Related papers: Teaching G{\"o}del's incompleteness theorems
An ultimate universal theory -- a complete theory that accounts, via few and simple first principles, for all the phenomena already observed and that will ever be observed -- has been, and still is, the aspiration of most physicists and…
When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them,…
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…
We develop first-order logic and some extensions for incomplete information scenarios and consider related complexity issues.
A detailed and rigorous analysis of G\"odel's proof of his first incompleteness theorem is presented. The purpose of this analysis is two-fold. The first is to reveal what G\"odel actually proved to provide a clear and solid foundation upon…
In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a…
A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel's incompleteness theorems. The truth value of a logical formula subject to a set of axioms…
The standard approach to logic in the literature in philosophy and mathematics, which has also been adopted in computer science, is to define a language (the syntax), an appropriate class of models together with an interpretation of…
This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness…
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…
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…
Algebraic logic studies algebraic theories related to proposition and first-order logic. A new algebraic approach to first-order logic is sketched in this paper. We introduce the notion of a quantifier theory, which is a functor from the…
A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive…
We introduce some early considerations of physical and mathematical impossibility as preludes to the Goedel incompleteness theorems. We consider some informal aspects of these theorems and their underlying assumptions and discuss some the…
Although the categorical arithmetic is not effectively axiomatizable, the belief that the incompleteness Theorems can be apply to it is fairly common. Furthermore, the so-called "essential" (or "inherent") semantic incompleteness of the…
This is an examination, a commentary, of links between some philosophical views ascribed to G\"odel and general proof theory. In these views deduction is of central concern not only in predicate logic, but in set theory too, understood from…
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…
G\"odel's second incompleteness theorem is standardly understood as showing that no sufficiently strong, consistent theory of arithmetic can prove its own consistency, a result typically interpreted against a model-theoretic background in…
I present the proof of Goedel's First Incompleteness theorem in an intuitive manner, while covering all technically challenging steps. I present generalizations of Goedel's fixed point lemma to two-sentence and multi-sentence versions,…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…