相关论文: A Simple and Elementary Proof of Zorn's Lemma

We formulate and discuss a general axiomatic theory of arbitrary objects. This theory is expressed in a simple first-order language without modal operators, and it is governed by classical logic.

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…

For most purposes, one can replace the use of Rolle's theorem and the mean value theorem, which are not constructively valid, by the law of bounded change. The proof of two basic results in numerical analysis, the error term for Lagrange…

A cyclic proof system is a proof system whose proof figure is a tree with cycles. The cut-elimination in a proof system is fundamental. It is conjectured that the cut-elimination in the cyclic proof system for first-order logic with…

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…

We give a categorical account of Arrow's theorem, a seminal result in social choice theory.

Superposition is an established decision procedure for a variety of first-order logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal term-generated model for the…

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos…

Knowledge representation is a popular research field in IT. As mathematical knowledge is most formalized, its representation is important and interesting. Mathematical knowledge consists of various mathematical theories. In this paper we…

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by…

We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully constructive, not using the axiom of…

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -- specifically that all partial equivalence relations have an index -- and explore the consequences of adding the…

A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries.…

We give an elementary proof of Brouwer's fixed-point theorem. The only mathematical prerequisite is a version of the Bolzano-Weierstrass theorem: a sequence in a compact subset of $n$-dimensional Euclidean space has a convergent subsequence…

The general use of subjective probabilities to model belief has been justified using many axiomatic schemes. For example, ?consistent betting behavior' arguments are well-known. To those not already convinced of the unique fitness and…

We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, thus it consists of alternatives that are optimal according to some…

Intuitive Set Theory (IST) is defined as the theory we get, when we add Axiom of Monotonicity and Axiom of Fusion to Zermelo-Fraenkel set theory. In IST, Continuum Hypothesis is a theorem, Axiom of Choice is a theorem, Skolem paradox does…

I introduce an approach for automated reasoning in first order set theories that are not finitely axiomatizable, such as $ZFC$, and describe its implementation alongside the automated theorem proving software E. I then compare the results…

We present an elementary proof of Fermat's Last Theorem. No ancillary results are used, not even the most basic ones. The proof directly leads to a contradiction of the Fermat equation in the set of integers.