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindström quantifiers such that B(ℒωω(Qα)) ≤ ℒωω(Q).

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].

Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the notion with work of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by [1], we investigate here the existence of benign or weakly benign sets.

Definition 0.1. (1) The set A is benign in M if for every α, β ∊ M if p = tp(α/A) = tp(β/A) then tp*(α/A) = tp*(β/A) where the *-type is the type in the language L* with a new predicate P denoting A.

(2) The set A is weakly benign in M if for every α,β ∊ M if p = stp(α/A) = stp(β/A) then tp*(α/A) = tp*(β/A) where the *-type is the type in language with a new predicate P denoting A.

Conjecture 0.2 (too optimistic). If M is a model of stable theory T and A ⊆ M then A is benign.

Shelah observed, after learning of the Baizhanov-Baldwin reductions of the problem to equivalence relations, the following counterexample.

Lemma 0.3. There is an ω-stable rank 2 theory T with ndop which has a model M and set A such that A is not benign in M.

Faris [1] presents a system for non-void classes employing four primitive functors with another defined, and based on ten asserted and eleven rejected axioms, with a special rule of rejection analogous to Słupecki's for the syllogistic. Primitive functors are here reduced to three, axioms are simplified, and it is shown that the system is decidable by the easier syllogistic means, e.g. the tabular method of Thomas [3].

From the theses (original numbering to the right with axioms starred)

we can deduce

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

Let n be a positive integer. By a βn-model we mean an ω-model which is elementary with respect to formulas. We prove the following βn-model version of Gödel's Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a βn-model of S, then there exists a βn-model of S + “there is no countable βn-model of S”. We also prove a βn-model version of Löb's Theorem. As a corollary, we obtain a βn-model which is not a βn+1-model.