Saharon Shelah

The monadic theory of order

We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are modeltheoretic, and we do not use automaton theory.

Fixed-point extensions of first-order logic

We prove that the three extensions of first-order logic by means of positive inductions, monotone inductions, and so-called non-monotone (in our terminology, inflationary) inductions respectively, all have the same expressive power in the case of finite structures. As a by-product, the collapse of the corresponding fixed-point hierarchies can be deduced.

Simple unstable theories

Abstract We point out a class of unstable theories which are simple, and develop for them an analog to the basic theorems on stable theories.

Finite diagrams stable in power

Abstract In this article we define when a finite diagram of a model is stable, we investigate what is the form of the class of powers in which a finite diagram is stable, and we generalize some properties of totally transcendental theories to stable finite diagrams. Using these results we investigate several theories which have only homogeneous models in certain power. We also investigate when there exist models of a certain diagram which are λ-homogenous and not λ + -homogeneous in various powers. We also have new results about stable theories and the existence of maximally λ-saturated models of power μ.

Categoricity for abstract classes with amalgamation

Abstract Let r be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS r . We prove that for a suitable Hanf number gc0 if χ0 r is categorical inλ1+ then it is categorical in λ0.

Every two elementarily equivalent models have isomorphic ultrapowers

We prove (without G.C.H.) that every two elementarily equivalent models have isomorphic ultrapowers, and some related results.

On Successors of Singular Cardinals

Publisher Summary This chapter discusses the successors of singular cardinals and explains the situation for the successor of a strong limit singular cardinal λ . The chapter finds a special subset S* ( λ + ), from which the stationary subsets of λ + can be found, which can be stopped from being stationary by μ -complete forcing. If λ is a singular strong limit, then for every normal two place function d from λ + to κ = cfλ, S o (d)≡ S l (d) ∪ CF(λ + ,≤ κ) ≡ λ + - S* ( λ+) mod Dλ+ . Therefore, S o ( d ) does not depend on d up to equivalence mod Dλ+ .

Choiceless polynomial time

Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, one of us conjectured the negative answer. The problem motivated a quest of stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras). The idea is to replace arbitrary choice with parallel execution. The resulting logic is more expressive than other PTime logics in the literature. A more difficult theorem shows that the logic does not capture all PTime.

Whitehead groups may not be free even assuming ch, II

AbstractWe prove some theorems on uncountable abelian groups, and consistency results promised in the first part, and also that a variant of