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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

The Life Of The Mathematician Georg Ferdinand Ludwig Philipp Cantor :: Essay Sam
OTHER infinities Cantor thought once you start dealing with infinities, everything is the same size. This did not turn out to be the case.
http://www.essaysample.com/essay/002869.html
4bitterguys.com - View Topic - Regarding The Economic Viability Of Hilbert's Hot
Save that for the ones that blew Georg Cantor s mind actual infinities. Cantor went on to show that there is an endless succession of infinities.
http://4bitterguys.com/phpBB2/viewtopic.php?t=5520
4.06: PHYSICS AND MATHEMATICS -- Logic And Computational Theory
research over the years indicates that these peculiar infinities are firmly One of 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
6.2 Finite Or Infinite?
a whole series of what he calls `transfinite numbers ), because in his proofs, Quine 1961 shows, Cantor presupposes that infinities were already definite.
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, 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.
Welcome To The Hotel Infinity
How Many Different Sizes of Infinity? Perhaps, thought Cantor, once you start dealing with infinities, everything is the same size.
http://www.cs.uidaho.edu/~casey931/mega-math/workbk/infinity/inbkgd.html

Extractions: Most everyone is familiar with the infinity symbolthe one that looks like the number eight tipped over on its side. The infinite sometimes crops up in everyday speech as a superlative form of the word many . But how many is infinitely many? How far away is "from here to infinity"? How big is infinity? You can't count to infinity. Yet we are comfortable with the idea that there are infinitely many numbers to count with: no matter how big a number you might come up with, someone else can come up with a bigger one: that number plus oneor plus two, or times two. Or times itself. There simply is no biggest number. Is there? Is infinity a number? Is there anything bigger than infinity? How about infinity plus one? What's infinity plus infinity? What about infinity times infinity? Children to whom the concept of infinity is brand new, pose questions like this and don't usually get very satisfactory answers. For adults, these questions don't seem to have very much bearing on daily life, so their unsatisfactory answers don't seem to be a matter of concern. At the turn of the century, in Germany, the Russian-born mathematician Georg Cantor applied the tools of mathematical rigor and logical deduction to questions about infinity in search of satisfactory answers. His conclusions are paradoxical to our everyday experience, yet they are mathematically sound. The world of our everyday experience is finite. We can't exactly say where the boundary line is, but beyond the finite, in the realm of the
Welcome To The Dana Centre
had finished my current task and then I turned to it, refreshed my recollection that it was George Cantor who had introduced multiple infinities and confirmed
http://193.71.79.98/default.aspx?DanaAction=ShowPost&PostID=326
Tennant On Compactness
No First Order axiomatization, then, can categorically describe a system whose 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.
What's A Number?
set of real numbers. One of the many Cantor s contributions was to establish various kinds of infinities. While it is true that
http://www.cut-the-knot.org/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. Using only arithmetic methods it's easy to prove that the number
To Infinity And Beyond
All these infinities are equal, but Cantor wondered whether there could be a truly larger infinity? One obvious candidate would seem to be the fractions.
http://www.laweekly.com/ink/printme.php?eid=42375
LookSmart - Directory - Guides And Directories Of Chaos And Complexity Theory
Farewell to Chaos, A Read articles discussing the order of deterministic chaos, Cantor s schizophrenic infinities, and Mandelbrot fractal dimensions.
http://search.looksmart.com/p/browse/us1/us317914/us328800/us10022550/us10108105
Http//www.newyorker.com
Others, however, dismissed Cantor s infinity of infinities as a fog on a fog and mathematical insanity. Cantor felt persecuted by these critics, which
http://flash.lakeheadu.ca/~kyu/E5111/Infinity.htm

