Church-Turing Thesis

Philosophical Computations, Part I, Course Notes. 65 pages.

ERRATA— In Appendix B, the links to UM6P France videos of Omega’s 50th birthday party have changed to https://youtube.com/watch?v=23wk87qrZts (Victoria Chaitin is actually Virginia Chaitin), and to https://youtube.com/watch?v=iImPhyubtFU

In the middle of page 15, “correspendence” should be “correspondence.”

Paradoxes and Inconsistent Mathematics es una defensa de la existencia de ciertas contradicciones como objetos verdaderos, y además ofrece un marco de trabajo lógico-matemático no-clásico apropiado para el estudio de las paradojas. Dicho marco de trabajo comprende el desarrollo de una lógica y una teoría de conjuntos deliberadamente inconsistentes, así como el desarrollo de una aritmética, álgebra, análisis real y topología inconsistentes pero no triviales. En este libro, Weber presenta su lógica dialeteica subDLQ de primer orden con identidad, y sugiere su adopción, primero como una herramienta útil para el estudio de argumentos válidos con conclusiones contradictorias (estos es, para analizar las instancias en que la lógica clásica parece fallar), y luego como candidata a una lógica universal que reemplace a la lógica clásica y sirva como base de un nuevo edificio matemático inconsistente.

Geometry in the works of mathematicians during the Islamic era included three basic parts: theoretical geometry, practical geometry, and geometry in astronomy. Theoretical geometry related to the tradition of Greek mathematicians like Euclid. Calculations of the area of lands, dividing land among inheritors, opening roads between lands, mensuration, and so on are examples of practical geometry. There is evidence that artists, carpenters, surveyors, blacksmiths, etc., used practical geometry to create their works during the Islamic era. There are references to the subject of practical geometry in science classification books by Ibn al-Akfānī and Ibn al-Nadīm.

We investigate the Church-Kalmár-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various "obstacles" to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the "design" of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.

We investigate the Church–Kalmár–Kreisel–Turing theses theoretical concerning (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church–Turing-type theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if

We examine the current status of the physical version of the Church-Turing Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic 6 (3) (2000) 265-283] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is intimately connected with the nature of time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impossible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing non-Turing computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability.

First of all, I would like to express my deep gratitude to my thesis supervisor, Prof. Alexander Leitsch, for accepting to supervise a thesis in such an interdisciplinary area, despite it was not of his primary research interest. He followed me, corrected me and kept giving support in every direction I moved to, in this interdisciplinary research. Without his scientific virtue and wideknowledge, I should admit that I would not be able to finish this work. I am also thankful for his great patience and understanding, since it needed revision so many times. I would like to thank to Prof. Karl Svozil, for his very inspiring talks and also lecture series in Quantum Computation. It helped me so much to discover my own interests. I am indebted to him for revising the parts related to physics, when the time was so critical. I would like to thank to anybody who participated to the small group of "logic and theoretical physics". In general, I am also indebted to all those people who created all those beautiful ideas that I studied, which gave me a true meaning in life. I count myself lucky for being witnessed to their work and understanding. Finally, I would like to thank to my wonderful family at most and good friends as well, for supporting me, whatever it took for them. Their never ending faith truly gave life into things. v

Listas y Funciones' •deListas (Una propu.esta de introducción de modelos .de cálculo). Resumen: Sobre una idea desarrollada por "G~. J,@COpini (Universidad de Roma), '.e introduce un modelo adicional a los clásicos de funciones recursivas y máquinas de Turing. Este modelo, no común• en• la literatura; es de una enorme potencia didáctica. En su aparente sencillez permite familiarizarse rápidamente con conceptos de gran profund!dad. El presente e~ .Ia base esltiJétural de un apunte de cátedra mas completo. , Los alumnos lI,eg~n a esta' 'parte de la materia con un 'manejo d~1 CC?~~pto clásico.' de funciones recursivas.. ". El resultado principal que se ofrece es que este nuevo modelo "al menos" incluye al de las funciones recursivas. El tema que sigue (que se esboza brevemente al final de este trabajo) es la presentación de las máquinas de Turing y la demostración de la capacidad de estas de "contemplar" entre sus posibilidades las funciones recursivas de listas. La clásica demostración de que las máquinas de Turing pueden ser "representadas" por las funciones recursivas, cierra en belleza la presentación de estos tres modelos de cálculo, mostrando su equivalencia profunda.

