A German translation with 2017 postscript of Floyd, Juliet. 2012. "Wittgenstein's Diagonal Argument: A Variation on Cantor and Turing." In Epistemology versus Ontology, Logic, Epistemology: Essays in Honor of Per Martin-Löf, edited by P. Dybjer, S. Lindström, E. Palmgren and G. Sundholm, 25-44. Dordrecht: Springer Science+Business Media. An analysis of philosophical aspects of Turing's diagonal argument in his (136) "On computable numbers, with an application to the Entscheidungsproblem" in relation to Wittgenstein's writings on Turing and Cantor.

In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that (...) there exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme. (shrink)

Computationalism is the claim that all possible thoughts are computations, i.e. executions of algorithms. The aim of the paper is to show that if intentionality is semantically clear, in a way defined in the paper, then computationalism must be false. Using a convenient version of the phenomenological relation of intentionality and a diagonalization device inspired by Thomson's theorem of 1962, we show there exists a thought that cannot be a computation.

Putnam construed the aim of Carnap’s program of inductive logic as the specification of a “universal learning machine,” and presented a diagonal proof against the very possibility of such a thing. Yet the ideas of Solomonoff and Levin lead to a mathematical foundation of precisely those aspects of Carnap’s program that Putnam took issue with, and in particular, resurrect the notion of a universal mechanical rule for induction. In this paper, I take up the question whether the Solomonoff–Levin proposal (...) is successful in this respect. I expose the general strategy to evade Putnam’s argument, leading to a broader discussion of the outer limits of mechanized induction. I argue that this strategy ultimately still succumbs to diagonalization, reinforcing Putnam’s impossibility claim. (shrink)

This paper explores the idea that Descartes’ cogito is a kind of diagonal argument. Using tools from modal logic, it reviews some historical antecedents of this idea from Slezak and Boos and culminates in an orginal result classifying the exact structure of belief frames capable of supporting diagonal arguments and our reconstruction of the cogito.

Computationalism is the claim that all possible thoughts are computations, i.e. executions of algorithms. The aim of the paper is to show that if intentionality is semantically clear, in a way defined in the paper, then computationalism must be false. Using a convenient version of the phenomenological relation of intentionality and a diagonalization device inspired by Thomson's theorem of 1962, we show there exists a thought that canno be a computation.

This book is about one of the most baffling of all paradoxes – the famous Liar paradox. Suppose we say: 'We are lying now'. Then if we are lying, we are telling the truth; and if we are telling the truth we are lying. This paradox is more than an intriguing puzzle, since it involves the concept of truth. Thus any coherent theory of truth must deal with the Liar. Keith Simmons discusses the solutions proposed by medieval philosophers and offers (...) his own solutions and in the process assesses other attempts to solve the paradox. Unlike such attempts, Simmons' 'singularity' solution does not abandon classical semantics and does not appeal to the kind of hierarchical view found in Barwise's and Etchemendy's The Liar. Moreover, Simmons' solution resolves the vexing problem of semantic universality – the problem of whether there are semantic concepts beyond the expressive reach of a natural language such as English. (shrink)

We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)

We consider structures of the form, where Φ and Ψ are non-empty sets and is a relation whose domain is Ψ. In particular, by using a special kind of a diagonal argument, we prove that if Φ is a denumerable recursive set, Ψ is a denumerable r.e. set, and R is an r.e. relation, then there exists an infinite family of infinite recursive subsets of Φ which are not R -images of elements of Ψ. The proof is a (...) very elementary one, without any reference even to e.g. the -theorem. Some consequences of the main result are also discussed. (shrink)

Gödel’s first incompleteness theorem is sometimes said to refute mechanism about the mind. §1 contains a discussion of mechanism. We look into its origins, motivations and commitments, both in general and with regard to the human mind, and ask about the place of modern computers and modern cognitive science within the general mechanistic paradigm. In §2 we give a sharp formulation of a mechanistic thesis about the mind in terms of the mathematical notion of computability. We present the argument (...) from Gödel’s theorem against mechanism in terms of this formulation and raise two objections, one of which is known but is here given a more precise formulation, and the other is new and based on the discussion in §1. (shrink)

The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical models of computation which claim to be able to solve the halting problem for Turing machines. We show that such arguments are flawed—a contradiction only occurs if a type of machine can compute its own diagonal function. We then demonstrate why such (...) a situation does not occur for the methods of hypercomputation under attack, and why it is unlikely to occur for any other serious methods. Introduction Issues with specific hypermachines Conclusions for hypercomputation. (shrink)

