Related papers: Real Analysis in Reverse
Analytic properties of function spaces over the real and the complex fields are different in some ways. This reflects in algebraic properties which are different at times and similar in some other respects. For instance, the ring of…
Real algebra is usually thought of as the study of certain kinds of preorders on fields and rings. Among its core themes are the separation theorems known as Positivstellens\"atze. However, there is a nascent subfield of real algebra which…
In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the…
Proper classes of extensions of real field was defined and topological properties of these extensions were studied. These extensions can be connected, in this case such set is not closed under binary operations (addition and…
We study the theory of a global field k as a k-vector space with a predicate for one of the absolute values on k. For example, we prove that in this language a global field with an ultrametric or real archimedean absolute value has a…
We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and…
Some aspects of the development of physics and the mathematics set one think about relation between complex numbers and reality around us. If number to spot as the relation of two quantities, from the fact of existence of complex numbers…
This paper is a part of ongoing research on order positive fields started some years ago. We prove that the real closure of an order positive field even in non-Archimedean case is also order positive.
We prove that an analogue of Rogers' theorem on sieving holds for an order if and only if the order is a Dedekind domain. We also prove that it holds for a finite commutative ring if and only if the ring is a direct product of local rings…
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…
We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger…
We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of ten equivalent statements} borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss…
Completion is one of the most studied techniques in term rewriting and fundamental to automated reasoning with equalities. In this paper we present new correctness proofs of abstract completion, both for finite and infinite runs. For the…
Over the last century, the principle of "induction on the continuum" has been studied by different authors in different formats. All of these different readings are equivalent to one of the three versions that we isolate in this paper. We…
Going back to Kreisel in the Sixties, hyperarithmetical analysis is a cluster of logical systems just beyond arithmetical comprehension. Only recently natural examples of theorems from the mathematical mainstream were identified that fit…
Let $(K,\nu)$ be an arbitrary-rank valued field, $R_\nu$ its valuation ring, $K(\alpha)/K$ a separable finite field extension generated over $K$ by a root of a monic irreducible polynomial $f\in R_\nu[X]$. We give necessary and sufficient…
We introduce and study in detail the notion of compatibility between valuations and orderings in real hyperfields. We investigate their relation with valuations and orderings induced on factor and residue hyperfields. Much of the theory…
We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality…
We introduce an approach to the foundations of physics that is more in line with the foundations of mathematics. The idea is to examine current theories and find a set of starting physical assumptions that are sufficient to rederive them,…
Cluckers and Lipshitz have shown that real closed fields equipped with real analytic structure are o-minimal. This generalizes the well-known subanalytic structure $\mathbb{R}_{\mathrm{an}}$ on the real numbers. We extend this line of…