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Cantor's Infinities: more detail

Satan, Cantor and Infinity: Mind-Boggling Puzzles (Dover Books on Mathematical & Word Recreations) by Raymond M. Smullyan, 2009-03-26

Satan, Cantor and Infinity and Other Mind-boggling Puzzles by Raymond M. Smullyan, 1993-02-18

Set Theory and the Sizes of Infinity: An entry from Gale's <i>Science and Its Times</i> by Sherri Chasin Calvo, 2000

Infinity: Countable set, Cantor's diagonal argument, Surreal number, Continuum hypothesis, Hyperreal number, Extended real number line

Transfinite number: Transfinite Number, Cardinal Number, Ordinal Number, Finite Set, Absolute Infinite, Georg Cantor, Infinity, Cardinality

Satan, Cantor & Infinity by Raymond M. Smullyan, 1994-06-12

lists with details

Mediev-L: Re: About Infinity
There are other systems besides cantor's of different kinds of infinities, for exampleone introduced around the same time Cantor put forth his version, by an
http://www.ku.edu/~medieval/melcher/matthias/t98/0084.html
Bounded Infinities
I've got ideas of bound infinities, sets whose elements are bound abstraction processes Figments - Semiotics - perfect language - cantor's paradox - Chaotic
http://homepages.which.net/~gk.sherman/baaaaaaa.htm

Extractions: mathematics 20 Sept 1998 Notes about the an idea of a bounded infinity (BI). Something that has infinite forms within the bound part of another space. An extremely simple BI is a line. It is constrained to a particular part of a nD space (n>1) but has infinite points within that line. Are there more complicated forms that can describe some of the interactions between two rival theories. For example, the difference between Value Added and Labour theories of value in economics. These are very rough notes... 2nd-3rd March 1998 Bits about Injective and Surjective: i.e with sets,
Infinity: You Can't Get There From Here -- Platonic Realms MiniText
that, after all, countable infinities are the only kind of infinities there are kindof proof, which has come to be called cantor's diagonalization argument
http://www.mathacademy.com/pr/minitext/infinity/index.asp

Extractions: Infinity, of course, infected our imaginations, and for some of us it cropped up in our conscious thoughts every now and then in new and interesting ways. I had nightmares for years in which I would think of something doubling in size. And then doubling again. And then doubling again. And then doubling again. And then until my ability to conceive of it was overwhelmed, and I woke up in a highly anxious state. and then I was awake, wide-eyed and perspiring. Only when I studied mathematics did I discover that my dream contained the seed of an important idea, an idea that the mathematician John Von Neumann had years before developed quite consciously and deliberately. It is called the Von Neumann heirarchy , and it is a construction in set theory.
Re 'Uncountable': How Many Reals *are* There? (mapping All Onto One Line)
born from a practical NEED, which IMHO cannot be said of cantor's gedanken experiments.Unless of course you admit playing with uncountable infinities to be a
http://home.iae.nl/users/benschop/reals.htm
"Uncountable": Finite Intuitions And Cantor's Diagonal (beyond Peano)
you may imagine that the whole lattice (of infinite width and hight ) 2^N is infinitelylarger (cantor's diagonal argument). Do the infinities cancel out
http://home.iae.nl/users/benschop/finite.htm