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of (...) a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers. (shrink)

I OFFER AN ANALYSIS OF DESCARTES'S COGITO WHICH IS RADICALLY NOVEL WHILE INCORPORATING MUCH AVAILABLE INSIGHT. BY ENLARGING FOCUS FROM THE DICTUM ITSELF TO THE REASONING OF DOUBT, DREAMING AND DEMON, I DEMONSTRATE A CLOSE PARALLEL TO THE LOGIC OF THE LIAR PARADOX. THIS HELPS TO EXPLAIN FAMILIAR PARADOXICAL FEATURES OF DESCARTES'S ARGUMENT. THE ACCOUNT PROVES TO BE TEXTUALLY ELEGANT AND, MOREOVER, HAS CONSIDERABLE INDEPENDENT PHILOSOPHICAL PLAUSIBILITY AS AN ACCOUNT OF MIND AND SELF.

We consider natural strengthenings of H. Friedman's Borel diagonalization propositions and characterize their consistency strengths in terms of the n -subtle cardinals. After providing a systematic survey of regressive partition relations and their use in recent independence results, we characterize n -subtlety in terms of such relations requiring only a finite homogeneous set, and then apply this characterization to extend previous arguments to handle the new Borel diagonalization propositions.

Cantor's diagonal argument provides an indirect proof that there is no one-one function from the power set of a set A into A. This paper provides a somewhat more constructive proof of Cantor's theorem, showing how, given a function f from the power set of A into A, one can explicitly define a counterexample to the thesis that f is one-one.

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. (shrink)

In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means (...) excluding as inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis. (shrink)

The article deals with Cantor's argument for the non-denumerability of reals somewhat in the spirit of Lakatos' logic of mathematical discovery. At the outset Cantor's proof is compared with some other famous proofs such as Dedekind's recursion theorem, showing that rather than usual proofs they are resolutions to do things differently. Based on this I argue that there are "ontologically" safer ways of developing the diagonal argument into a full-fledged theory of continuum, concluding eventually that famous semantic (...) paradoxes based on diagonal construction are caused by superficial understanding of what a name is. (shrink)

The authors investigated the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine PROVER9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. PROVER9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the (...) argument into better focus. Also, the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a ?diagonal argument? and, moreover, one used to establish a positive conclusion rather than a paradox. (shrink)

The authors investigated the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine PROVER9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. PROVER9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the (...) argument into better focus. Also, the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a ‘diagonal argument’ and, moreover, one used to establish a positive conclusion rather than a paradox. (shrink)

The exploration of metaphysical arguments in the symbolic AI environment provides clarification and raises unexpected questions about notions in philosophy of religion and theology. Recent attempts to apply automatic theorem prover technology to Anselm's ontological argument have led to a simplification of the argument. This computationally discovered simplification has given rise to logical observations. The article assesses one of these observations: the application of the diagonal method (in Cantor's version) to Anselm's argument. The evaluation of the (...) applications of theorem provers to metaphysical and theological arguments contributes to the following topics in philosophy of religion: the limits of natural theology, the relationship between religion and STEM, and theology's scientificity. (shrink)

This paper deploys a Cantor-style diagonal argument which indicates that there is more possible mathematical content than there are propositional functions in Russell and Whitehead's Principia Mathematica and similar formal systems. This technical result raises a historical question: "How did Russell, who was himself an expert in diagonal arguments, not see this coming?" It turns out that answering this question requires an appreciation of Russell's understanding of what logic is, and how he construed the relationship between logic (...) and Principia Mathematica. (shrink)

We consider mainly the following version of set theory: “ZF+DC and for every λ,λℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda, \lambda^{\aleph_0}}$$\end{document} is well ordered”, our thesis is that this is a reasonable set theory, e.g. on the one hand it is much weaker than full choice, and on the other hand much can be said or at least this is what the present work tries to indicate. In particular, we prove that for a sequence δ¯=⟨δs:s∈Y⟩,cf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\delta} = \langle\delta_{s}: s \in Y\rangle, {\rm cf}}$$\end{document} large enough compared to Y, we can prove the pcf theorem with minor changes.We then deduce the existence of covering numbers and define and prove existence of a class of true successor cardinals. Using this we give some diagonalization arguments on Abelian groups, chosen as a characteristic case.We end by showing that some such consequences hold even in ZF above. (shrink)

