Diagonal Argument Research Articles

From Ancient Greece to Modern Education: Universality and Lack of Generalization of the Socratic Dialogue

Two thousand four hundred years ago Socrates gave a remarkable lesson of geometry, perhaps the first detailed record of a pedagogical method in vivo in history [Plato. (2008). Apología de Sócrates. Menón. Crátilo. Madrid: Alianza Editorial]. Socrates asked Meno's slave 50 questions requiring simple additions or multiplications. At the end of the lesson the student discovered by himself how to duplicate a square using the diagonal of the given one as the side of the new square. We studied empirically the reproducibility of this dialogue in educated adults and adolescents of the 21st century. Our results show a remarkable agreement between Socratic and empiric dialogues. Even in questions in which Meno's slave made a mistake, within an unbounded number of possible erred responses, the vast majority of participants produced the same error as Meno's slave. Our results show that the Socratic dialogue is built on a strong intuition of human knowledge and reasoning which persists more than 24 centuries after its conception, providing one of the most striking demonstrations of universality across time and cultures. At the same time, they also emphasize its educational failure. After following every single question including Socrates' “diagonal argument,” almost 50% of the participants failed to learn the simplest generalization when asked to double the area of a square of different size.

A Computationally-Discovered Simplification of the Ontological Argument

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.

Roads to infinity: the mathematics of truth and proof

The Diagonal Argument Counting and Countability Does One Infinite Size Fit All? Cantor's Diagonal Argument Transcendental Numbers Other Uncountability Proofs Rates of Growth The Cardinality of the Continuum Historical Background Ordinals Counting Past Infinity The Countable Ordinals The Axiom of Choice The Continuum Hypothesis Induction Cantor Normal Form Goodstein's Theorem Hercules and the Hydra Historical Background Computability and Proof Formal Systems Post's Approach to Incompleteness Godel's First Incompleteness Theorem Godel's Second Incompleteness Theorem Formalization of Computability The Halting Problem The Entscheidungs problem Historical Background Logic Propositional Logic A Classical System A Cut-Free System for Propositional Logic Happy Endings Predicate Logic Completeness, Consistency, Happy Endings Historical Background Arithmetic How Might We Prove Consistency? Formal Arithmetic The Systems PA and PAomega Embedding PA in PAomega Cut Elimination in PAomega The Height of This Great Argument Roads to Infinity Historical Background Natural Unprovable Sentences A Generalized Goodstein Theorem Countable Ordinals via Natural Numbers From Generalized Goodstein to Well-Ordering Generalized and Ordinary Goodstein Provably Computable Functions Complete Disorder Is Impossible The Hardest Theorem in Graph Theory Historical Background Axioms of Infinity Set Theory without Infinity Inaccessible Cardinals The Axiom of Determinacy Largeness Axioms for Arithmetic Large Cardinals and Finite Mathematics Historical Background

Cantor's Other Proofs that R Is Uncountable

SummaryThis expository note describes some of the history behind Georg Cantor's proof that the real numbers are uncountable. In fact, Cantor gave three different proofs of this important but initially controversial result. The first was published in 1874 and the famous diagonalization argument was not published until nearly two decades later. We explore the different ideas used in each of his three proofs.

On Cantor's First Uncountability Proof, Pick's Theorem, and the Irrationality of the Golden Ratio

In Cantor's original proof of the uncountability of the reals (not the diagonalization argument), he constructs, given any countable sequence of real numbers, a real number not in the sequence. When we apply this argument to a certain standard enumeration of the rationals, the real number we produce will necessarily be irrational. Using some planar geometry, including Pick's theorem on the number of lattice points enclosed within certain polygonal regions, we show that this number is the reciprocal of the golden ratio, whence follows the well-known fact that the golden ratio is irrational.