Consulté le 5 mars 2013. Résumé L'accès à la connaissance scientifique est une construction de l'objectivité qui nécessite l'aperçu critique de «résultats négatifs». Ceux-ci consistent en la construction explicite des limites internes aux théories et les méthodes actuelles. Nous ferons allusion au rôle de certains résultats qui, en logique, en physique ou en informatique, ont ouvert de nouveaux domaines de connaissances en affirmant : «Non, nous ne pouvons pas calculer cela, nous ne pouvons pas décider que ...». L'idée est que les sciences de la vie et de la cognition, en particulier dans le cadre des mathématiques et de l'informatique, ont besoin de résultats similaires afin de fixer des limites au transfert passif de méthodes physico-mathématiques dans leur construction autonome de la connaissance, et d'ouvrir la voie à de nouveaux outils et perspectives. Nous comparerons cette perspective avec l'exigence, tant au niveau national qu'européen, de finaliser la plupart des activités de recherche (toutes ?) pour des applications industrielles prévisibles.

While Hegel is generally not known as a philosopher of mathematics, he maintained a deep interest in the history of mathematics, especially in its transformations between antiquity and the modern age. Charles S. Peirce, who was the son of a distinguished mathematician and was involved in developments in mathematics in the second half of the nineteenth century, was critical of what he perceived as Hegel’s lack of mathematical acumen. Nevertheless, he
recognized in Hegel’s Science of Logic, structural features of his own mathematically
informed philosophy.
In this paper I look to Hegel’s discussion of magnitude in The Science of Logic, and especially to his conception of the relation between continuous and discrete magnitudes, in order to articulate a solution he might offer to difficulties encountered by Peirce in his opposition to Cantor’s set-theoretical analysis of the continuum. It is argued that Hegel’s interest in the ancient Platonic/Pythagorean tradition in mathematics provided him with crucial resources in this regard.

In this article, we are going to solve the problem P=NP for a particular kind of problems called basic problems of numerical determination. We are going to propose 3 fundamental Axioms permitting to solve the problem P=NP for basic problems of numerical determination, those Axioms can also be considered as pure logical assertions, intuitively evident and never contradicted, permitting to understand the solution of the problem P=NP for basic problems of numerical determination. We will see that those Axioms imply that the problem P=NP in undecidable for basic problems of numerical determination. Nonetheless we will see that it is possible to give a theoretical justification (which is not a classical proof) of the proposition “P≠NP”. We will then study a 2nd problem, named “PN=DPN problem” analogous to the problem P=NP but which is fundamental in mathematics.

Single-head tapes are used to simulate multihead Turing machine tape units without loss. For one-dimensional tapes, the new simulations are simpler and use far fewer tapes than prevj known simulations. For multidimensional tapes, the results are entirely new.

The recent debate on hyper-computation has raised new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics We propose here the idea of “geometry of effective physical process” as the essentially physical notion of ...

According to some philosophers, computational explanation is proprietary to psychology-it does not belong in neuroscience. But neuroscientists routinely offer computational explanations of cognitive phenomena. In fact, computational explanation was initially imported from computability theory into the science of mind by neuroscientists, who justified this move on neurophysiological grounds. Establishing the legitimacy and importance of computational explanation in neuroscience is one thing; shedding light on it is another. I raise some philosophical questions pertaining to computational explanation and outline some promising answers that are being developed by a number of authors.

SOMMARIO.-La teoria dei linguaggi formali ~ una nuova disciplina matematica con applicazioni nei settori della linguistica e dell'informatica. Ne vengono presentati i fondamenti e i problemi pifi significativi nell'intento principale di suscitare interesse per la teoria dei linguaggi nell'ambiente delle matematiche classiche. Una bibliografia commentata ~ stata redatta per il lettore che desideri addentrarsi neUa letteratura specializzata, i.-INTRODUZIONE STORICA.

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical "algorithm" for one of the insoluble problems of mathematics, the Hilbert's tenth and equivalently the Turing halting problem. The key element of this algorithm is the computability and measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values. The fact is that quantum computers can prove theorems by methods that neither a human brain nor any other Turing-computational arbiter will ever be able to reproduce. What if a quantum algorithm delivered a theorem that it was infeasible to prove classically. No such algorithm is yet known, but nor is anything known to rule out such a possibility, and this raises a question of principle: should we still accept such a theorem as undoubtedly proved? We believe that the rational answer ot this question is yes, for our confidence in quantum proofs rests upon the same foundation as our confidence in classical proofs: our acceptance of the physical laws underlying the computing operations.

