Related papers: A case for weakening the Church-Turing Thesis
One of the elegant achievements in the history of proof theory is the characterization of the provably total recursive functions of an arithmetical theory by its proof-theoretic ordinal as a way to measure the time complexity of the…
We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where alpha is a…
In 1995, David Chalmers opined as implausible that there may be parts of our arithmetical competence that no sound formal system could ever duplicate. We prove that the recursive number-theoretic relation x=Sb(y 19|Z(y)) - which is…
Classical theory proves that every primitive recursive function is strongly representable in PA; that formal Peano Arithmetic, PA, and formal primitive recursive arithmetic, PRA, can both be interpreted in Zermelo-Fraenkel Set Theory, ZF;…
We give an effective procedure that produces a natural number in its output from any natural number in its input, that is, it computes a total function. The elementary operations of the procedure are Turing-computable. The procedure has a…
The overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will…
Let $\mathcal{T}$ be any of the three canonical truth theories $\textsf{CT}^-$ (Compositional truth without extra induction), $\textsf{FS}^-$ (Friedman--Sheard truth without extra induction), and $\textsf{KF}^-$ (Kripke--Feferman truth…
We offer a mathematical proof of consistency for Peano Arithmetic PA formalizable in PA. This result is compatible with Goedel's Second Incompleteness Theorem since our consistency proof does not rely on the representation of consistency as…
It is quite well-known from Kurt Godel's (1931) ground-breaking result on the Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are…
One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of…
We prove that if our calculating capability is that of a universal Turing machine with a finite tape, then Church's thesis is true. This way we accomplish Post (1936) program.
Church's thesis claims that all effecticely calculable functions are recursive. A shortcoming of the various definitions of recursive functions lies in the fact that it is not a matter of a syntactical check to find out if an entity gives…
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…
We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…
Standard interpretations of Goedel's "undecidable" proposition, [(Ax)R(x)], argue that, although [~(Ax)R(x)] is PA-provable if [(Ax)R(x)] is PA-provable, we may not conclude from this that [~(Ax)R(x)] is PA-provable. We show that such…
Based on the MRDP theorem concerning the Hilbert tenth problem, there is a corresponding Diophantine equation called proof equation for every formula of the First-order Peano Arithmetic (PA). A formula is provable in PA, if and only if the…
"Church's thesis" ($\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\mathbb{N} \to \mathbb{N}$ is computable, i.e. definable in a model of computation. $\mathsf{CT}$ is inconsistent in both…
While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set $X$ may be coded as a…
The Church-Turing Thesis confuses numerical computations with symbolic computations. In particular, any model of computability in which equality is not definable, such as the lambda-models underpinning higher-order programming languages, is…
Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the…