相关论文: Is mathematics consistent?

This paper investigates the mathematical representation of time in physics. In existing theories time is represented by the real numbers, hence their formal properties represent properties of time: these are surveyed. The central question…

Choices in the semantics and the signature of a theory are integral in determining how the theory is used and how challenging it is to reason over it. Our interest in this paper lies in the SMT theory of sequences. Various versions of it…

Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and…

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a…

Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…

Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over…

In this paper I want to suggest a new solution to the problem of musical tuning. On one hand, I see it as a generalization of Just Intonation (JI) to inharmonic timbers, on another, as a unification of spectral interference and harmonicity…

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…

The concept of emergence, or synergy in its simplest form, is widely used but lacks a rigorous definition. Our work connects information and set theory to uncover the mathematical nature of synergy as the failure of distributivity. It…

We discuss why Type Theory is preferable as foundation of Mathematics compared to set theory.

We consider the problem of deciding the satisfiability of quantifier-free formulas in the theory of finite sets with cardinality constraints. Sets are a common high-level data structure used in programming; thus, such a theory is useful for…

We present a system of axioms motivated by a topological intuition: The set of subsets of any set is a topology on that set. On the one hand, this system is a common weakening of Zermelo-Fraenkel set theory ZF, the positive set theory GPK…

We offer a mathematical proof of consistency for Peano Arithmetic PA formalizable in PA. This result is compatible with Goedel's Second Incompleteness Theorem since our consistency proof does not rely on the representation of consistency as…

We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that…

We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some…

The study of theory combination in Satisfiability Modulo Theories (SMT) involves various model theoretic properties (e.g., stable infiniteness, smoothness, etc.). We show that such properties can be partly captured by the natural density of…

It was pointed out recently [A. Kent, Phys. Rev. Lett. 78 (1997) 2874] that the consistent histories approach allows contrary inferences to be made from the same data. These inferences correspond to projections $P$ and $Q$, belonging to…

G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…

We describe and explain the desire, common among mathematicians, both for unity and independence in its major themes. In the dialogue that follows, we express our spontaneous and considered judgment and reservations by contrasting the…

In this paper, we present a proof of the consistency of the New Foundations set theory ($\mathit{NF}$). $\mathit{NF}$'s main idea is to permit very large sets (including the Universal Set) by restricting set formation to stratified…