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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

41. Infinity
    The result, confusing though it may seem, is that some infinities are bigger Cantor s work represented a threat to the entrenched complacency of the old  
    http://www.simonsingh.com/Infinity.html
    
    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:

42. Infinity -- Facts, Info, And Encyclopedia Article
    A concise introduction to Cantor s mathematics of infinite sets. Large cardinals are quantitative infinities defining the number of things in a (Several  
    http://www.absoluteastronomy.com/encyclopedia/i/in/infinity.htm
    
    Extractions: Infinity is a term with very distinct, separate meanings which arise in (The rational and systematic study of religion and its influences and of the nature of religious truth) theology (The rational investigation of questions about existence and knowledge and ethics) philosophy (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin : "In-finite", is not ended In (The rational and systematic study of religion and its influences and of the nature of religious truth) theology , for example in the work of (Someone who is learned in theology or who speculates about theology (especially Christian theology)) theologians such as (Scottish theologian who was very influential in the Middle Ages (1265-1308)) Duns Scotus , the infinite nature of (The supernatural being conceived as the perfect and omnipotent and omniscient originator and ruler of the universe; the object of worship in monotheistic religions)

43. What's A Number?
    One of the many Cantor s contributions was to establish various kinds of infinities. The approach I employ is Cantor s famous diagonal process.  
    http://www.cut-the-knot.com/do_you_know/numbers.shtml
    
    Extractions: Philosophical Library, 1965 Indeed there are many different kinds of numbers. Let's talk a little about each of these in turn. A number r is rational if it can be written as a fraction r = p/q where both p and q are integers. In reality every number can be written in many different ways. To be rational a number ought to have at least one fractional representation. For example, the number may not at first look rational but it simplifies to 3 which is 3 = 3/1 a rational fraction. On the other hand, the number 5 by itself is not rational and is called irrational. This is by no means a definition of irrational numbers. In Mathematics, it's not quite true that what is not rational is irrational. Irrationality is a term reserved for a very special kind of numbers. However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational). Much of the scope of the theory of rational numbers is covered by Arithmetic. A major part belongs to Algebra. The theory of irrational numbers belongs to Calculus.

44. Archimedes Plutonium
    Thus in this fashion any and every Real becomes a flipped over adic. Each correspond oneto-one. There, Cantor s orders of infinities disappear forever.  
    http://www.iw.net/~a_plutonium/File118.html

45. THE LIFE OF THE MATHEMATICIAN GEORG FERDINAND LUDWIG PHILIPP CANTOR. Essay Sampl
    Today, Cantor s work is widely used in the many fields of mathematics. Cantor thought once you start dealing with infinities, everything is the same  
    http://www.essaysample.com/essay/002869.html
    
    Extractions: These one-to-one correspondence sets show that even though we add an unknown variable, multiply by two, and square a set, the upper and lower sets still remain equal. Since we will never run out of numbers any correspondence set with two infinite values will be equal. All these sets clearly have the same cardinality since its members can be put in a one-to-one correspondence with each other on and on forever. These sets are said to be countably infinite and their cardinality is denoted by the Hebrew letter aleph with a subscript nought, .

46. Math Links
    Presented here are Zeno s Paradox and Cantor s infinities. The Problem of Points An age-old gambling problem led to the development of probability by  
    http://www.narragansett.k12.ma.us/nrhs/math/mathlinks.htm
    
    Extractions: MATH LINKS Narragansett Regional High School Math Department 1999-2000 About the Staff Course Offerings Mathematical Buds ... Math Links Professional Organizations National Education Association Massachusetts Teachers' Association National Council of Teachers of Mathematics International Society for Technology Education ... Society for Industrial and Applied Mathematics National and State Organizations U. S. Department of Education Massachusetts Department of Education Other Interesting Mathematics Sites T he Mathematics Teacher Education Resource Place American Journal of Mathematics the oldest mathematics journal in America An Overview of the History of Mathematics Famous Problems in the History of Mathematics The Bridges of Konigsberg - This problem inspired the great Swiss mathematician Leonard Euler to create graph theory, which led to the discovery of topology. The Value of Pi - Throughout the history of civilization various mathematicians have been concerned with discovering the value of and different expressions for the ratio of the circumference of a circle to its diameter. Puzzling Primes - To fully comprehend our number system, mathematicians need to understand the properties of the prime numbers. Finding them isn't so easy, either.