It is argued that Yablo’s Paradox is not strictly paradoxical, but rather ‘ω-paradoxical’. Under a natural formalization, the list of Yablo sentences may be constructed using a diagonalization argument and can be shown to be ω-inconsistent, but nonetheless consistent. The derivation of an inconsistency requires a uniform fixed-point construction. Moreover, the truth-theoretic disquotational principle required is also uniform, rather than the local disquotational T-scheme. The theory with the local disquotation T-scheme applied to individual sentences from the Yablo list is (...) also consistent. (shrink)

§1. Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done (...) to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.In 1995 an engineer named William Dilworth, who had published a refutation of Cantor's argument in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, sued for libel a mathematician named Underwood Dudley who had called him a crank. (shrink)

In this thesis I investigate the theoretical possibility of a universal method of prediction. A prediction method is universal if it is always able to learn from data: if it is always able to extrapolate given data about past observations to maximally successful predictions about future observations. The context of this investigation is the broader philosophical question into the possibility of a formal specification of inductive or scientific reasoning, a question that also relates to modern-day speculation about a fully automatized (...) data-driven science. I investigate, in particular, a proposed definition of a universal prediction method that goes back to Solomonoff and Levin. This definition marks the birth of the theory of Kolmogorov complexity, and has a direct line to the information-theoretic approach in modern machine learning. Solomonoff's work was inspired by Carnap's program of inductive logic, and the more precise definition due to Levin can be seen as an explicit attempt to escape the diagonal argument that Putnam famously launched against the feasibility of Carnap's program. The Solomonoff-Levin definition essentially aims at a mixture of all possible prediction algorithms. An alternative interpretation is that the definition formalizes the idea that learning from data is equivalent to compressing data. In this guise, the definition is often presented as an implementation and even as a justification of Occam's razor, the principle that we should look for simple explanations. The conclusions of my investigation are negative. I show that the Solomonoff-Levin definition fails to unite two necessary conditions to count as a universal prediction method, as turns out be entailed by Putnam's original argument after all; and I argue that this indeed shows that no definition can. Moreover, I show that the suggested justification of Occam's razor does not work, and I argue that the relevant notion of simplicity as compressibility is already problematic itself. (shrink)

We prove a generalization of the hyper-game theorem by using an abstract version of inductively generated formal topology. As applications we show proofs for Cantor theorem, uncountability of the set of functions from N to N and Gödel theorem which use no diagonal argument.

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory.

By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov (...) complexity. We shall show also that Vopěnka's proof can be reformulated in arithmetic by using the arithmetized completeness theorem. (shrink)

This paper details an algorithm for a binary, primitive recursive function that apparently computes, for any $i$ and $n$, $f_i\left(i,n\right)$. The algorithm works by exploiting the fact that, in the formal system described, the index assigned to a p.r. function codes the definitional composition of the function. The algorithm exploits such a code to generate a "canonical proof" of $f_i\left(i,n\right)=m$. Since this kind of algorithm is shown impossible by diagonal arguments, the algorithm must be in error. But the error (...) is worth investigating, for it is unclear upon sustained, earnest reflection where it lies. I thus offer this material to readers with a humble request for assistance. (shrink)

In my paper , I noted that Fitch's argument, which purports to show that if all truths are knowable then all truths are known, can be blocked by typing knowledge. If there is not one knowledge predicate, ‘ K’, but infinitely many, ‘ K 1’, ‘ K 2’, … , then the type rules prevent application of the predicate ‘ K i’ to sentences containing ‘ K i’ such as ‘ p ∧¬ K i⌜ p⌝’. This provides a motivated (...) response to Fitch's argument so long as knowledge typing is itself motivated. It was the burden of my paper to explore the case that knowledge typing is as motivated as truth typing by drawing on the parallels between epistemic paradoxes generated by sentences of the kind ‘this sentence is unknown’ and semantic paradoxes generated by sentences such as ‘this sentence is untrue’. Given that typing truth is one of the acknowledged options for solving semantic paradoxes, if the parity argument succeeds it follows that epistemic typing is as well-motivated as truth typing and that the typing response to Fitch's argument is correspondingly strong.Halbach presents an apparent problem for this argument. Let ‘ N’ and ‘ P’, respectively, denote the necessity and possibility predicates, ‘ K 1’ the knowledge predicate of first type, and let ‘γ’ be a K-free sentence such that γ ↔¬ P⌜ K 1⌜γ⌝⌝; we know such a ‘γ’ exists by a standard diagonalization argument. If we assume that γ is knowable at the next knowledge type, that is, at type 1, and that the possibility typing does not interfere with the knowledge typing, a contradiction quickly ensues. Formally: γ ↔¬ P⌜ …. (shrink)

