Related papers: On the Constructive Truth and Falsity in Peano Ari…
We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…
The concept of informal mathematical proof considered in intuitionism is apparently vulnerable to a version of the liar paradox. However, a careful reevaluation of this concept reveals a subtle error whose correction blocks the…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
We show that numerous distinctive concepts of constructive mathematics arise automatically from an "antithesis" translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations,…
Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically…
According to Chaitin, G\"odel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable…
Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the…
In Feferman's work, explicit mathematics and theories of generalized inductive definitions play a central role. One objective of this article is to describe the connections with Martin-Lof type theory and constructive Zermelo-Fraenkel set…
Let N be the set all of non-negative integers, let A be a finite subset of N, and let (2A) be the set of all numbers of form a+b for each a and b in A. The arithmetic structure of A was accurately characterized by Freiman when (i)…
Non-compact proofs are a class of reasoning that is used in mathematics but overlooked in the analysis of (un)provability of consistency. We focus on proofs of arithmetical statements (*) "for any natural number n, F(n)." A proof of (*) is…
For Hilbert, the consistency of a formal theory T is an infinite series of statements "D is free of contradictions" for each derivation D and a consistency proof is i) an operation that, given D, yields a proof that D is free of…
We prove some constructive results that on first and maybe even on second glance seem impossible.
The usual definition of the set of constructible reals is $\Sigma ^1_2$. This set can have a simpler definition if, for example, it is countable or if every real is constructible. H. Friedman asked if the set of constructible reals can be…
We provide a systematic, thorough treatment of the foundations of probability theory and stochastic processes along the lines of E. Bishop's constructive analysis. Every existence result presented shall be a construction; and the input…
General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might…
The paper proposes and studies new classical, type-free theories of truth and determinateness with unprecedented features. The theories are fully compositional, strongly classical (namely, their internal and external logics are both…
We investigate the complexity of explicit construction problems, where the goal is to produce a particular object of size $n$ possessing some pseudorandom property in time polynomial in $n$. We give overwhelming evidence that $\bf{APEPP}$,…
We present axioms for the real numbers by omitting the field axioms and then derive the field properties of the real numbers. We prove all our theorems constructively.
In this paper we present a proof of Goodman's Theorem, a classical result in the metamathematics of constructivism, which states that the addition of the axiom of choice to Heyting arithmetic in finite types does not increase the collection…
The proofs of Kleene, Chaitin and Boolos for G\"odel's First Incompleteness Theorem are studied from the perspectives of constructivity and the Rosser property. A proof of the incompleteness theorem has the Rosser property when the…