47. Set Theory
    Until then, no one envisioned the possibility that infinities come in different Motivation for Cantor s discovery of Set Theory came from his work on  
    http://plato.stanford.edu/entries/set-theory/
    
    Extractions: Please Read How You Can Help Keep the Encyclopedia Free The objects of study of Set Theory are sets . As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. The language of set theory is based on a single fundamental relation, called membership . We say that A is a member of B (in symbols A B ), or that the set

48. Symmetry Forms The Basis Of Mathematical Truth
    The completed infinities, mathematician Georg Cantor s infinite sets, could be explained as cardinal identities, akin to qualia from which finite subsets,  
    http://www.webspawner.com/users/monolithiclogic/
    
    Extractions: Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness. Predicates like "red" are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity "natural number" or "real number" describes an entire collection of individual objects. Symmetry is analogous to a self evident truth and is distributive via the laws of nature, being distributed over the entire set called universe. A stratification of Cantorian alephs with varying degrees of freedom. More freedom = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called universe and gives it "identity".

49. 6.2 Finite Or Infinite?
    The mathematical problem of `actual infinities is whether certain classes (the We are not obliged to accept Cantor s construction of definite infinite  
    http://www.generativescience.org/books/pnb/finite-infinite.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

50. [FOM] As To Strict Definitions Of Potential And Actual Infinities.
    FOM As to strict definitions of potential and actual infinities. Cantor s and any modern axiomatic set theory have to do with just infinite sets,  
    http://www.cs.nyu.edu/pipermail/fom/2003-January/006124.html
    
    Extractions: Thu Jan 16 17:19:32 EST 2003 http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html ) I have given a quite impressive list of Cantor's opponents as regards the rejection of the actual infinite who, according to W.Hodges' classification, "must be <AZ: considered as> ... dangerously unsound minds" (see his famous paper "An Editor Recalls Some Hopeless Papers." - The Bulletin of Symbolic Logic, Volume 4, Number 1, March 1998. Pp. 1-17, http://www.math.ucla.edu/~asl/bsl/0401-toc.htm ). Now I would like to remind some of appropriate statements of such the "dangerously unsound minds". For example, Solomon Feferman writes (in his recent remarkable book "In the light of logic. - Oxford University Press, 1998."): "[...] there are still a number of thinkers on the subject (AZ: on Cantor's transfinite ideas) who in continuation of Kronecker's attack, object to the panoply of transfinite set theory in mathematics [.] In particular, these opposing <AZ: anti-Cantorian> points of view reject the assumption of the actual infinite (at least in its non-denumarable forms) [...]Put in other terms: the actual infinite is not required for the mathematics of the physical world." The same view as to rejection of the actual infinite is clearly expressed by Ja.Peregrin (see his "Structure and meaning" at:

51. [FOM] As To Strict Definitions Of Potential And Actual Infinities.
    FOM As to strict definitions of potential and actual infinities. Today every metamathematician and set theorist knows well that Cantor s naive set  
    http://www.cs.nyu.edu/pipermail/fom/2002-December/006072.html

52. Fatal_Mistake_of_Cantor_ENG
    such differentiation of infinities is the famous George Cantor s theorem about Against this background, the proof of Cantor s theorem looks simply  
    http://www.com2com.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html

53. SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS
    No doubt, it is the famous G.Cantor s theorem concerning the only one this Cantor s theorem, and the talk itself about differing infinities will become  
    http://www.com2com.ru/alexzen/papers/ng-02/contr_rev.htm

54. Powell's Books - Review-a-Day - Everything And More: A Compact History Of Infini
    The book culminates with Georg Cantor s creation of a set theory of trying to parse Cantor s diagonal proof establishing that some infinities are larger  
    http://www.powells.com/csm/review/2003_12_01.html

55. Proof Of Infinities
    Proof That Not All infinities Are The Same Size Cantor s first proof is complicated, but his second is much nicer and is the standard proof today.  
    http://math.bu.edu/INDIVIDUAL/jeffs/cantor-proof.html
    
    Extractions: has three elements in it; we match each bullet with a natural number: and the last number is 3. In infinite sets, we do the same thing. For example, the number of squares and the number of natural numbers is the same. To show this rather odd result, consider the following matching: You can continue this matching; as you can see, every natural number gets a unique square, so the number of squares and the number of natural numbers is the same. (Another way of putting this is that every natural number has a square, and every square is associated with a natural number). How about the rational numbers? It might seem that the set of all rationals, like 3/8 and 5/9 and 12321/98732 would be much larger than the set of natural numbers. However, you can match them up. The proof is fairly simple, but difficult to format in html. But here's a variant, which introduces an important idea: matching each number with a natural number is equivalent to writing an itemized list. Let's write our list of rationals as follows: and so on. Notice that first we list all the fractions whose numerator and denominator add to 1, then those that add to 2, then those that add to three, etc. Every fraction is somewhere on this list (and a little elementary arithmetic sequence calculation can tell you

56. Philosophy - The Uniform Solution Of The Paradoxes
    that Cantor s Paradox demonstrates the possibility of infinities with no to stay with Cantor s belief that at least some infinities are numerable.  
    http://www.philosophy.uwa.edu.au/staff/slater/publications/the_uniform_solution_
    