Propagation of singularities for Cauchy problems of semilinear thermoelastic systems with microtemperatures

AbstractThe propagation of singularities of solutions to the Cauchy problem of a semilinear thermoelastic system with microtemperatures in one space variable is studied. First, by using a diagonalization argument of phase space analysis, the coupled thermoelastic system with microtemperatures will be decoupled microlocally. Second, using a classical bootstrap argument, the property of finite propagation speed of singularities for the semilinear thermoelastic system is obtained. Finally, it is also shown that the microlocal weak singularities propagate along the null bicharacteristics of the hyperbolic operators of the coupled semilinear system (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Diagonalization in Double Frames

We consider structures of the form (Φ, Ψ, R), where Φ and Ψ are non-empty sets and \({R\subseteq \Psi\times \Phi}\) 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 \({S_{n}^{m}}\)-theorem. Some consequences of the main result are also discussed.

L p − L q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

Lp − Lq decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, Lp − Lq decay estimate of parabolic type of solution to the Cauchy problem is obtained.

Structural properties of oracle classes

Denote by C the class of oracles relative to which P = NP (collapsing oracles), and by S the class of oracles relative to which P ≠ NP (separating oracles). We present structural results on C and S . Using a diagonalization argument, we show that neither C nor S is closed under disjoint union, also known as join. We show that this implies that neither C nor S is closed under union, intersection, or symmetric difference. Consequently EL 1 , the first level of the extended low hierarchy, is not closed under join.

Continuum, name and paradox

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.

A parabolic problem arising in Financial Mathematics

We study a parabolic problem arising in Financial Mathematics. Under suitable conditions, we prove the existence and uniqueness of solutions in a general domain using the method of upper and lower solutions and a diagonal argument.

Boundary value problems on the half-line for a generalised Painlevé II equation

Two types of boundary value problem on the infinite half-line I = [ 0 , + ∞ ] are investigated for a class of generalised Painlevé II equations. Existence results are obtained via the method of upper and lower solutions together with a diagonal argument.

Penguins And Pandas: A Note On Teaching Cantors Diagonal Argument

This note describes contexts that have been used by the author in teaching Cantors diagonal argument to fine arts and humanities students.

Cantor’s Diagonal Argument: An Extension to the Socialist Calculation Debate—A Comment

In a recent article Robert P. Murphy (2006) uses Cantor’s diagonal argument to prove that market socialism could not function, since it would be impossible for the Central Planning Board to complete a list containing all conceivable goods (or prices for them). In the present paper we argue that Murphy is not only wrong in claiming that the number of goods included in the list should be uncountable, but also that the number of equations/prices is irrelevant from the point of view of market socialism.

Existence Results on Semi-Infinite Intervals for Nonlocal Evolutions Equations and Inclusions via Semigroup Theory

In this paper, we prove existence and controllability results on semi-infinite intervals for first- and second-order evolutions equations and inclusions in Banach spaces with nonlocal conditions. Our theory makes use of semigroups, cosine families, nonlinear alternative of Leray–Schauder, and a diagonalization argument.

Undecidability of the existence of dictator for strongly candidate stable voting procedures in an infinite society and Cantor's diagonal argument

The strong candidate stability theorem by Dutta et al. [ 4], one of the major theoremsof social choice theory, states that, with a finite number of voters, there exists a dictator for any voting procedure which satisfies strong candidate stability, strong unanimity and independence of irrelevant alternatives (IIA). This paper investigates a decidability problem of voting procedures in a society with an infinite number of individuals (infinite society) using Cantor's diagonal argument presented by Yanofsky [ 19] which is based on Lawvere [ 10]. We will show the following result. The problem whether a strongly candidate stable voting procedure has a dictator or has no dictator in an infinite society is undecidable. It is proved using the arguments similar to those used to prove an extended version of Cantor's theorem that there cannot be an onto function from (the set of natural numbers) to its power set (). This undecidability means that for any strongly candidate stable voting procedure we can not decide whether or not it has a dictator in finite steps by some program. A dictator of a voting procedure is a voter such that if he strictly prefers a candidate (denoted by x) to another candidate (denoted by y), then the voting procedure does not choose y. Strong candidate stability requires that there be no change in the outcome of an election if a candidate withdraws who would lose ifevery candidate stood for office.

