A dichotomy theorem for mono-unary algebras
We study the isomorphism relation of invariant Borel classes of countable mono-unary algebras and prove a strong dichotomy theorem.
A failure of quantifier elimination.
We show that log is needed to eliminate quantifiers in the theory of the real numbers with restricted analytic functions and exponentiation.
A family of logarithmic functions of distinct growth rates
We construct a totally ordered set Γ of positive infinite germs (i.e. germs of positive real-valued functions that tend to +∞), with order type being the lexicographic product ℵ1 × ℤ2. We show that Γ admits order preserving automorphisms of pairwise distinct growth rates.
A few Remarks on n-infinite Forcing Companions
A few Remarks on Reduced Ideal-products
A finitely Axiomatizable complete theory with atomless F1 (T).
A first-order version of Pfaffian closure
The purpose of this paper is to extend a theorem of Speissegger [J. Reine Angew. Math. 508 (1999)], which states that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Specifically, we display a collection of properties possessed by the real numbers that suffices for a version of the proof of this theorem to go through. The degree of flexibility revealed in this study permits the use of certain model-theoretic arguments for the first time, e.g. the compactness theorem....
A fixed point theorem in o-minimal structures
Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.
A fully equational proof of Parikh’s theorem
We show that the validity of Parikh’s theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of -term equations of continuous commutative idempotent semirings.
A Fully Equational Proof of Parikh's Theorem
We show that the validity of Parikh's theorem for context-free languages depends only on a few equational properties of least pre-fixed points. Moreover, we exhibit an infinite basis of μ-term equations of continuous commutative idempotent semirings.
A further glance at classifiable 1-ary functions
A generalization of a formalized theory of fields of sets on non-classical logics [Book]
A generalization of abstract model theory
A generalized Łoś ultraproduct theorem
A gradient inequality at infinity for tame functions.
Let f be a C1 function defined over Rn and definable in a given o-minimal structure M expanding the real field. We prove here a gradient-like inequality at infinity in a neighborhood of an asymptotic critical value c. When f is C2 we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level.
A graphical representation of relational formulae with complementation
We study translations of dyadic first-order sentences into equalities between relational expressions. The proposed translation techniques (which work also in the converse direction) exploit a graphical representation of formulae in a hybrid of the two formalisms. A major enhancement relative to previous work is that we can cope with the relational complement construct and with the negation connective. Complementation is handled by adopting a Smullyan-like uniform notation to classify and decompose...
A graphical representation of relational formulae with complementation∗
We study translations of dyadic first-order sentences into equalities between relational expressions. The proposed translation techniques (which work also in the converse direction) exploit a graphical representation of formulae in a hybrid of the two formalisms. A major enhancement relative to previous work is that we can cope with the relational complement construct and with the negation connective. Complementation is handled by adopting a Smullyan-like...
A group in a group.
A Hanf number for saturation and omission
Suppose t = (T,T₁,p) is a triple of two countable theories T ⊆ T₁ in vocabularies τ ⊂ τ₁ and a τ₁-type p over the empty set. We show that the Hanf number for the property ’there is a model M₁ of T₁ which omits p, but M₁ ↾ τ is saturated’ is essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some distinctions between ’first order’ and ’second order quantification’. In particular, we show that if κ is uncountable,...
A linear extension operator for Whitney fields on closed o-minimal sets
A continuous linear extension operator, different from Whitney’s, for -Whitney fields (p finite) on a closed o-minimal subset of is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.