Extractions: Few ideas have had a racier history than the idea of infinity. It arose amid ancient paradoxes, proceeded to baffle philosophers for a couple of millennia, and then, by a daring feat of intellect, was finally made to yield its secrets in the late nineteenth century-though not without leaving a new batch of paradoxes. You don't need any specialized knowledge to follow the plot: the main discoveries, despite the ingenuity behind them, can be conveyed with a few strokes of a pen on a cocktail napkin. All of this makes infinity irresistible meat for the popularizer, and quite a few books in that vein have appeared over the years. Now, in "Everything and More: A Compact History of infinity " (Norton; $23.95), the celebrated author David Foster Wallace has set out to initiate readers into its mysteries. It might seem odd that finite beings like us could come to know anything about infinity, given that we have no direct experience of it. Descartes thought that the idea of infinity was innate, but the behavior of children suggests otherwise; in one study, children in their early school years "reported 'counting and counting' in an attempt to find the last number, concluding there was none after much effort." As it happens, the man who did the most to capture infinity in a theory claimed that his insights were vouchsafed to him by God and ended his life in a mental asylum.
Kluwer Online Internet Publishing System - Educational Studies In Mathematics
http://www.kluweronline.com/article.asp?PIPS=394705
ETHOLOGY Archives -- March 1996 (#85)
I claim Cantor s transfinite infinities are false because he was not guided by quantum physics. Renormalization procedures work
http://segate.sunet.se/cgi-bin/wa?A2=ind9603&L=ethology&F=&S=&P=10236
Bret Willet's Paper On Infinity
Cantor went on to prove that there is a hierarchy of infinities; there are an infinite number of transfinite cardinal numbers. In
http://www.facstaff.bucknell.edu/udaepp/090/w3/bretw.htm

Extractions: Kenneth Bret Willet "To infinity and beyond!" These were the inspired words of Buzz Lightyear in the Disney movie Toy Story. Granted, one would not expect to find much mathematical content in an animated film directed toward children, but these words raise an interesting issue that mathematicians and the general public struggled with for many years. Can one go beyond infinity? How can such a concept be possible or even imaginable? These questions led to the development of many new theories and even a new system of numbers. Disney's Buzz Lightyear A study of infinity must begin with an introduction to set theory. A set is merely a collection of objects. Georg Cantor was the sole creator of set theory; he published an article in 1874 that marks the beginning of set theory and has come to change the course of mathematics. Cantor's theory was met with a great deal of opposition due to its assertion of infinite numbers. The famous mathematician Leopold Kronecker was especially opposed to Cantor's revolutionary new way of looking at numbers. Kronecker believed only in constructive mathematics, those objects that can be constructed from a finite set of natural numbers. Despite this opposition from influential thinkers, set theory laid the foundation for twentieth century mathematics. Although there were some flaws in Cantor's theory, sets became an essential part of new mathematics and therefore set theory was adapted to eliminate its original paradoxes [2].
Sciforums.com - Infinity And Beyond(s)
He then showed, using what s become known as Cantor s theorem, that there s a hierarchy of infinities of which aleph0 is the smallest.
http://www.sciforums.com/archive/index.php/t-32934

Extractions: I've heard mention a time or two of multiple (infinite?) definitions of infinity. Meaning: On a number line there is an infinite number of points between 1 and 2. There is also an infinite number of points between 1 and 100, 1 and 100000000, and so on. Is the infinite number of points between 1 and 10000 a bigger infinity than that between 1 and 2? Can the infinities between each and every point be added all together to make one ginormous infinity? Is this sum "true" infinity, where all the others are smaller and portionate, though stilll infinite? I've heard mention a time or two of multiple (infinite?) definitions of infinity. Meaning: On a number line there is an infinite number of points between 1 and 2. There is also an infinite number of points between 1 and 100, 1 and 100000000, and so on. Is the infinite number of points between 1 and 10000 a bigger infinity than that between 1 and 2?
Science And Technology Unit
Why study such problems when irrational numbers do not exist. Certainly Cantor s array of different infinities were impossible under this way of thinking.
http://www.sabah.gov.my/ust/math04.html