Undecidability of the existence of dictator for strongly candidate stable voting procedures in an infinite society and Cantor's diagonal argument

The strong candidate stability theorem by Dutta et al. [ 4], one of the major theoremsof social choice theory, states that, with a finite number of voters, there exists a dictator for any voting procedure which satisfies strong candidate stability, strong unanimity and independence of irrelevant alternatives (IIA). This paper investigates a decidability problem of voting procedures in a society with an infinite number of individuals (infinite society) using Cantor's diagonal argument presented by Yanofsky [ 19] which is based on Lawvere [ 10]. We will show the following result. The problem whether a strongly candidate stable voting procedure has a dictator or has no dictator in an infinite society is undecidable. It is proved using the arguments similar to those used to prove an extended version of Cantor's theorem that there cannot be an onto function from <img src="/img/revistas/cam/v27n3/n_bastao.gif" align=absmiddle>(the set of natural numbers) to its power set <img src="/img/revistas/cam/v27n3/p1_curs.gif" align=absmiddle>(<img src="/img/revistas/cam/v27n3/n_bastao.gif" align=absmiddle>). This undecidability means that for any strongly candidate stable voting procedure we can not decide whether or not it has a dictator in finite steps by some program. A dictator of a voting procedure is a voter such that if he strictly prefers a candidate (denoted by x) to another candidate (denoted by y), then the voting procedure does not choose y. Strong candidate stability requires that there be no change in the outcome of an election if a candidate withdraws who would lose ifevery candidate stood for office.

On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’,Philosophy

In a recent article (‘The Continuum: Russell’s Moment of Candour’), Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set.1 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 mysterious, 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.2 The aim of this discussion note is to explain why Ormell’s arguments should not convince philosophers that classical mathematics is wrong about the continuum. All that Ormell manages to show is that once the concept of a real number is restricted to include only those definable (and hence constructively acceptable), the set of such real numbers is denumerable (i.e., countably infinite). ‘Definability’ seems to be an intuitive informal notion, suggesting

Arity and alternation: a proper hierarchy in higher order logics

We study the effect of simultaneously bounding the maximal-arity of the higher-order variables and the alternation of quantifiers in higher-order logics, as to their expressive power on finite structures (or relational databases). Let $\mathit{AA}^i(r,m)$ be the class of (i?+?1)-th order logic formulae where all quantifiers are grouped together at the beginning of the formulae, forming m alternating blocks of consecutive existential and universal quantifiers, and such that the maximal-arity (a generalization of the concept of arity, not just the maximal of the arities of the quantified variables) of the higher-order variables is bounded by r. Note that, the order of the quantifiers in the prefix may be mixed. We show that, for every i???1, the resulting $\mathit{AA}^i(r,m)$ hierarchy of formulae of (i?+?1)-th order logic is proper. This extends a result by Makowsky and Pnueli who proved that the same hierarchy in second-order logic is proper. In both cases the strategy used to prove the results consists in considering formulae which, represented as finite structures, satisfy themselves. As the well known diagonalization argument applies here, this gives rise, for each order i and each level of the $\mathit{AA}^i(r,m)$ hierarchy of arity and alternation, to a class of formulae which is not definable in that level, but which is definable in a higher level of the same hierarchy. We then use a similar argument to prove that the classes of $\Sigma^i_m \cup \Pi^i_m$ formulae in which the higher-order variables of all orders up to i?+?1 have maximal-arity at most r, also induce a proper hierarchy in each higher-order logic of order i???3. It is not known whether the correspondent hierarchy in second-order logic is proper. Using the concept of finite model truth definitions introduced by M. Mostowski, we give a sufficient condition for that to be the case.

Some results on the forced pendulum equation

This paper is devoted to the study of the forced pendulum equation in the presence of friction, namely u ″ + a u ′ + sin u = f ( t ) with a ∈ R and f ∈ L 2 ( 0 , T ) . Using a shooting type argument, we prove the existence of at least two essentially different T -periodic solutions under appropriate conditions on T and f . We also prove the existence of solutions decaying with a fixed rate α ∈ ( 0 , 1 ) by the Leray–Schauder theorem. Finally, we prove the existence of a bounded solution on [ 0 , + ∞ ) using a diagonal argument.