We prove the equivalence of the semantic version of Tarski’s theorem on the undefinability of truth with the semantic version of the diagonal lemma and also show the equivalence of a syntactic version of Tarski’s undefinability theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the literature and show that the syntactic version of Tarski’s theorem can deliver Gödel–Rosser’s incompleteness theorem.

A cornerstone of modern mathematical logic is the diagonal lemma of Gödel and Carnap. It is used in e.g. the classical proofs of the theorems of Gödel, Rosser and Tarski. From its first explication in 1934, just essentially one proof has appeared for the diagonal lemma in the literature; a proof that is so tricky and hard to relate that many authors have tried to avoid the lemma altogether. As a result, some so called diagonal-free proofs have been given for (...) the above mentioned fundamental theorems of logic. In this paper, we provide new proofs for the semantic formulation of the diagonal lemma, and for a weak version of the syntactic formulation of it. (shrink)

In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic. I also explain how this model can be used to determine the referent of a first-order theory such as First-Order Arithmetic. By reference or referent I mean the non-empty set of objects that the syntactical terms of a well-formed formula pick out given a particular interpretation of the language. To do so, I will first draw on previous work to make explicit (...) the notion of reference and its hyperintensional features. Then I present HYPER-REF and offer a heuristic method for determining the reference of any formula. Then I discuss some of the benefits and most salient features of HYPER-REF, including some remarks on the nature of self-reference in formal languages. (shrink)

Hypercomputation—the hypothesis that Turing-incomputable objects can be computed through infinitary means—is ineffective, as the unsolvability of the halting problem for Turing machines depends just on the absence of a definite value for some paradoxical construction; nature and quantity of computing resources are immaterial. The assumption that the halting problem is solved by oracles of higher Turing degree amounts just to postulation; infinite-time oracles are not actually solving paradoxes, but simply assigning them conventional values. Special values for non-terminating processes are likewise (...) irrelevant, since diagonalization can cover any amount of value assignments. This should not be construed as a restriction of computing power: Turing’s uncomputability is not a ‘barrier’ to be broken, but simply an effect of the expressive power of consistent programming systems. (shrink)

Most paradoxes of self-reference have a dual or ‘hypodox’. The Liar paradox (Lr = ‘Lr is false’) has the Truth-Teller (Tt = ‘Tt is true’). Russell’s paradox, which involves the set of sets that are not self-membered, has a dual involving the set of sets which are self-membered, etc. It is widely believed that these duals are not paradoxical or at least not as paradoxical as the paradoxes of which they are duals. In this paper, I argue that some paradox’s (...) duals or hypodoxes are as paradoxical as the paradoxes of which they are duals, and that they raise neglected and interestingly different problems. I first focus on Richard’s paradox (arguably the simplest case of a paradoxical dual), showing both that its dual is as paradoxical as Richard’s paradox itself, and that the classical, Richard-Poincaré solution to the latter does not generalize to the former in any obvious way. I then argue that my argument applies mutatis mutandis to other paradoxes of self-reference as well, the dual of the Liar (the Truth-Teller) proving paradoxical. (shrink)

Alonzo Church (1903–1995) was a renowned mathematical logician, philosophical logician, philosopher, teacher and editor. He was one of the founders of the discipline of mathematical logic as it developed after Cantor, Frege and Russell. He was also one of the principal founders of the Association for Symbolic Logic and the Journal of Symbolic Logic. The list of his students, mathematical and philosophical, is striking as it contains the names of renowned logicians and philosophers. In this article, we focus primarily on (...) Church’s philosophical contributions. (For an account of his life and academic history see the Introduction to The Collected Works of Alonzo Church (2019).) However, we also discuss his mathematical results when it is desirable to do so in order to pursue a philosophical issue. (shrink)

This dissertation addresses a variety of foundational issues pertaining to the notion of algorithm employed in mathematics and computer science. In these settings, an algorithm is taken to be an effective mathematical procedure for solving a previously stated mathematical problem. Procedures of this sort comprise the notional subject matter of the subfield of computer science known as algorithmic analysis. In this context, algorithms are referred to via proper names of which computational properties are directly predicated )). Moreover, many formal results (...) are traditionally stated by explicitly quantifying over algorithms. These observations motivate the view that algorithms are themselves abstract mathematical objects -- a view I refer to as algorithmic realism. The status of this view is clearly related to that of Church's Thesis -- i.e. the claim that a function is computable by an algorithm just in case it is partial recursive. But whereas Church's Thesis is widely accepted on the basis of several well-known mathematical analyses of algorithmic computability, there is presently no consensus on how we can uniformly identify individual algorithms with mathematical objects. In the first chapter of this work, I attempt to illustrate the theoretical significance of algorithmic realism. I suggest that this view is not only of foundational significance to computer science, but also worth highlighting due to the role algorithms play in the fixation of mathematical knowledge and their relationship to intensional entities such as propositions and properties. Chapter Two presents a formal framework for reducing computational discourse to mathematical discourse modeled on contemporary discussion of mathematical nominalism. Chapter Three is centered on a case study intended to illustrate the technical exigencies involved with defending algorithmic realism. Chapter Four explores various ways in which algorithms might be identified with models of computation. And finally, Chapter Five argues that no such identification can uniformly satisfy all statements of algorithmic identity and non-identity affirmed by computational practice. I suggest that the technical crux of algorithmic realism lies in the necessity of reducing recursive specifications of algorithms to iterative models of computation in a manner which preserves algorithmic identity. (shrink)

In this paper I would like to indicate that this interpretation of Gödel goes far beyond what he really proved. I would like to show that to get from his result to a conclusion of the above kind requires a train of thought which is fuelled by much more than Gödel's result itself, and that a great deal of the excessive fuel should be utilized with an extra care.