相关论文: Logical Dreams
This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets $V$, and it is in this…
In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
The article introduces the concept of uniformity, which is formulated as a scheme of axioms. The connection of this concept with ordered sets is studied. The effectiveness of using axiom schemes as a convenient and short way of replacing…
We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more…
Dynamical systems theory is especially well-suited for determining the possible asymptotic states (at both early and late times) of cosmological models, particularly when the governing equations are a finite system of autonomous ordinary…
We express the Segre class of a monomial scheme in projective space in terms of log canonical thresholds of associated ideals. Explicit instances of the relation amount to identities involving the classical polygamma functions.
I first comment on some recent advances in computing equipment for CPU-intensive numerical simulations, and on possible developments in the near future. I then discuss some particularly important and yet unsolved problems in dynamics and…
The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
The game of SET is one of the best mathematical games ever. It is no wonder that people have tried to generalize it. We discuss existing generalizations of the game of SET to different groups. We concentrate on two types of generalization:…
This article highlights historical achievements in the partition theory of countable homogeneous relational structures, and presents recent work, current trends, and open problems. Exciting recent developments include new methods involving…
We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of…
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 introduce string diagrams as a formal mathematical, graphical language to represent, compose, program and reason about games. The language is well established in quantum physics, quantum computing and quantum linguistic with the…
We develop the theory of logarithmic p-divisible groups and the theory of logarithmic finite locally free commutative group schemes.
Classical probability theory is formulated using sets. In this paper, we extend classical probability theory with propositional computability logic. Unlike other formalisms, computability logic is built on the notion of events/games, which…
Mathematical reasoning---a core ability within human intelligence---presents some unique challenges as a domain: we do not come to understand and solve mathematical problems primarily on the back of experience and evidence, but on the basis…
The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the…