This thesis argues that there is no sense to the idea of an absolutely general physical domain, a view I call physical anti-absolutism. Whilst there are reasons from the logic and metaphysics of totality to cast doubt upon the notion of totality, notably via Cantorian diagonal arguments and Russell’s paradox, these results have hitherto had little influence on the physics of totality—the nature of totality as viewed from our best physical theories. The totality of all physical things, we are (...) told—from loose change to distant galaxies—does comprise an absolutely general domain. After all, the universe is the whole world, the totality of all things, complete; literally, turned into one. How could it be otherwise? If there are entities which lie beyond the total, the total must subsume those entities too; totality never fails to include. Extant discussion in both metaphysics and physics has typically failed to connect the absolutely general domain as viewed by the metaphysician and logician with the absolutely general domain as viewed by the physicist. My aim is to bridge this gap, suggesting why we should consider logical results and metaphysical ideas to apply in the physical setting too. I propose that there are reasons from the physics of Newtonian mechanics, Einstein’s relativistic theories and quantum mechanics to think that that one can always add more to a particular ontology, and thus there is no sense to the idea of an all-inclusive physical domain. (shrink)

This work is, in large part, a series of refutations; it is also the author's Ph.D. thesis. First to be refuted is Russell's vicious circle principle as a general remedy for the solution of the paradoxes. The author rejects the classification of paradoxes into syntactic and semantic, since in his view there are no purely syntactic paradoxes. The distinction in logic between the uninterpreted syntactical aspect of a system and the system when given a determinate interpretation is held to be (...) untenable. Tarski's distinction between object-language and meta-language and his concept of semantically closed language are considered irrelevant for the solution of the Liar paradox. The author claims that the usual versions of the Liar paradox have the same structure as the Barber paradox, viz., [S ↔ ~S]. The author solves the Liar paradox by pointing out that it does not have a proper reference. Cantor's diagonal argument for the indenumerability [[sic]] of the real numbers is labeled as unsatisfactory. Since the diagonal number is dependent upon the real numbers in the constructed list, the author claims that this makes the diagonal number to be of a different nature and status than the real numbers in the list; thus we have what the author calls the dependence fallacy. The author also refuses to accept Cantor's nested interval proof of the indenumerability [[sic]] of the real numbers. Within the proof, two infinite sequences are constructed, each of which converges to a limit. Because the proof does not give a "definite rule of convergence," the author is not satisfied that the infinite sequences converge. Also rejected is Cantor's theorem. Other paradoxes analyzed are the Berry, Richard, heterological, "Richardian", Russell, Cantor, and Burali-Forti paradoxes.--T. G. N. (shrink)

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable (...) numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability. (shrink)

One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem — employed by Cantor himself in showing that the set of real numbers is uncountable — is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. (...) Classically, these two methods are equivalent. But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e., employs intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. I describe the ways in which these two methods.. (shrink)

In this paper we will discuss the active part played by certain diagonal arguments in the genesis of computability theory. 1 In some cases it is enough to assume the enumerability of Y while in others the effective enumerability is a substantial demand. These enigmatical words by Kleene were our point of departure: When Church proposed this thesis, I sat down to disprove it by diagonalizing out of the class of the λ–definable functions. But, quickly realizing that the diagonalization (...) cannot be done effectively, I became overnight a supporter of the thesis. (1981, p. 59) The title of our paper alludes to this very work, a task on which Kleene claims to have set out after hearing such a remarkable statement from Church, who was his teacher at the time. There are quite a few points made in this extract that may be surprising. First, it talks about a proof by diagonalization in order to test—in fact to try to falsify—a hypothesis that is not strictly formal. Second, it states that such a proof or diagonal construction fails. Third, it seems to use the failure as a support for the thesis. Finally, the episode we have just described took place at a time, autumn 1933, in which many of the results that characterize Computability Theory had not yet materialized. The aim of this paper is to show that Church and Kleene discovered a way to block a very particular instance of a diagonal construction: one that is closely related to the content of Church's thesis. We will start by analysing the logical structure of a diagonal construction. Then we will introduce the historical context in order to analyse the reasons that might have led Kleene to think that the failure of this very specific diagonal proof could support the thesis. This is a joint paper. We have both attempted to add a small piece to an amazing historical jigsaw puzzle at a juncture we feel to be appropiate. In the paper by Manzano 1997 the aforementioned words by Kleene were quoted, and since then several logicians, Enrique Alonso first and foremost, have questioned her on this issue. Here we both submit our reply. (1999, pp. 249--273). (shrink)