The classical view of computing positions computation as a closed-box transformation of inputs (rational numbers or finite strings) to outputs. According to the interactive view of computing, computation is an ongoing interactive process rather than a function-based transformation of an input to an output. Specifically, communication with the outside world happens during the computation, not before or after it. This approach radically changes our understanding of what is computation and how it is modeled. The acceptance of interaction as a new paradigm is hindered by the Strong Church-Turing Thesis (SCT), the widespread belief that Turing Machines (TMs) capture all computation, so models of computation more expressive than TMs are impossible. In this paper, we show that SCT reinterprets the original Church-Turing Thesis (CTT) in a way that Turing never intended; its commonly assumed equivalence to the original is a myth. We identify and analyze the historical reasons for the widespread belief in SCT. Only by accepting that it is false can we begin to adopt interaction as an alternative paradigm of computation. We present Persistent Turing Machines (PTMs), that extend TMs to capture sequential interaction. PTMs allow us to formulate the Sequential Interaction Thesis, going beyond the expressiveness of TMs and of the CTT. The paradigm shift to interaction provides an alternative understanding of the nature of computing that better reflects the services provided by today's computing technology.

Iniziamo occupandoci dei concetti base di alfabeto e stringa. Definizione 1. 1. Definiamo alfabeto un insieme finito non vuoto di simboli. L' alfabeto si indica abitualmente, ma non necessariamente, con la lettera greca Σ. I simboli sono di solito lettere, cifre, segni di interpunzione ed altri caratteri tipografici, ma in generale un alfabeto può essere un insieme qualsiasi, purché finito ; chiamandolo alfabeto intendiamo avvertire che useremo i suoi elementi come usiamo le lettere della lingua italiana, cioè per formare parole non necessariamente di senso compiuto. Per questa ragione, gli elementi di Σ sono anche detti caratteri o lettere , e li intenderemo atomici, cioè non costruiti con altri elementi (in altre parole, non sono a loro volta insiemi). Alcuni esempi di alfabeti sono l' alfabeto binario β = { 0, 1 } , l' insieme dei caratteri ASCII o il suo sottoinsieme dei soli caratteri stampabili Σ = {

In this article we discuss the proof in the short unpublished paper appeared in the 3rd volume of Gödel's Collected Works entitled "On undecidable sentences" (*1931?), which provides an introduction to Gödel's 1931 ideas regarding the incompleteness of arithmetic. We analyze the meaning of the negation of the provability predicate, and how it is meant not to lead to vicious circle. We show how in fact in Gödel's entire argument there is an omission regarding the cases of non-provability, which, once taken into consideration again, allow a completely different view of Gödel's entire argument of incompleteness. Previous results of the author are applied to show that the definition of a contradiction is included in the argument of *1931?. Furthermore, an examination is also briefly presented in order to the application of the substitution in the well-known Gödel formula as a violation of the uniqueness, calling into question its very derivation.

Following Post program, we will propose a linguistic and empirical interpretation of Gödel's incompleteness theorem and related ones on unsolvability by Church and Turing. All these theorems use the diagonal argument by Cantor in order to find limitations in finitary systems, as human language, which can make "infinite use of finite means". The linguistic version of the incompleteness theorem says that every Turing complete language is Gödel incomplete. We conclude that the incompleteness and unsolvability theorems find limitations in our finitary tool, which is our complete language.

Goodstein's argument is essentially that the hereditary representation m [b] of any given natural number m in the natural number base b can be mirrored in Cantor Arithmetic, and used to well-define a finite decreasing sequence of transfinite ordinals each of which is not smaller than the ordinal corresponding to the corresponding member of Goodstein's sequence of natural numbers G(m). The standard interpretation of this argument is first that G(m) must therefore converge; and second that this conclusion-Goodstein's Theorem-is unprovable in Peano Arithmetic but true under the standard interpretation of the Arithmetic. We argue however that even assuming Goodstein's Theorem is indeed unprovable in PA, its truth must nevertheless be an intuitionistically unobjectionable consequence of some constructive interpretation of Goodstein's reasoning. We consider such an interpretation, and construct a Goodstein functional sequence to highlight why the standard interpretation of Goodstein's argument ought not to be accepted as a definitive property of the natural numbers.

