Related papers: Refutability as Recursive as Provability
We distinguish finitarily between algorithmic verifiability, and algorithmic computability, to show that Goedel's 'formally' unprovable, but 'numeral-wise' provable, arithmetical proposition [(Ax)R(x)] can be finitarily evidenced as:…
First-order Goedel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Goedel logics G_V (sets of those…
Goedel Incompleteness Theorem leaves open a way around it, vaguely perceived for a long time but not clearly identified. (Thus, Goedel believed informal arguments can answer any math question.) Closing this loophole does not seem obvious…
Reproducibility is a confused terminology. In this paper, I take a fundamental view on reproducibility rooted in the scientific method. The scientific method is analysed and characterised in order to develop the terminology required to…
The purpose of this paper is to develop and study recursive proofs of coinductive predicates. Such recursive proofs allow one to discover proof goals in the construction of a proof of a coinductive predicate, while still allowing the use of…
Primitive recursion, mu-recursion, universal object and universe theories, complexity controlled iteration, code evaluation, soundness, decidability, G\"odel incompleteness theorems, inconsistency provability for set theory, constructive…
This paper studies the modal logical aspects of provability predicates and consistency statements for theories of arithmetic. First, we provide an overview of previous works on the correspondence between various derivability conditions for…
We explore from an algebraic viewpoint the properties of the tree languages definable with a first-order formula involving the ancestor predicate, using the description of these languages as those recognized by iterated block products of…
The fact that the famous Godel incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of logician community.…
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…
Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable.…
Current concerns about reproducibility in many research communities can be traced back to a high value placed on empirical reproducibility of the physical details of scientific experiments and observations. For example, the detailed…
A new viewpoint of the G\"odel's incompleteness theorem be given in this article which reveals the deep relationship between the logic and computation. Upon the results of these studies, an algorithm be given which shows how to search a…
Godel's theory T can be understood as a theory of the simply-typed lambda calculus that is extended to include the constant 0, the successor function S, and the operator R_tau for primitive recursion on objects of type tau. It is known that…
We show that a partial-correctness assertion about an iterative program is provable in Hoare Logic iffit is provable in standard second-order logic with comprehension restricted to first-order predicates. This equivalence was claimed twice…
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…
Established frameworks to understand problems with reproducibility in science begin with the relationship between our understanding of the prior probability of a claim and the statistical certainty that should be demanded of it, and explore…
In this paper, we provide a complete classification for the first-order Goedel logics concerning the property that the formulas admit logically equivalent prenex normal forms. We show that the only first-order Goedel logics that admit such…
For any functions $f(x)$, $g(x)$ from $\mathbb {N}$ to $\mathbb {R}$ we call repeats $uvu$ such that $g(|u|)\le |v|\le f(|u|)$ as {\it $f,g$-gapped repeats}. We study the possible number of $f,g$-gapped repeats in words of fixed length~$n$.…
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…