The number of countable generic models for finite forcing
The number of countable isomorphism types of complete extensions of the theory of Boolean algebras
There is a conjecture of Vaught [17] which states: Without The Generalized Continuum Hypothesis one can prove the existence of a complete theory with exactly nonisomorphic, denumerable models. In this paper we show that there is no such theory in the class of complete extensions of the theory of Boolean algebras. More precisely, any complete extension of the theory of Boolean algebras has either 1 or nonisomorphic, countable models. Thus we answer this conjecture in the negative for any complete...
The number of countable models of a theory of one unary function
The number of -equivalent nonisomorphic models for κ weakly compact
For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are -equivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ¹₁-definable over . By [SV] it is possible to have a generic extension where the possible numbers of equivalence classes...
The observational predicate calculus and complexity of computations (Preliminary communication)
The ordering of commutative terms
By a commutative term we mean an element of the free commutative groupoid of infinite rank. For two commutative terms , write if contains a subterm that is a substitution instance of . With respect to this relation, is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has...
The permutation group induced on a moiety.
The positive and generalized discriminators don't exist
In this paper it is proved that there does not exist a function for the language of positive and generalized conditional terms that behaves the same as the discriminator for the language of conditional terms.
The prime spectrum of an infinite product of copies of Z
The rational field is not universally definable in pseudo-exponentiation
We show that the field of rational numbers is not definable by a universal formula in Zilber's pseudo-exponential field.
The rationality of the Poincaré series associated to the p-adic points on a variety.
The real closure of a commutative regular f-ring
The real field with the rational points of an elliptic curve
We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.
The regular inverse Galois problem over non-large fields
By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field over which the regular inverse Galois problem can be shown to be solvable, but such that does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.
The role of the omitting types theorem in infinitary logic.
The small index property for algebras.
The small index property for quadratic spaces.
The smallest common extension of a sequence of models of ZFC
In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length is sometimes the smallest common extension of this sequence and very often it is not.
The strong amalgamation property
The Strong Lefschetz Principle in Algebraic Geometry.