Related papers: From Incompleteness Towards Completeness
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…
In this paper we will try to provide a solid form of intrinsic set theoretical optimism. In other words, we will try to vindicate G\"odel's views on phenomenology as a method for arriving at new axioms of ZFC in order to decide independent…
Different from the view that information is objective reality, this paper adopts the idea that all information needs to be compiled by the interpreter before it can be observed. From the traditional complexity definition, this paper defines…
When a proposition has no proof in an inference system, it is sometimes useful to build a counter-proof explaining, step by step, the reason of this non-provability. In general, this counter-proof is a (possibly) infinite co-inductive proof…
We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove…
A rather easy yet rigorous proof of a version of G\"odel's first incompleteness theorem is presented. The version is "each recursively enumerable theory of natural numbers with 0, 1, +, *, =, logical and, logical not, and the universal…
This is the transcript of a lecture given at UMass-Lowell in which I compare and contrast the work of Godel and of Turing and my own work on incompleteness. I also discuss randomness in physics vs randomness in pure mathematics.
We investigate the eliminability of the absoluteness operator Delta in Goedel logics. While Delta is not definable from the standard connectives and disrupts important proof-theoretic properties, we show that it becomes eliminable at the…
A completeness theorem is proved involving a system of integro-differential equations with some $\lambda$-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.
We investigate relationships between versions of derivability conditions for provability predicates. We show several implications and non-implications between the conditions, and we discuss unprovability of consistency statements induced by…
We introduce the completeness problem for Modal Logic and examine its complexity. For a definition of completeness for formulas, given a formula of a modal logic, the completeness problem asks whether the formula is complete for that logic.…
In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.
Our earlier publications showed semantic tableau admits partial exceptions to the Second Incompleteness Theorem where a formalism recognizes its self consistency and views multiplication as a 3-way relation (rather than as a total…
Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under…
The concept of coherent states in explanation of a nature of nonlinear phenomena in optics will be inevitably replaced by the concept of inequality of forward and reversed transitions.
Generalized uncertainty relations may depend not only on the commutator relation of two observables considered, but also on mutual correlations, in particular, on entanglement. The equivalence between the uncertainty relation and Bohr's…
Many papers have been published over the years that either conjecture or even (claim to) prove the universality of the form of Maxwell's equations. We present yet another derivation of Maxwell's equations and discuss the conclusions…
In this paper we propose an interpretation for self-referential propositions in a "meta-model" N* of ZF. This meta-model N* is considered as an informal model of arithmetic that mathematicians often use when working with number theory.…
It is proved that if $T$ is a $\Sigma_{n+1}$ Definable theory which is $\Sigma_n$-sound and extends $PA$, then $T$ can not prove the sentence $\Sigma_n-sound(T)$ that expresses the $\Sigma_n$-soundness of $T$. Optimality of this result is…
The no-supervenience theorem limits the capacity of physicalist theories to provide a comprehensive account of human consciousness. The proof of the theorem is difficult to formalize because it relies on both alethic and epistemic notions…