Abstract We show that under the assumption of existence canonical inner model with one Woodin cardinal $M_1$ , there is a $\mathsf {ZFC}$ in which $\mbox {NS}_{\omega _{1}}$ $\aleph _2$ -saturated and ${\Delta }_{1}$ -definable $\omega _1$ as parameter answers question S. D. Friedman L. Wu. also starting from an arbitrary universe cardinal, saturated ladder system $\vec {C}$ full Suslin tree T ...

A direct application of Zorn’s lemma gives that every Lipschitz map f : X ⊂ Qp → Qp has an extension to a Lipschitz map f̃ : Qp → Qp . This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps S ⊂ Rn → R`. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the p-adic ...

Let X be a definably compact definable C manifold and 2 ≤ r < ∞. We prove that the set of definable Morse functions is open and dense in the set of definable C functions on X with respect to the definable C topology.

Eilenberg and Mac Lane [1] explained the notion of a 'natural' embedding by giving a categorical definition. Starting from their examples, we argue that one could equally well explain natural as meaning 'uniformly definable in set theory'. But do the categorically natural embeddings coincide with the uniformly definable ones? This is partly a technical question about whether certain well-known ...

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an a...

The vast power of iterated recurrence is tamed by data ramification: if a function over words is definable by ramified recurrence and composition, then it is feasible, i.e. computable in polynomial time, i.e. any computation using the first n input symbols can have at most p(n) distinct configurations, for some polynomial p. Here we prove a dual result for coinductive data: if a function over s...

We first show that the projection image of a discrete definable set is again for an arbitrary definably complete locally o-minimal structure. This fact together with results in previous paper implies tame dimension theory and decomposition theorem into good-shaped subsets called quasi-special submanifolds. Using this fact, latter part paper, we investigate expansions ordered groups when restric...

R l definable in R such that F (t) = G(t, F (t)) for all t ∈ (a, b) and each component function Gi : R 1+l → R is independent of the last l− i variables (i = 1, . . . , l). If R is o-minimal and F : (a, b) → R is R-Pfaffian, then (R, F ) is o-minimal (Proposition 7). We say that F : R → R is ultimately R-Pfaffian if there exists r ∈ R such that the restriction F ↾(r,∞) is R-Pfaffian. (In genera...

A central question in Banach space theory has been to identify the class of Banach spaces that contain almost isometric copies of the classical sequence spaces `p and c0. Banach space theory entered a new era in the mid 1970’s, when B. Tsirelson [34] constructed the first space not containing isomorphic copies any of the classical sequence spaces. Tsirelson’s space has been called “the first tr...

We first define the trace on a domain $\Omega$ which is definable in an o-minimal structure. then show that every function $u\in W^{1,p}(\Omega)$ vanishing boundary sense satisfies Poincar\'e inequality. finally show, given family of domains $(\Omega_t)_{t\in \mathbb{R}^k}$, constant this inequality remains bounded, if so does volume $\Omega_t$.