Related papers: (Seemingly) Impossible Theorems in Constructive Ma…

A very short proof of G\"odel's second incompleteness theorem (for set theory, second order arithmetic etc.)

We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…

We reveal a contradiction in measure-theoretic probability. The contradiction is an "equation" $1/2 = 0$ with its two sides representing probabilities. Unlike known paradoxes in mathematics, the revealed contradiction cannot be explained…

The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical methods uncovers a hidden computational content for numerous…

In this paper we demonstrate that the class of basic feasible functionals has recursion theoretic properties which naturally generalize the corresponding properties of the class of feasible functions. We also improve the Kapron - Cook…

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of…

We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability…

The assumptions needed to prove Cox's Theorem are discussed and examined. Various sets of assumptions under which a Cox-style theorem can be proved are provided, although all are rather strong and, arguably, not natural.

We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts…

This book is an introductory course to basic commutative algebra with a particular emphasis on finitely generated projective modules, which constitutes the algebraic version of the vector bundles in differential geometry. We adopt the…

We show that some mathematical results and their negations are both deducible. The derived contradictions indicate the inconsistency of current mathematics. This paper is an updated version of arXiv:math/0606635v3 with additional results…

An example of constructive (in A.A.Markov's sense) real-valued function, which is integrable by Riemann, but is not integrable by Darboux, is constructed.

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

For any $n\geq 6$ we construct almost strongly minimal geometries of type $\bullet \overset{n}{-} \bullet \overset{n}{-}\bullet$ which are $2$-ample but not $3$-ample.

This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular…

We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…

G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing…

Practicing mathematicians often assume that mathematical claims, when they are true, have good reasons to be true. Such a state of affairs is "unreasonable", in Wigner's sense, because basic results in computational complexity suggest that…

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have "nice" representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already…