We show that the classical interpretations of Tarski's inductive definitions actually allow us to define the satisfaction and truth of the quantified formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two essentially different ways: (a) in terms of algorithmic verifiabilty; and (b) in terms of algorithmic computability. We show that the classical Standard interpretation I_PA(N, Standard) of PA essentially defines the satisfaction and truth of the formulas of the first-order Peano Arithmetic PA in terms of algorithmic verifiability. It is accepted that this classical interpretation---in terms of algorithmic verifiabilty---cannot lay claim to be finitary; it does not lead to a finitary justification of the Axiom Schema of Finite Induction of PA from which we may conclude---in an intuitionistically unobjectionable manner---that PA is consistent. We now show that the PA-axioms---including the Axiom Schema of Finite Induction---are, however, algorithmically computable finitarily as satisfied / true under the Standard interpretation I_PA(N, Standard) of PA; and that the PA rules of inference do preserve algorithmically computable satisfiability / truth finitarily under the Standard interpretation I_PA(N, Standard). We conclude that the algorithmically computable PA-formulas can provide a finitary interpretation I_PA(N, Algorithmic) of PA from which we may classically conclude that PA is consistent in an intuitionistically unobjectionable manner. We define this interpretation, and show that if the associated logic is interpreted finitarily then (i) PA is categorical and (ii) Goedel's Theorem VI holds vacuously in PA since PA is consistent but not omega-consistent.

If we assume the Thesis that any classical Turing machine T, which halts on every n-ary sequence of natural numbers as input in a determinate time t(n), determines a PA-provable formula, whose standard interpretation is an n-ary arithmetical relation f(x 1 , ..., x n) that holds if, and only if, T halts, then we can define Laplace's formula recursively such that it can express the state of a deterministic quantum universe which is irreducibly probabilistic. Contents

We consider the thesis that an arithmetical relation, which holds for any, given, assignment of natural numbers to its free variables, is Turing-decidable if, and only if, it is the standard representation of a PA-provable formula. We show that, classically, such a thesis is, both, unverifiable and irrefutable, and, that it implies the Turing Thesis is false; that Gödel's arithmetical predicate R(x), treated as a Boolean function, is in the complexity class NP, but not in P; and that the Halting problem is effectively solvable, albeit not algorithmically.

We clarify the confusion, misunderstanding and misconception that the physical finiteness of the universe, if the universe is indeed finite, would rule out all hypercomputation, the kind of computation that exceeds the Turing computability, while maintaining and defending the validity of Turing computation and the Church-Turing thesis. Through private communication with some individuals, we have encountered some confusion, misunderstanding and misconception that the physical finiteness of the universe, if the universe is indeed finite, would rule out all hypercomputation, the kind of computation that exceeds the Turing computability. And now this misleading thinking has somehow made its way to formal presentation in [1]. We would like to take this as an opportunity to publicly present our arguments, for the record, against such misconception. For that purpose, we pose below three questions and then give our answer to each one.

The NP-complete problem of the travelling salesman (TSP) is considered in the framework of quantum adiabatic computation (QAC). We first derive a remarkable lower bound for the computation time for adiabatic algorithms in general as a function of the energy involved in the computation. Energy, and not just time and space, must thus be considered in the evaluation of algorithm complexity, in perfect accordance with the understanding that all computation is physical. We then propose, with oracular Hamiltonians, new quantum adiabatic algorithms of which not only the lower bound in time but also the energy requirement do not increase exponentially in the size of the input. Such an improvement in both time and energy complexity, as compared to all other existing algorithms for TSP, is apparently due to quantum entanglement. We also appeal to the general theory of Diophantine equations in a speculation on physical implementation of those oracular Hamiltonians.

To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically nondecidable problems, those of a modified version of the Hilbert's tenth problem and of the computation of the Chaitin's Ω number, which is a representation of the Gödel's Incompletness theorem. Some interesting connection to Quantum Field Theory is pointed out.

We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle-that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal-quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.

We study a set of truncated matrices, given by Smith [8], in connection to an identification criterion for the ground state in our proposed quantum adiabatic algorithm for Hilbert's tenth problem. We identify the origin of the trouble for this truncated example and show that for a suitable choice of some parameter it can always be removed. We also argue that it is only an artefact of the truncation of the underlying Hilbert spaces, through showing its sensitivity to different boundary conditions available for such a truncation. It is maintained that the criterion, in general, should be applicable provided certain conditions are satisfied. We also point out that, apart from this one, other criteria serving the same identification purpose may also be available.

We review the proposal of a quantum algorithm for Hilbert's tenth problem and provide further arguments towards the proof that: (i) the algorithm terminates after a finite time for any input of Diophantine equation; (ii) the final ground state which contains the answer for the Diophantine equation can be identified as the component state having better-than-even probability to be found by measurement at the end time-even though probability for the final ground state in a quantum adiabatic process need not monotonically increase towards one in general. Presented finally are the reasons why our algorithm is outside the jurisdiction of no-go arguments previously employed to show that Hilbert's tenth problem is recursively non-computable.