Extractions: Subject: Re: Discrete, continuous. It's all the same Date: Thu, 07 Mar 2002 From: Nico Benschop Newsgrp: sci.math - "Paul P. Budnik Jr." wrote: > "Mark" wrote > > >> "Paul P. Budnik Jr." Newsgrp: sci.math Subject: Re: Why are rationals countable? Date: Fri, 19 Apr 2002 16:06:05 +0200 From: Nico Benschop "Zdislav V. Kovarik" wrote: > > In article , > Agapito Martinez Author: benschop_nf@my-dejanews.com Date: 1998/11/20 Forum: sci.math - Re: Cantor's Diagonal Proof: FLAWED! - In article Date: Fri, 12 Mar 1999 10:49:26 GMT From: Nico Benschop Newsgroup: sci.math In article , Mike Deeth Strict reasoning Date: 5 Feb 99 07:50:18 -0500 (EST) sci.math Jim Trek Subject: Re: Can the "crackpot" theories of the infinite be formalized? Date: Thu, 15 Apr 1999 13:56:30 GMT From: Nico Benschop Re: 1/0 . . . (sci.math 10jan2001)
Cantor's Conjecture
infinities. But all these definitions and their considerations are beautiful (butquite empty) words having no legitimate attitude to mathematics. cantor's
http://www.ontologystream.com/beads/Cantor/Zenkin/bead1.htm

Extractions: Back Send comments to review committee. Forward The bead game is under development. The interactive function of the game comes from clicking the forward and back links above and from game players sending in Remarks. These Remarks are often edited to produce a distinct separation of concepts. Remarks edited in a way that is not faithful to the particapant's meaning can be revised. Linking in additional comments can be made via submission of beads. E-mailed Remarks from Alex Zenkin edited into three beads (This one) Humankind, from Aristotle's' time, discusses the problems concerning the nature of potential ("mathematical") infinity versus actual ("metaphysical") infinity. But an end of these discussions isn't seen even today. I am not sure that we are able to complete that discussion now. Cantor himself produced a lot of texts as to potential, actual, absolute, "in concreto", "in abstracto", metaphysical, theological, etc. infinities. But all these "definitions" and their considerations are beautiful (but quite empty) words having no legitimate attitude to mathematics. Cantor's Theorem on the uncountability of the set X of all real numbers x, belonging to the segment [0,1], is the only place where the actuality of a set X having the property of being infinite is really used as a mathematical (i.e., not philosophical) object.
Visual Inspection And Sense Analysis Of Data
popular form. It is proved that the main G.cantor's Theorem as tothe existence of different infinities is wrong. So, all modern
http://www.ontologystream.com/beads/visad/bead1-1.htm

Extractions: Back Send comments to review committee. Forward The bead game is underdevelopment, however part of the function of the game comes from clicking the forward and back links above and from game players sending in Remarks. 1. Alexander Zenkin, FATAL MISTAKE OF G.CANTOR'S THEORY. - "Voprosy Filosolii" ("Problems of Philosophy"), ), 2000, No. 2 (February), 165-168. (link) (in Engish) The journal is one of the most professional ones on Philosophy of Science in Russia and the World. Here, main G.Cantor's mistakes are discribed in popular form. It is proved that the main G.Cantor's Theorem as to the existence of different infinities is wrong. So, all modern meta-mathematics which is based on that Cantor Theorem is incorrect too. Also, all modern large computer proving systems are based on a doubtful basis. It is a fact of not pure scientific interest. 2. Andrey Vaganov (interview with A.Zenkin) MULTIMEDIA VERSION OF ROCK-GRAPHICS ("Nezavisimaya Gazenta" ("Independent Newspaper"), the "NG-SCIENCE"-supplement, 22 March, 2000): (link) (in Russian) The "NG-SCIENCE"-supplement is one of the most serious and wide-readable ones in Russia. Here, a description of the CCG-approach and its usage in creative human-being's activities is given.
The Continuum Hypothesis
There were others, such as David Hilbert, Ernst Zermelo and Leibniz, that wouldgo on to show that cantor's higher infinities are in fact necessary for
http://www.math.rutgers.edu/courses/436/436-s00/Papers2000/brazza.html

