In this paper we study transitive group actions.:G × X → X, definable in an ω-stable theory, where G is a connected group and X a set of Morley rank 2, with respect to sharp transitivity on qα. Here q is the generic type of X (X is of degree 1 by Proposition (1)), for ordinals α > 0, qα is the αth power of q, i.e. (aβ)β < α, ⊨ qα iff (aβ)β < α is an independent sequence (in the sense of forking) of realizations of q, and G is defined to be sharply transitive on qα iff for all (aβ)β < α, (bβ)β < α ⊨ qα there is one and only one g ∈ G with g.aβ = bβ for all β < α. The question studied here is: For which powers α of q are there group actions subject to the above conditions with G sharply transitive on qα?

In §1 we will see that for group actions satisfying the above conditions, G can be sharply transitive only on finite powers of q. Moreover, if G is sharply transitive on qn for some n ≥ 2, then the action of the stabilizer Ga on a certain subset Y of X satisfies the conditions above with Ga being sharply transitive on qm−1, where q′ is the generic type of Y (Proposition (8)). Thus, there would be a complete answer to the question if one could find some n < ω such that there is no group action as above with G sharply transitive on qn, but for n – 1 there is. In this paper we prove that such n exists and that it is either 5 or 6. More precisely, in §2 we prove that there is no group action satisfying the above conditions with G sharply transitive on q6. This is the main result of this paper. It is not known to the author whether the same also holds for q5 instead of q6. However, it does not hold for q4, as is seen in §3. There we give an example provided from projective geometry, for a group action satisfying the above conditions with G sharply transitive on q4; for G we choose PGL(3, K) and for X the projective plane over K, where K is some algebraically closed field.

We introduce the notion of ${\cal C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.

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.

Determining the truth value of self-referential sentences is an interesting and often tricky problem. The Gödel sentence, asserting its own unprovability in P (Peano arithmetic), is clearly true in N(the standard model of P), and Löb showed that a sentence asserting its own provability in P is also true in N (see Smorynski [Sm, 4.1.1]). The problem is more difficult, and still unsolved, for sentences of the kind constructed by Kreisel [K1], which assert their own falsity in some model N* of P whose complete diagram is arithmetically defined. Such a sentence χ has the property that N ⊨ iff N* ⊭ χ (note that ¬χ has the same property).

We show in §1 that the truth value in N of such a sentence χ, after a certain normalization that breaks the symmetry between it and its negation, is determined by the parity of a natural number, called the rank of N, for the particular construction of N* used. The rank is the number of times the construction can be iterated starting from N and is finite for all the usual constructions. We also show that modifications of, e.g., Henkin's construction (in his completeness proof of predicate calculus) allow arbitrary finite values for the rank of N. Thus, on the one hand the truth value of χ in N, for a given “nice” construction of N*, is independent of the particular (normalized) choice of χ, and we shall see that χ is unique up to (provable) equivalence in P. On the other hand, the truth value in question is sensitive to minor changes in the definition of N* and its determination seems to be largely a combinatorial problem.

We consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after ω innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent n moves of ONE only, then TWO has a winning strategy depending on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] and clarifies some of the results in [B-J-S] and in [S1].

We use the spaces and n of complete types and of existential types to investigate various notions which appear in the theory of the algebraic structure of models.

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.

In 1930, A. Heyting first specified a formal system for part of intuitionistic mathematics. Although his rules were presumably motivated by the “intended interpretation” or meaning of the logical symbols, over the years a number of other possible interpretations have been discovered for which the rules are also valid. In particular, one might mention the realizablity interpretation of Kleene, the (Dialectica) interpretation of Gödel, and various semantic interpretations, such as Kripke models. (Each of these has several variants or close relatives.) Each such interpretation can be regarded as defining precisely a certain “notion of constructivity”, the study of which may illuminate the still rather vague notions which underlie the intended interpretation; or, if one doubts that there is a single interpretation “intended” by all constructivist mathematicians, the study of precisely defined interpretations may help to delineate and distinguish the possibilities.

In the last few years, Heyting's systems have been vastly extended, in order to encompass the large and growing body of constructive mathematics. Several kinds of new systems have been put forward and studied. The present author has extended the various realizability interpretations to several of these systems [B1], [B2] and drawn a number of interesting applications. The mathematical content of the present paper is an interpretation in the style of Godel's Dialectica interpretation, but applicable to the new systems put forward by Feferman [Fl]. The original motivation for this work was to obtain certain metamathematical applications: roughly speaking, Markov's rule and its variants.

We prove that if holds for a singular cardinal μ, then any collection of fewer than cf(μ) stationary subsets of μ+ must reflect simultaneously.

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.

We give upper and lower bounds for the consistency strength of the failure of a combinatorial principle introduced by Jensen. Square on singular cardinals.

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot )$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

We study nonorthogonality of symmetric, regular types and show that it preserves generic stability and is an equivalence relation on the set of all generically stable, regular types. We prove that some of the nice properties from the stable context hold in general. In the case of strongly regular types we will relate to the global Rudin–Keisler order.