    Extractions: Employment after Philosophy THE UNIFORM SOLUTION 1. Introduction 2. Indeterminate Sense true? It is true, of course, if the sentence named 's' is translated 'p' - but then, why isn't this presumption made explicit? The logical truth is that which immediately resolves the many paradoxes involving Truth, in the same manner as above (see, e.g. Slater [18], [21]). In fact it also resolves Curry's Paradox, which Priest classifies differently. The central question for Tarskians is thus why they take to be necessary something which is plainly contingent, and whose very contingency removes the paradoxes that have bedevilled them. 3. Indeterminate Reference holds for all sets x, and so, in particular, it holds for where DN19 is the set of finite ordinals definable in less than 19 words. But he also wants to say since 'the least ordinal not in the set of finite ordinals definable in less than 19 words' defines a finite ordinal in less than 19 words - because of its length.

57. Aristotle's Potential And Actual Infinite And Cantor
    According to Aristotle, actual infinities cannot exist, but potential infinities exist in KRONECKER I don t know what predominates in Cantor s theory  
    http://www.mlahanas.de/Greeks/Infinite.htm
    
    Extractions: Aristotle's Potential and Actual Infinite and Cantor Michael Lahanas Zeno's Paradoxa such as the Achilles and the tortoise were the reason that ancient Greeks tried to avoid the Infinite. Common logic is that parts of a set are less than all the parts of it. Archytas Thought Experiment about the size of the Universe there is no end and we say that we have an infinite set. But is it true that the subsets of this sets are less numerous than the entire set? Consider all the even numbers 2,4,6,8.... and all the natural numbers 1,2,3,4,.... We may think that there are more natural numbers than even natural numbers since the even numbers are a subset. But is this true? Let us count the number of even natural numbers. We have to find a mapping from the natural numbers to the even natural numbers. For each natural number m there is a even natural number 2m. Therefore there are as many even number as there are even and odd numbers! Consider the 2 concentric circles in the Figure above. Has the larger outer circle more points than the smaller circle? We think that the answer is probably yes! But is this true? We have to count the number of points in both circles. If we can find a mapping that assigns for each point on the larger circle a point in the smaller circle then both circles have the same number of points. Every point B of the larger circle can be connected with a line going through the center. This line will pass through a point A of the smaller circle. We could consider this as the mapping that we wanted. Every point of the larger circle can be mapped to a point on the smaller circle. How is this possible? Should the larger circle not contain more points than the smaller circle?

58. Certainty, Infinity, Impossibility
    orders of infinities, Cantor s diagonalization argument, R = Rn ; n=2, 3, Does the hierarchy of infinities have any implications for the structure  
    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

59. Archive Of Astronomy Questions And Answers
    infinity can be classified in terms or Cantor s Transfinite Numbers. There are countable and uncountable infinities, and who knows what other kinds  
    http://www.astronomycafe.net/qadir/q1493.html
    
    Extractions: Infinity is many things to many people. To mathematicians, it is something that is practically 'concrete' and the many different kinds of 'infinity' can be classified in terms or Cantor's Transfinite Numbers. There are countable and un-countable infinities, and who knows what other kinds in between. To physicists and astronomers, there are no infinities in the physical world. Not even space and time itself need to be 'infinite' in the purely mathematical sense. Whenever 'infinity' appears as a prediction by a theory of the physical world, it is deemed a flaw, and must be eliminated. It is considered a signal that the particular theory has been extended beyond its domain of applicability. Newtonian mechanics had no problem with bodies attaining infinite velocity, or acting through space with infinite speed. These expectations led to falsifiable predictions and that is, in part, why we now have special and general relativity. In the few cases where general relativity predicts 'infinity' in the form of singularities, this is regarded as a defect in general relativity theory that will be healed when a fully quantum theory of gravity is developed. For more on singularities, see a Previous Question number 1503 Return to Ask the Astronomer

60. Ziring Book Review Pages - Current
    Some of the other topics covered in less depth include Turing machines, Post production systems, Cantor s infinities, NPcompleteness, Maxwell s demon,  
    http://www.erols.com/ziring/bookrev.htm
    
    Extractions: (New: The Advent of the Algorithm by Dr. David Berlinsk) Book Reviews Master Index Old Review Pages: For other book news and views, check out the Usenet: rec.arts.books.* newsgroups Here are some books that you might find interesting. Go Figure! Using Math to Answer Everyday Imponderables by Clint Brookhart ISBN 0-8092-2608-1 Educators and social critics noticed, in the late 1980s, that general familiarity with basic arithmetical concepts and techniques was low among the American population. J.A. Paulos even coined a word to describe it: innumeracy . The real key to improving numeracy, many of its advocates point out, is to make mathematics seem more real and applicable to people's everyday lives.

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