Extractions: Rutgers, Spring 2000 "Infinity is up on trial..." (Bob Dylan, Visions of Johanna , cited in In the Light of Logic ), pg. 28. These five words suffice to summarize the essence of Cantor's work. Cantor was tormented by opposition throughout his career. After conceiving and then proving his theorems on infinite sets, Cantor struggled against the negative reactions of his peers. It was not until the end of his lifetime that Cantor received the recognition he deserved. Cantor, a devout Christian, always held to his beliefs because to him, they came directly from G-d. "Where G-d was concerned, it was impossible to entertain hypotheses. There were no alternatives to be considered" (p. 238, Georg Cantor ). Georg Ferdinand Ludwig Philipp Cantor contributed greatly not only to discrete mathematics, but to every science based in mathematics. "Whatever the disappointments Cantor was to suffer, his transfinite set theory represented a revolution in the history of mathematics. Not a revolution in the sense of returning to ear lier starting points, but more a revolution in the sense of overthrowing older, established prejudices against the infinite in any actual, completed form." (Pg. 118, Georg Cantor). With his theory of sets and his introduction of the concept of infinite nu mbers, Cantor broke through the barriers of previous generations, and has allowed for the further exploration of areas that were previously unattainable.
6.2 Finite Or Infinite?
in the philosophy of mathematics itself to reject cantor's assumptions, most Thesearguments tie together the problems of natural and mathematical infinities.
http://www.generativescience.org/books/pnb/node24.html

Extractions: Finite or Infinite? There are two principal options open to us. If something is to be actual then we can either maintain that it must be finite, or that can be infinite. Actual things must be determinate, but is not clear whether infinite things can be determinate too. On the face of it, infinite things are unlimited and indefinite, and hence not fully determinate in the required sense. Mathematics since Cantor, however, has succeeded in giving some kind of determinacy to the notion of infinite sets, and hence it is no longer clear whether actual things are not allowed to be infinite. This may seem a rather academic point, but it turns out the the whole difference between classical physics and quantum physics can be made to depend on this decision! If we have actual events, for instance, then the two options are either allowing actual events to succeed each other continuously in time, or requiring events to have non-zero time intervals between them. The purpose of this book to show how this latter choice paves the way for a realistic understanding of quantum mechanics. The fact that quantum theory has proved a good theory therefore provides some kind of evidence to support the idea that all actualities must be finite.
Spicerack.sr.unh.edu/~dvf/Pathways/inf
to difficult problems before they developed careful rules for manipulating infinities. Theelaboration of cantor's ideas becomes a major theme of mathematics
http://spicerack.sr.unh.edu/~dvf/Pathways/inf
The Power Set
consisitent until you have an infinite number of infinities embedded in infinities. upwith a number system that looks an awful like cantor's cardinal numbers.
http://descmath.com/diag/power.html

Extractions: The goal of transfinite theory is to create a dichotomy between the rational and real numbers and to use this dichotomy as the foundation for a definition of the continuous line. Transfinite theorist claim to have accomplished this goal with the diagonal method . However, I have found the diagonal proof lacking. For example, using the diagonal method, I am able to show that the set of namable numbers is both denumerable and non-denumerable. It appears that the diagonal method itself it not sufficient to establish the claimed distinction between rational and reals, and I am unable to even begin a study of transfinite theory. Transfinite theorists claim that the dichotomy that exists between the rational and the real numbers also exists between the set of ordered pairs and the power set of the integer. Exploring these sets might give me better answers to the nature of the transfinite. n elements. The distribution of the elements follow the binomial pattern (Pascal's Triangle):
Rich Theory
of a rich infinity, I think you would find that cantor's transfinite numbers It simplyshows that Galileo's conjecture that all infinities are the same size
http://descmath.com/diag/rich.html