A mahāvidyā inference is used for establishing another inference. Its Reason is normally an omnipresent property. Its Target is defined in terms of a general feature that is satisfied by different properties in different cases. It assumes that there is no case that has the absence of its Target. The main defect of a mahāvidyā inference μ is a counterbalancing inference that can be formed by a little modification of μ. The discovery of its counterbalancing inference can invalidate such an (...) inference. This paper will argue that Cantor’s diagonal argument too shares some features of the mahāvidyā inference. A diagonal argument has a counterbalanced statement. Its main defect is its counterbalancing inference. Apart from presenting an epistemological perspective that explains the disquiet over Cantor’s proof, this paper would show that both the mahāvidyā and diagonal argument formally contain their own invalidators. (shrink)

To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing (...) these versions of Yablo's paradox, we prove some theorems in LTL. This is the first time that Yablo's paradox(es) become new(ly discovered) theorems in mathematics and logic. (shrink)

We claim that a recent article of P. Cotogno ([2003]) in this journal is based on an incorrect argument concerning the non-computability of diagonal functions. The point is that whilst diagonal functions are not computable by any function of the class over which they diagonalise, there is no ?logical incomputability? in their being computed over a wider class. Hence this ?logical incomputability? regrettably cannot be used in his argument that no hypercomputation can compute the Halting problem. (...) This seems to lead him into a further error in his analysis of the supposed conventional status of the infinite time Turing machines of Hamkins and Lewis ([2000]). Theorem 1 refutes this directly. The diagonalisation misunderstanding Infinite computation Conclusion. (shrink)

Turing’s famous 1936 paper “On computable numbers, with an application to the Entscheidungsproblem” defines a computable real number and uses Cantor’s diagonal argument to exhibit an uncomputable real. Roughly speaking, a computable real is one that one can calculate digit by digit, that there is an algorithm for approximating as closely as one may wish. All the reals one normally encounters in analysis are computable, like π, √2 and e. But they are much scarcer than the uncomputable reals (...) because, as Turing points out, the computable reals are countable, whilst the uncomputable reals have the power of the continuum. Furthermore, any countable set of reals has measure zero, so the computable reals have measure zero. In other words, if one picks a real at random in the unit interval with uniform probability distribution, the probability of obtaining an uncomputable real is unity. One may obtain a computable real, but that is in- finitely improbable. But how about individual examples of uncomputable reals? We will show two: H and the halting probability Ω, both contained in the unit interval. Their construction was anticipated in.. (shrink)

The article deals with Cantor’s diagonal argument and its alleged philosophical consequences such as that there are more reals than integers and, hence, that some of the reals must be independent of language because the totality of words and sentences is always count-able. My claim is that the main flaw of the argument for the existence of non-nameable objects or truths lies in a very superficial understanding of what a name or representation actually is.

It would be an understatement to say that Russell was interested in Cantorian diagonal paradoxes. His discovery of the various versions of Russell’s paradox—the classes version, the predicates version, the propositional functions version—had a lasting effect on his views in philosophical logic. Similar Cantorian paradoxes regarding propositions—such as that discussed in §500 of The Principles of Mathematics—were surely among the reasons Russell eventually abandoned his ontology of propositions.1 However, Russell’s reasons for abandoning what he called “denoting concepts”, and his (...) rejection of similar “semantic dualisms” such as Frege’s theory of sense and reference—at least in “On Denoting”—made no explicit mention of any Cantorian paradox. My aim in this paper is to argue that such paradoxes do pose a problem for certain theories such as Frege’s, and early Russell’s, about how definite descriptions are meaningful. My first aim is simply to lay out the problem I have in mind. Next, I shall turn to arguing that the theories of descriptions endorsed by Frege and by Russell prior to “On Denoting” are susceptible to the problem. Finally, I explore what responses a.. (shrink)