Assuming Jensen's principle ◊+ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todorčević.

A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with a review (...) of some generalizations of results from descriptive set theory concerning Baire category which are then used in non-trivial Souslin tree constructions that yield Souslin algebras with a remarkable subalgebra structure. In particular, we use this method to prove that under the diamond principle there is a bi-embeddable though not isomorphic pair of homogeneous Souslin algebras. (shrink)

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We (...) also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved. (shrink)

Fix a cardinal κ. We can ask the question: what kind of a logic L is needed to characterize all models of cardinality κ up to isomorphism by their L-theories? In other words: for which logics L it is true that if any models A and B of cardinality κ satisfy the same L-theory then they are isomorphic?It is always possible to characterize models of cardinality κ by their Lκ+,κ+-theories, but we are interested in finding a “small” logic L, i.e., (...) the sentences of L are hereditarily of smaller cardinality than κ. For any cardinal κ it is independent of ZFC whether any such small definable logic L exists. If it exists it can be second order logic for κ=ω and fourth order logic or certain infinitary second order logic Lκ,ω source for uncountable κ. All models of cardinality κ can always be characterized by their theories in a small logic with generalized quantifiers, but the logic may be not definable in the language of set theory. Our work continues and extends the work of Ajtai [Miklos Ajtai, Isomorphism and higher order equivalence, Ann. Math. Logic 16 181–203]. (shrink)

We give some restrictions for the search for a model of the club principle with no Souslin trees. We show that ${\diamondsuit(2^\omega, [\omega]^\omega}$ , is almost constant on) together with CH and “all Aronszajn trees are special” is consistent relative to ZFC. This implies the analogous result for a double weakening of the club principle.

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...) with this feature in the sense that any other $\Sigma _1^2 $ -statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of $V_\lambda $ . If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies¬CH. (shrink)

Any model of ZFC + GCH has a generic extension (made with a poset of size ℵ 2 ) in which the following hold: MA + 2 ℵ 0 = ℵ 2 +there exists a Δ 2 1 -well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on ω 1 . Therefore, the study of such ladders is a main concern of this article.

We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...) measure and CATEGORY = 1. Several important new metamathematical patterns among classical statements of set theory are pointed out. (shrink)

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model L, we obtain the inner model of hereditarily ordinal definable sets [33]. In this paper...

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.

We define the concept of a logic frame , which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete , if every finite consistent theory has a model. We show that for logic frames built from the cardinality quantifiers “there exists at least λ ” completeness always implies .0-compactness. On (...) the other hand we show that a recursively compact logic frame need not be ℵ0-compact. (shrink)

Certain set theoretical notions cannot be split into finer subnotions.

Given any subset A of ω1 there is a proper partial order which forces that the predicate xA and the predicate xω1A can be expressed by -provably incompatible Σ3 formulas over the structure Hω2,,NSω1. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of Hω2 definable over Hω2,,NSω1 by a provably antisymmetric Σ3 formula with two free variables. The proofs of these results involve a technique for manipulating the (...) guessing properties of club-sequences defined on stationary subsets of ω1 at will in such a way that the Σ3 theory of Hω2,,NSω1 with countable ordinals as parameters is forced to code a prescribed subset of ω1. On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy. (shrink)