Extractions: Email : ust@sabah.gov.my A history of set theory The history of set theory is rather different from the history of most other areas of mathematics. For most areas a long process can usually be traced in which ideas evolve until an ultimate flash of inspiration, often by a number of mathematicians almost simultaneously, produces a discovery of major importance. Set theory however is rather different. It is the creation of one person, Georg Cantor . Before we take up the main story of Cantor's development of the theory, we first examine some early contributions.
Plato And Cantor Vs. Wittgenstein And Brouwer
notion of infinite totalities, at least implicitly. Without infinite totalities, or actual infinities, Cantor s paradise would fall.
http://www.angelfire.com/az3/nfold/plato.html

Extractions: Prove all things. Hold fast to that which is good. I Thes 5:21 [This page was begun in January 2002; as of Aug. 1, 2002, it remains a work in progress.] Without going into an extensive examination of phenomenology and the psychology of learning, perception and cognition, let us consider the mind of a child. Think of Mommy controlling a pile of lollipops and crayons, some of which are red. In this game, the child is encouraged to pick out the red objects and transfer them to 'his' pile. The child employs a mental act of separation (some might call this 'intuition') to select out an item, in this case by direct awareness of the properties of redness and of ease of holding with his hands. This primal separation ability is necessary for the intuition of replication. Crayon and lollipop are 'the same' by virtue of redness. In turn, this intuition of replication, or iteration, requires a time sense, whereby if the child hears 'more' he associates the word with an expectation of a craving being satisfied ('more milk').
Mathematical Quotes
Then there were the transfinitists like Cantor himself, who ascribe the same degree of reality to actual completed infinities as they did to finite quantities.
http://www.braungardt.com/Mathematica/mathematical_quotes.htm

Extractions: Up Schrödinger Quotes by Georg Cantor The first 10.000 digits of Pi ... Moebius strip Mathematical Quotes The Klein bottle Incompleteness Theorem A history of set theory Golden Ratio and Fibonacci series ... The First 10000 Primes As for everything else, so for a mathematical theory: beauty can be perceived but not explained. Arthur Cayley How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? Max Wilhelm Dehn Number is the within of all things. Pythagoras Numbers are the highest degree of knowledge. It is knowledge itself. Plato From Pi in the Sky, by John Barrow. Oxford University Press, 1992. p. 216 Poets do not go mad, but chess players do. Mathematicians go mad, and cashiers, but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination. G. K. Chesterton In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)
The Spiritual Function Of Mathematics, By Thomas J McFarlane
.It is an Infinity of some higher order, that is an INFINITY which comprehends lesser infinities. 58 Here Wolff refers to Cantor s mathematical theory of
http://www.integralscience.org/sacredscience/SS_spiritual.html

Extractions: If one searches the historical record for evidence of rational and logical thought, one finds among the most highly developed intellects the spiritual philosophers such as Shankara, Nagarjuna, and Plato, for whom the primary function of the intellect is to serve the ends of spiritual realization. Moreover, mathematics, perhaps the most subtle and rigorous form of thought, traces its origins back to Pythagoras and Plato, for whom mathematics is first and foremost a spiritual activity. Although today the spiritual function of mathematics, and rational thought in general, has been largely forgotten, yet there are a small number among us who remember; perhaps the most notable to live in our century is Franklin Merrell-Wolff. This essay presents Merrell-Wolff's writings on the spiritual function of mathematics, selected from his three major works, Pathways Through To Space The Philosophy of Consciousness Without An Object , and Introceptualism . Mathematics, according to Wolff, functions as a bridge between the relative and transcendent states of consciousness. It serves, on the one hand, as a vehicle for crossing from the transcendent to the relative by providing a highly subtle and precise language for expressing the immediate contents of transcendent states with minimal distortion. On the other hand, it also serves as a vehicle for crossing from the relative to the transcendent by providing highly abstract and universal symbols for generating insights through contemplation. Wolff emphasizes, however, that although the structure of this mathematical bridge is provided by the highly subtle forms of thought, an actual crossing of the bridge requires the motivating power of love and devotion.
M I L L E N N I U M :: View Topic - Proof Of God - Error In Mathematics
Here is another page that deals with Cantor and infinities http//users.forthnet.gr/ath/kimon/Continuum.htm This is the crucial quote from the website But
http://www.childrenofmillennium.org/phpBB2/viewtopic.php?t=103&start=0

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