Extractions: Curing the "disease" of transfinite theory is not the simple matter of forbidding speculation on the infinite. It is the challenge to come up with a theory that is richer than transfinite theory. It is the challenge to create a new theory that can preserve the good while cutting out the rot. Personally, I think Georg Cantor and set theorists have made some tremendous strides in our understanding of mathematics. Ultimately we should hope to create a new theory of the infinite that encapsulates these benefits. This essay called Rich Theory is my amateurish stab at the dragon. The first thing I want to mention is my firm belief that all men and women are capable of reason. Both the classicist and transfinite theorists are guilty of trying to prevent their opponents from speaking. Kronecker's behavior in stifling Cantor's career was a black eye on the integrity of mathematics, just as Hilbert's censoring of LEJ Brouwer was a disgrace of the following generation. In almost all cases, there is more than one way to prove a given theorem in mathematics. There is not a pure way of thinking. We are all struggling with trying to find our voices, and trying to find the best way to communicate our ideas. The dictatorship of the Bourbaki has been no better for mankind that the dictatorship of the scholastics.
BBC - Radio 4 - 5 Numbers - Infinity
The result, confusing though it may seem, is that some infinities are bigger thanothers! cantor's work represented a threat to the entrenched complacency of
http://www.bbc.co.uk/radio4/science/5numbers5.shtml

Extractions: This quote could almost be an epithet for the mathematician Georg Cantor, one of the fathers of modern mathematics. Born in 1845, Cantor obtained his doctorate from Berlin University at the precocious age of 22. His subsequent appointment to the University of Halle in 1867 led him to the evolution of Set Theory and his involvement with the until-then taboo subject of infinity. Within Set Theory he defined infinity as the size of the never-ending list of counting numbers (1, 2, 3, 4.). Within this he proved that sub-sets of numbers that should be intuitively smaller (such as even numbers, cubes, primes etc) had as many members as the counting numbers and as such were of the same infinite size. By pairing off counting and even numbers together, we see that the number of counting and even numbers must be the same:
Search AZ Directory Contacting People About Us
No First Order axiomatization, then, can categorically describe a systemwhose size is one of cantor's higher infinities. (Systems
http://www.philosophy.unimelb.edu.au/handouts/161016/tennant.html

Extractions: You are here: Arts Dept Philosophy Handouts tennant (My original intention was to give this as a talk to the University of Melbourne Philosophy Department Colloquium in August, 2000. As often happens, I found the hour was not long enough: trying to ensure that everything was clear to the non-logicians in the audience, I got through about half before it was time to go to lunch. Emulating Tully, I went of and wrote out the oration I wished I had delivered.) 1. History. The decades around 1900 saw a concentration of studies of the axiomatic method of an intensity unmatched since Aristotle. Mathematicians and logicians in Germany (Dedekind, Hilbert, Frege), Italy (Peano's school) and the United States (1) formulated axiomatic descriptions of a variety of mathematical systems and studied the general theory of axiom systems. Two distinct goals were identified for an axiomatization. One was descriptive An axiomatization should describe a system in enough detail to specify it uniquely: the intended system should be the only one compatible with the axioms.
LA Weekly: Columns: Quark Soup: To Infinity And Beyond
has led to the discovery of numbers that dwarf even cantor's remotest dreams Theclasses of infinities now under study sound indeed like refugees from Lewis
http://www.laweekly.com/ink/03/16/quark-wertheim.php

Extractions: The new guy in town Bill Bratton has a good idea. He wants to know just how deep Rampart-style sins go in L.A. Here are two cases that might be worth another look by the chief. CELESTE FREMON records the final moments of the life of 19-year-old Flavio Aragon , who bled to death in 1994 in the presence of police officers. CHARLES RAPPLEYE examines the death of Erik Vega , who was shot in 1996 after officers dropped him off near rival gang territory. And ROBERT GREENE measures up Bratton THUGS ON FILM
Certainty, Infinity, Impossibility
Key ideas Numbers, The Pythagoreans and the irrationality of 2 , ordersof infinities, cantor's diagonalization argument, R = R n ; n=2, 3
http://cs.wwc.edu/~aabyan/CII/book.html