The paper is devoted to the problem of describing reality in the language of mathematics and logic in connection with the intellectual intuition corresponding to a certain stage of knowledge development. The question is raised as to how the basic requirements to mathematical theory and logic will change if some of the multiverse models of modern physics are taken as the basis. Mathematics is considered in the context of various historical approaches, and mutual criticism of intuitionism, logicism, and formalism is analyzed. It is shown that some of the well-known requirements to a formal theory (such as consistency) may begin to play a different role if the multiverse hypothesis is accepted. In the framework of theories based on the idea of multiple worlds, the logical consequence, the natural law of Duns Scotus, the law of excluded middle, and other well-known facts of classical logic which in some cases cause controversy due to their intuitive unacceptability are resolved. The paper discusses an approach based on paraconsistent logics-such logics can be considered the first ones to correspond to multiverse theories. The author raises the problem of universality of the mathematical language and the corresponding intellectual intuition-the issue of whether mathematics is able to describe any of the physically possible worlds and, consequently, to become the basis for a "theory of everything" (not so much in the sense of the theory of quantum gravity as in the sense of describing all possible worlds), and of what epistemological consequences that can lead to. It is shown that in a unified theory claiming to describe multiverse models, the classical intuitive requirement of consistency becomes restrictive and serves the purpose of approximate description of a particular world but not the totality of all possible ones. This calls for a shift in the general methodology of describing the world in the framework of such a theory, and for a revision of existing standards.

We think that computational models at the quantum level can solve some problems (at least in the sense of computational complexity) that are hard to calculate by classical models.

We will simply browse that topic considered by many authors, that is, the computational results of computational models with special structures that can send resources back to the past provided by quantum theory and general relativity [24]. These results mainly revolve around Deutschian CTCs and post-selected CTCs based on quantum theory. Compared with this, there are far fewer results for computational results in general relativity, for example, in [7].

Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel's little-known assertion that the human mind has a power "converging to infinity," and as an anchoring problem Rado's (1963) Turing-uncomputable "busy beaver" (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel's intuition to a greater level of precision, and places it within a sensible case against computationalism.

DO we need a fundamental change in our professional culture and knowledge foundation for control and automation? If so, what are necessary and critical steps we must take to ensure such a change would take place effectively and efficiently, or more general, smoothly and sustainably?
This question has started circulating in my mind two decades ago right after my first two decades of research and development in automation, robotics, intelligent control, and artificial intelligence. By the end of the 20th century, the effort of achieving the initial goal of intelligent control for complex systems, such as intelligent robotic systems and smart human-machine interaction, as envisioned by its pioneers, such as K. S. Fu [1] and G. N. Saridis [2], had seemed run into and stopped by an invisible wall, as witnessed by the fact that the well-known flagship academic gathering in the field, the annual IEEE International Symposium on Intelligent Control (IEEE ISIC), was losing steam after a decade’s rapid rising since 1985 [3]. For those of you interested, related historical reviews and future perspectives can be found in [3]–[5].

for their precious collaboration, and the Congress Service of the University of Siena for the administrative aspects of the conference. The high scientific quality of the conference was possible through the conscientious work of the Programme Committee, the Special Session organisers, and the referees. We are grateful to all members of the Programme Committee for their efficient evaluations and extensive debates, which established the final programme. We also thank the following referees:

In this paper, a biographical overview of Alan Turing, the 20th century mathematician, philosopher and early computer scientist, is presented. Turing has a rightful claim to the title of 'Father of modern computing'. He laid the theoretical groundwork for a universal machine that models a computer in its most general form before World War II. During the war, Turing was instrumental in developing and influencing practical computing devices that have been said to have shortened the war by up to two years by decoding encrypted enemy messages that were generally believed to be unbreakable. After the war, he was involved with the design and programming of early computers. He also wrote foundational papers in the areas of what are now known as Artificial Intelligence (AI) and mathematical biology shortly before his untimely death. The paper also considers Turing's subsequent influence, both scientifically and culturally.

We give a simple finitary proof that every Goodstein Sequence G(m) over the natural numbers must terminate finitely. We note first that if G(p) terminates finitely then so does G(m) for all natural numbers m < p. We then show that for every natural number k there is a natural number u such that the (k − 1) th term of G(u) is k k , and that G(u) terminates finitely.

We conclude from Gödel's Theorem VII of his seminal 1931 paper that every recursive function f (x1, x2) is representable in the first-order Peano Arithmetic PA by a formula [F (x1, x2, x3)] which is algorithmically verifiable, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counterexample under the standard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence.