Related papers: The axiom of choice and model-theoretic structures
In order to accommodate the empirical fact that population structures are rarely simple, modern studies of evolutionary dynamics allow for complicated and highly-heterogeneous spatial structures. As a result, one of the most difficult…
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$.…
We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be…
Relational structures are emerging as ubiquitous mathematical machinery in the semantics of open systems of various kinds. Cartesian bicategories are a well-known categorical algebra of relations that has proved especially useful in recent…
We investigate different set-theoretic constructions in Residuated Logic based on Fitting's work on Intuitionistic Set Theory. We start by stating some results concerning constructible sets within valued models of Set Theory. We present two…
We study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, G\"{o}del's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their…
A knowledge system S describing a part of real world does in general not contain complete information. Reasoning with incomplete information is prone to errors since any belief derived from S may be false in the present state of the world.…
Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.
We use decision theory to confront uncertainty that is sufficiently broad to incorporate "models as approximations." We presume the existence of a featured collection of what we call "structured models" that have explicit substantive…
The mechanism of spontaneous symmetry breaking in quantum systems is briefly reviewed, rectifying part of the standard wisdom on logical and mathematical grounds. The crucial role of the localization properties of the time evolution for the…
The Higgs mechanism of mass generation is the main ingredient in the contemporary Standard Model and its various generalizations. However, there is no comprehensive theory of spontaneous symmetry breaking. We summarize the relevant…
We give a simple example of a set that is weakly Dedekind infinite (= can be mapped onto omega) but dually Dedekind finite (=cannot be mapped noninjectively onto itself), namely, the power set of a superamorphous set. (A infinite set is…
This article presents an elementary proof of Zorn's Lemma under the Axiom of Choice, simplifying and supplying necessary details in the original proof by Paul R. Halmos in his book, Naive Set Theory. Also provided, is a preamble to Zorn's…
This paper deals with the question of whether uncertainty regarding model structure, especially in climate modeling, exhibits a kind of "chaos." Do small changes in model structure, in other words, lead to large variations in ensemble…
We isolate several classes of stationary sets of kappa^omega and investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.
An integer part I of a real closed field K is a discretely ordered subring with minimal element 1 such that, for every x in K, I contains some i such that x is between i and i+1 in the ordering of K. Mourgues and Ressayre showed that every…
We address a recent proposal concerning 'surplus structure' due to Nguyen et al. ['Why Surplus Structure is Not Superfluous.' Br. J. Phi. Sci. Forthcoming.] We argue that the sense of 'surplus structure' captured by their formal criterion…
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and G\"odel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed…
We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth (even in a relative sense). Our method is to…
In this paper, a model for understanding the effects of selection using systems- level computational approaches is introduced. A number of concepts and principles essential for understanding the motivation for constructing the model will be…