Cantor's Theorem Research Articles

A stationary point theorem characterizing metric completeness

We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.

Russell, His Paradoxes, and Cantor's Theorem: Part I

AbstractIn these articles, I describe Cantor's power‐class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell's work. These include Russell's paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class‐intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor's theorem, its proof, how it can be used to manufacture paradoxes, Frege's diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes.

Russell, His Paradoxes, and Cantor's Theorem: Part II

AbstractSequel to Part I. In these articles, I describe Cantor's power‐class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell's work. These include Russell's paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class‐intensions and equivalence classes of coextensional properties. Part II addresses Russell's own various attempts to solve these paradoxes, including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead.

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 topological set theory

This paper is concerned with topological set theory, and particularly with Skala's and Manakos' systems for which we give a topological characterization of the models. This enables us to answer natural questions about those theories, reviewing previous results and proving new ones. One of these shows that Skala's set theory is in a sense compatible with any ‘normal’ set theory, and another appears on the semantic side as a ‘Cantor theorem’ for the category of Alexandroff spaces. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A Negation-free Proof of Cantor's Theorem

A Negation-free Proof of Cantor's Theorem

Cantor Diagrams: A Unifying Discussion of Self-Reference

Within the framework of category theory, Cantor diagrams are introduced as the common structure of the self-reference constructions by Cantor, Russell, Richard, Godel, Peter, Turing, Kleene, Tarski, according to the so-called Cantor diagonal method. Such diagrams consist not only of diagonal arrows but also of idempotent, identity and shift arrows. Cantor theorem states that no Cantor diagram is commutative. From this theorem, all the constructions above can be systematically retrieved. We do this by grouping them into two main classes: the class based on Cantor diagrams with a numerical shift function and the class based on Cantor diagrams with a Boolean shift function.

Mathematical chestnuts from around the world

1. Problems from Ireland 2. Three solutions to an old chestnut 3. Problems from Eotvos-Kurschak competitions 4. Polish math olympiads 5. East German olympiads 6. Problems from Pi Mu Epsilon Journal 7. Austrian-Polish math olympiads 8. Problems from Quantum 9. Bulgarian problems for 11-14 year olds 10. Cusumano's challenge 11. Five easy problems from Leningrad 12. An arithmetic puzzle 13. Gleanings from the Mathematical Gazette 14. Problems from the Putnam contest 15. A second look at a problem from Romania 16. 32 miscellaneous problems 17. Two problems in combinatorics 18. An unused problem from the 1988 International Olympiad 19. Four problems from the 1995 International Olympiad 20. Two geometry problems 21. An unlikely perfect square 22. The nine-point circle and Coolidge's theorem, the De Longchamps point of a triangle, Cantor's theorem, and Napoleon's theorem 23. A problem from the Philippines 24. Four solutions by George Evagelopoulos 25. A Canadian problem 26. A function of exponential order Solutions.

Designing a Calculational Proof of Cantor's Theorem

Designing a Calculational Proof of Cantor's Theorem

Designing a Calculational Proof of Cantor's Theorem

Cantor's Diagonalization The one purpose of this little Note is to show that formal arguments need not be lengthy at all; on the contrary, they are often the most compact rendering of the argument. Its other purpose is to show the strong heuristic guidance that is available to us when we design such calculational proofs in sufficiently small, explicit steps. We illustrate our approach on Georg Cantor's classic diagonalization argument [chosen because, at the time, it created a sensation]. Cantor's purpose was to show that any set S is strictly smaller than its powerset g9S (i.e., the set of all subsets of S). Because of the 1-1 correspondence between the elements of S and its singleton subsets, which are elements of gPS, S is not larger than gPS, and our proof can now be focussed on the strictly, i.e., we have to show that there is no 1-1 correspondence between S and gUS. We can confine ourselves to non-empty S.

A Game-Like Activity for Learning Cantor's Theorem

A Game-Like Activity for Learning Cantor's Theorem

Nonuniqueness for a bounded, differentiable complete orthonormal system inL 2[0, 1

The argument of Musegian and Ovsepjan is adapted to produce a complete orthonormal system on [0, 1] of uniformly bounded functions, differentiable on [0, 1], andC∞ on [0, 1], for which the analogue of Cantor's uniqueness theorem is false. We also construct a complete orthonormal system ofC∞ functions which vanish to infinite order at both endpoints.

ON THE ORDER-THEORETIC CANTOR THEOREM

In this report1, we present an order-theoretic version of the Cantor theorem. This result, which is based on the interplay of the notions of partial order and of completeness, permits to give a unified and simplified account to a long list of results related to the Bishop– Phelps theorem. We survey briefly only its simplest applications and refer the reader to [10] for a complete presentation of the results.

An Intuitionistic Version of Cantor's Theorem

AbstractAn intuitionistic version of Cantor's theorem, which shows that there is no surjective function from the type of the natural numbers N into the type N → N of the functions from N into N, is proved within Martin‐Löf's Intuitionistic Type Theory with the universe of the small types.Mathematics Subject Classification: 03B15, 03B20.

Utility functions and the order type of the continuum

Debreu proved that there exists a continuous isotone real-valued function on any given second countable topological pre-ordered space. It is shown that this result implies and is implied by Cantor's Theorem characterising the linear continuum.

Fundamental Schemes of Discretized Formulation of Continuously Distributed Models

For continuums such as elastically vibrating substances, e. g., beams and plates, mathematical models are discussed with respect to the major problems involving the discretized formulations : (1) To decrease the order of spatial dimension. (Cantor's theorem of the infinite set theory and the continuous cardinality are referred to help imagining the idea.) The example of decreasing is given to show how the principle applies by means of the comparison of two cases which are formulations of modal controls of bending vibrations. (2) Finite freedom forms for infinite eigenmodes of distributed parameter systems. An approximating method using spectral decompositions and resolvent representations of the linear operator is shown. The necessities of the conditions for the state observability and the state controllability in the designs of finite freedom controls provided with the eigenmodes for continuously distributed objects are pointed out.

Set theory for verification: I. From foundations to functions

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations, and functions and discusses interactive proofs of Cantor's Theorem, the Composition of Homomorphisms challenge [9], and Ramsey's Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics.

?-match: An inference rule for incrementally elaborating set instantiations

In this paper we describe a new inference rule, called −-match, which is used for finding set instantiations within an automated reasoning program. We have implemented −-match within a theorem prover called & and have used the system to prove some non-trivial theorems in mathematics, including Cantor's theorem and the correctness of transfinite induction. While not complete, we believe that −-match is a generally useful inference rule for finding set instantiations. One of the major contributions of the −-match rule is the ability to instantiate a term as an incompletely specified set abstraction, and then subsequently elaborate the identity of this set by considering other subgoals in the proof. This elaboration happens as a consequence of the deduction rules of the prover, and requires no additional machinery in the prover.

Automated deduction in von Neumann-Bernays-G�del set theory

I present a new clausal version of NGB set theory, and compare my version with that first given by Boyer et al. [4]. A complete set of reductions for Boolean rings is given, derived from those of Hsiang [7]. I list over 400 theorems proved semiautomatically in elementary set theory, and supply the proofs of several of these, including Cantor's theorem. I present a semiautomated proof that the composition of homomorphisms is a homomorphism, thus solving a challenge problem given in [4]. Using the clauses and heuristics presented, there is no apparent obstacle to the semiautomated development of set theory through considerably more difficult theorems.