Arithmetic of Dedekind cuts of ordered Abelian groups

The elementary theory of Dedekind cuts in polynomially bounded structures

Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M. We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M, which yields a description of all subsets of M^n, definable in the expanded structure.

Ramsey for $\mathcal{R}_{1}$ ultrafilter mappings and their Dedekind cuts

Associated to each ultrafilter $\mathcal{U}$ on $ω$ and each map $p:ω\rightarrow ω$ is a Dedekind cut in the ultrapower $ω^ω/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space $\mathcal{R}_{1}$. Associated to the space $\mathcal{R}_{1}$ is a notion of Ramsey ultrafilter for $\mathcal{R}_{1}$ generalizing the familiar notion of Ramsey ultrafilter on $ω$. We characterize, under CH, the cuts …

On the number of Dedekind cuts and two-cardinal models of dependent theories

For an infinite cardinal $κ$, let $dedκ$ denote the supremum of the number of Dedekind cuts in linear orders of size $κ$. It is known that $κ<dedκ\leq 2^κ$ for all $κ$ and that $dedκ<2^κ$ is consistent for any $κ$ of uncountable cofinality. We prove however that $2^κ\leq ded ( ded ( ded ( dedκ)))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as…

Another Proof of the Existence a Dedekind Complete Totally Ordered Field

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.

Quick cut-elimination for strictly positive cuts

In this paper we show that the intuitionistic theory for finitely many iterations of strictly positive operators is a conservative extension of the Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic.

可数递归饱和皮亚诺算术模型的自同构群和强切口

The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts

本文研究可数递归饱和皮亚诺算术模型的自同构群子群，证明其小索引性质、余合集为不可数集，并指出非平凡正规子群在其中为渺小集，特别说明无限循环群不是其同态像。