Extractions: Syllabus Schedule Project Reports ... Course Rationale Instructors: Anthony Aaby Thomas Thompson These documents are written using XHTML1.1 and MathML. For those documents that include MathML, Amaya Opera , or Mozilla are appropriate browsers. (Social and cultural implications of mathematics and computer science.) Course Description: Reading, reflection, and discussion of the implications of the limits of reason and objective reality that underlie rational western civilization. The topics are an eclectic collection selected from 2,500 years of mathematics and logic. Prerequisites: Upper division standing, general studies mathematics, and strong curiosity about the limits of reason. Option: Upon request, graded S/NC. Distance learning: This course is avaliable on the internet as a distance learning course. The lecture outlines are provided online. In lieu of oral presentations, three peer evaluations for each essay are required. Goals: To paraphrase Bertrand Russell, "A course should have either intelligibility or correctness; to combine the two is impossible" thus our choice is to focus on intelligibility. We hope to avoid the situation described by Clifford Allen: "More intellectual `ticking off' from B.[ertrand] R.[ussell] at dinner because I used the word 'sentence' when I should have used `phrase'. I'm dead sick of it." Upon completion of this course you will be aware of and understand some of the philosophical implications of: The paradoxes of naive set theory
CII Paper
Key ideas The Pythagoreans and the irrationality of 2 , orders of infinities,cantor's diagonalization argument, R = R n ; n=2, 3, , describability.
http://cs.wwc.edu/~aabyan/CII/paperCII.html

Extractions: Abstract: A chance remark to a collegue about the need for a course dealing with the philosophical and spiritual implications of notions that are fundamental to logic, mathematics and computer science led to the creation of a course for majors and nonmajors. Such a course would focus on some of the fundamental limits of language and logic that are part of western civilization's rational heritage. The course topics would include the finite and the infinite, the relationship between language, truth, and proof, computability including determinism and nondeterminism and the limits of computation, and Gödel's incompleteness theorem. This paper provides an overview of the course that resulted from that remark. Keywords and Phrases: The Pythagoreans and the irrationality of n A chance remark to a collegue: "We need a course titled Infinity and Impossibility" growing out of mutual interests in the foundational concepts in mathematics, logic, and computation and their implications for faith led to the creation of a course which would focus on some of the fundamental limits of language and logic that are part of western civilization's rational heritage. This paper reports on the design of that course.
4.06: PHYSICS AND MATHEMATICS -- Logic And Computational Theory
research over the years indicates that these peculiar infinities are firmly Oneof the most successful attempts used cantor's original conceptual framework
http://www.imprint-academic.demon.co.uk/SPECIAL/04_06.html

Extractions: This work is motivated by the desire to model the degree to which one must consciously attend to a problem to solve it. This `degree' of conscious attention is not the time or space (or any of the usual resources) needed to solve the problem, as is the case in complexity theory. Rather, it is based on the degree to which a problem solver can monitor and control himself. A problem will be called more complex here (or `deeper') if one must have a higher degree of consciousness to solve it. `Problems' will be modelled by recursive functions from the natural numbers to the natural numbers, `problem solvers' by Turing Machines, and `degrees' of consciousness by constructible ordinals. For any constructible ordinal a , an a -self-monitoring machine, or a -machine, (as they will be called) behaves as follows: Before seeing the input, it places the initial ordinal a into an ordinal clock. This is the degree of consciousness by which it must compute the function on all f g such that For all x g x f x sane machines are defined which are well-behaved in the sense that any two a -machines will behave similarly. The class of sane functions that are
EMail Msg <9305050249.AA05043@turing.pacss.binghamton.edu>
I agree, analysis does not explicitly depend on cantor's construction, but I havea preference for avoiding talk about completed infinities unless absolutely
http://www-ksl.stanford.edu/email-archives/interlingua.messages/310.html
EMail Msg <9305241234.AA14945@turing.pacss.binghamton.edu>
does seem to lead to the dreaded swamp of confusions or cantor's paradise Therefore,I believe that talk of countable infinities in foundational studies is OK
http://www-ksl.stanford.edu/email-archives/interlingua.messages/338.html

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