We identify precisely which singular ordinals are definable by monadic second order formulae over the ordinals, assuming knowledge of the definable regular cardi-

In this article we give strong versions for preparation theorems $\mathbb{R}_{\textnormal{an,exp}}$-definable functions outgoing from methods of Lion and Rolin ($\mathbb{R}_{\textnormal{an,exp}}$ is the o-minimal structure generated by all restricted analytic global exponential function). By a deep model theoretic fact Van den Dries, Macintyre Marker every function piecewise given $\mathcal{L}_...

Let T be a complete first order theory in a language L satisfying uniform finiteness and let TP be the model companion of T in an expanded language LP with a new unary predicate P . We show that: TP satisfies uniform finiteness; if T is an expansion of the theory of dense linear order then every cut definable in a model M of TP is definable in M L; and, if T is an expansion of the theory dense ...

Tree paths are investigated using first-order logic. The following results are obtained: (i) every definable path can be defined by a firstorder formula using at most one parameter chosen from the path itself; (ii) a canonical representation of the formulas that define definable paths is obtained; and (iii) every tree that has only finitely many paths that are not definable is n-equivalent to a...

We consider implicit definability over the natural number system N,+,×,=. We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of N which are not explicitly definable from each other. The second theorem says that there exists a subset of N which is not implicitly definable but belongs to a countable, explicitly definable ...

An arithmetical system is presented with the property that from every proof a realizing term can be extracted that is definable in a certain affine linear typed variant of Gödel’s and therefore defines a non-size-increasing polynomial time computable function.

Goldblatt and Thomason’s theorem on modally definable classes of Kripke frames and Venema’s theorem on modally definable classes of Kripke models are generalised to coalgebras.