Related papers: Inconsistency of Primitive Recursive Arithmetic
The main purpose of this note is to pose a couple of problems which are easily formulated thought some seem to be not yet solved. These problems are of general interest for discrete mathematics including a new twig of a bough of theory of…
The aim of this essay is to propose a conception of mathematics that is fully consonant with naturalism. By that I mean the hypothesis that everything that exists is part of the natural world, which makes up a unitary whole.
We define instantiational and algorithmic completeness for a formal language. We show that, in the presence of Church's Thesis, an alternative interpretation of Goedelian incompleteness is that Peano Arithmetic is instantiationally…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
In this note, we show that, despite the widespread assumption, the consistency formula for Peano Arithmetic PA, Con(PA), "for all x, x is not a code of a derivation of (0=1)," is not equivalent in PA to the consistency of PA. Specifically,…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
The "variance method" has been used to prove many classical inequalities in design theory and coding theory. The purpose of this expository note is to review and present some of these inequalities in a unified setting. I will also discuss…
This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a…
This paper exposes a contradiction in the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While Godel's incompleteness theorems state that a consistent system cannot prove its consistency, they do not eliminate proofs using a…
A question is proposed whether or not set theory is consistent.
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…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
The first aim of this work is to establish a Peano type existence theorem for an initial value problem involving complex fractional derivative and the second is, as a consequence of this theorem, to give a partial answer to the local…
Formal theories of arithmetic have traditionally been based on either classical or intuitionistic logic, leading to the development of Peano and Heyting arithmetic, respectively. We propose to use $\mu$MALL as a formal theory of arithmetic…
Cone regression is a particular case of quadratic programming that minimizes a weighted sum of squared residuals under a set of linear inequality constraints. Several important statistical problems such as isotonic, concave regression or…
There have been a number of developments in measuring inconsistency in logic-based representations of knowledge. In contrast, the development of inconsistency measures for computational models of argument has been limited. To address this…
The goal of this paper is introduction of a concept of natural multidimensional numbers and to construct a generalized Peano arithmetic of these multidimensional numbers. For this purpose we define a polymultiset as a special set-like form…
Primitive recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, primitive corecursion, is underappreciated and still seen as exotic. We aim to put them both on equal footing by giving a…
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…
The recent advancements in Deep Learning models and techniques have led to significant strides in performance across diverse tasks and modalities. However, while the overall capabilities of models show promising growth, our understanding of…