相关论文: Is Complexity a Source of Incompleteness?
In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the…
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…
It is a widespread belief that results like G\"odel's incompleteness theorems or the intrinsic randomness of quantum mechanics represent fundamental limitations to humanity's strive for scientific knowledge. As the argument goes, there are…
The celebrated Trakhtenbrot's theorem states that the set of finitely valid sentences of first-order logic is not computably enumerable. In this note we will extend this theorem by proving that the finite satisfiability problem of any…
In this short paper, I present a few theorems on sentences of arithmetic which are related to Yablo's Paradox as G\"odel's first undecidable sentence was related to the Liar paradox. In particular, I consider two different arithemetizations…
The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of…
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…
This paper continues the author's previous study \cite{Kura20}, showing that several weak principles inspired by non-normal modal logic suffice to derive various refined forms of the second incompleteness theorem. Among the main results of…
This paper gives a counterexample to the impossibility, by G\"odel's second incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a fragment of arithmetic on the assumption that the latter is consistent.…
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…
In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however,…
A formula $\phi$ is called \emph{$n$-provable} in a formal arithmetical theory $S$ if $\phi$ is provable in $S$ together with all true arithmetical $\Pi_{n}$-sentences taken as additional axioms. While in general the set of all $n$-provable…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
Little effort has been devoted to studying generalised notions or models of (un)predictability, yet is an important concept throughout physics and plays a central role in quantum information theory, where key results rely on the supposed…
Recently it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and…
We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer…
A Henkin-style proof of completeness of first-order classical logic is given with respect to a very small set (notably missing cut rule) of Genzten deduction rules for intuitionistic sequents. Insisting on sparing on derivation rules,…
Complex classifiers may exhibit "embarassing" failures in cases where humans can easily provide a justified classification. Avoiding such failures is obviously of key importance. In this work, we focus on one such setting, where a label is…
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…
The likelihood principle makes strong claims about the nature of statistical evidence but is controversial. Its claims are undermined by the existence of several examples that are assumed to show that it allows, with unity probability,…