21. Math Links the notion that mathematics is a selfconsistent system of knowledge. Presented here are Zeno s Paradox and Cantor s infinities. http://www.narragansett.k12.ma.us/nrhs/math/mathlinks.htm

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.

22. Science News: Infinite Wisdom: A New Approach To One Of Mathematics' Most Notori Cantor s hierarchy of infinities was such a revolutionary concept that many of his contemporaries rejected it out of hand. Their http://articles.findarticles.com/p/articles/mi_m1200/is_9_164/ai_108050571

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); Advanced Search Home Help IN all publications this publication Reference Automotive Business Computing Entertainment Health News Reference Sports YOU ARE HERE Articles Science News August 30, 2003 Content provided in partnership with Print friendly Tell a friend Find subscription deals Infinite wisdom: a new approach to one of mathematics' most notorious problems Science News August 30, 2003 by Erica Klarreich How many numbers are there? For children, the answer might be a millionthat is, until they discover a billion, or a trillion, or a googol. Then, maybe they notice that a googol plus one is also a number, and they realize that although the names for numbers run out, the numbers themselves never do. Yet to mathematicians, the idea that there are infinitely many numbers is just the beginning of an answer. Counterintuitive as it seems, there are many infinitiesinfinitely many, in fact. And some are bigger than others. In the late 19th century, mathematicians showed that most familiar infinite collections of numbers are the same size. This group includes the counting numbers (1, 2, 3, ...), the even numbers, and the rational numbers (quotients of counting numbers, such as 3/4 and 101/763). However, in work that astonished the mathematicians of his day, the Russian-born Georg Cantor proved in 1873 that the real numbers (all the numbers that make up the number line) form a bigger infinity than the counting numbers do.

23. Talk:Cantor's Diagonal Argument - Wikipedia, The Free Encyclopedia Cantor s Diagonal argument is my favourite piece of Mathematics Andre Engels. If we accept the existence of uncountable infinities (and i guess you do if you http://en.wikipedia.org/wiki/Talk:Cantor's_diagonal_argument

Talk:Cantor's diagonal argument From Wikipedia, the free encyclopedia. I'm glad there is an article on this in Wikipedia. Cantor's Diagonal argument is my favourite piece of Mathematics Andre Engels OK, the two "notes" on the page as it currently stands is annoying. We can prove this property of the *reals*, and not just their decimal expansions if we use the following rule: The digit x is increased by 1, unless it is 8 or 9, and then the digit becomes 1. This will gaurantee the number is different from every other number on the list and would not create a different representation of any of those numbers. The discussion of 0.99999... = 1 should most certainly be put on its own page to lay to rest the assumption that decimal expansions are unique. But please, let's not get sloppy on the page cluttering it with "notes." quote Its result can be used to show that the notion of the [set of all sets]? is an inconsistent notion; if S would be the set of all sets then P(S) would at the same time be bigger than S and a subset of S. end quote Can the diagonal argument really do this? If we accept the existence of uncountable infinities (and i guess you do if you accept the diagonal argument) then it is pretty clear that the set of all sets must be of uncountable size if it exists. How do you define an ordered list of elements of an uncountable set? If you can't define an ordered list, how do you apply the diagonal argument? (Maybe you can, but it's not immediately obvious how. I think a bit more needs to be written about this if the above quote is to remain.)

24. INTEGRITY - Robust Non-Fractal Complexity -  NECSI Journal Paper In an era of Relativity, Quantum Mechanics, Cantor s infinities, holography, Zadeh Logic, quark architecture, spread spectrum transmission, Complexity, and the http://www.ceptualinstitute.com/uiu_plus/necsij1send.htm

Presented at the NECSI / ICCS International Conference on Complex Systems September 21-26, 1997; Nashua New Hampshire USA Robust Non-Fractal Complexity James N Rose Ceptual Institute, 1271 Bronco Circle, Minden NV 89423 http://www.ceptualinstitute.com email: integrity@ceptualinstitute.com Section Links Abstract Complexity: Orchestration of Multiple Stochastic Systems Redefining the general structure of Mathematics The new Weltanschauung ... Summary Abstract: Primary versions of complexity to date have been considered relative to fractal models. They have tended to show that complexly ordered patternings arise or emerge after massive iterations of some relatively simple functions which, on their face, do not indicate that important relational and temporal patternings are nascently inherent in them. Corollary work (Prigogine, et al) has shown that in some cases contra-entropy plateaus of stability exist far from initial equilibrium conditions, giving secondary and tertiary conditions on which to build complex systems. These are important and pervasive factors of complexity. Yet, complexity can also be seen in situations which do not involve inordinate membership or interaction samples, and also, in situations that are not easily assessable by equilibrium statistics. It is the author's contention that there also exists a more general and robust form of complexity generating mechanism denotable in simple systems with non-homogeneous construction (that is, in systems which have independent yet interactive sub-components). These sub-components can be evaluated with their own behavior-space, independent from yet interactive with the behavior-space of the system at large.

25. QBasics definities, verwijzen ze wel degelijk naar wat Engelsen ook als infinities spellen. Met Cantor gaat er een wereld open van oneindig veel graden van oneindigheid http://home.pi.be/~pin12499/infiniti.htm

Wugi's Infinities Oneindige waarnemingen - Infinite perception Guido "Wugi" Wuyts Dilbeek, Belgium, Europe, World, Solar System, Milky Way, Local Cluster, ... Taalzaak @ 2 Denkzaak ... O, redactie NPN Readers' Indigest NewsThink Infinities Oneindig simpel? Hoewel mijn infinities bedoeld zijn te klinken ende luiden als definities, verwijzen ze wel degelijk naar wat Engelsen ook als infinities spellen. Een uitnodiging dus tot nadenken over oneindigheid. Ze wordt niet opgediend als een afgewerkt stuk overtuiging, daar weet ondergetekende zich te zeer leek voor, maar als samenraapsel van bedenkingen en twijfels die tijdens mijn studie en latere lectuur opdoken en daar niet door weggenomen of opgelost raakten. Mijn laatste leesvoer was The Mystery of the Aleph van Amir D.Aczel, een interessante bron voor de stroom van mijn bedenkingen die ik hierna zijn weg laat zoeken. Met Cantor gaat er een wereld open van oneindig veel graden van oneindigheid, alefs genoemd, de ene onbereikbaar ver boven de andere. Daarbij lijkt de eerste graad van oneindigheid, alef-0 "de aftelbare", een vertrouwd buurtjongetje dat iedereen goed meent te kennen. Mijn bewering is nu echter dat deze aftelbare oneindigheid zelf al niet als een "uniek gegeven" bestaat, laat staan een basis zou bieden voor hogere alefs. Een rode draad van het boek is dat oneindigheid enerzijds al sinds de oudheid gekend is als een potentieel gegeven, iets waarnaar je bij middel van lijstjes, reeksen... toe kan streven zonder het ooit te bereiken. Anderzijds, dat Cantor de eerste was die er geen graten in zag om oneindigheid, namelijk een oneindige verzameling, te beschouwen als iets afs, een kant en klaar, afgerond geheel waar je een "aantal" op kan plakken.

26. "Uncountable": Finite Intuitions And Cantor's Diagonal (beyond Peano) Do the infinities cancel out? 1}*. Ciao, Nico Benschop http//home.iae.nl/users/benschop/finite.htm http//home.iae.nl/users/benschop/cantor.htm http http://home.iae.nl/users/benschop/finite.htm

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." WSM discussion group (msg #1960, 22apr04) Your construction of a 1-1 mapping integers reals can be further simplified, without loss of generality, as follows. One needs only to consider the reals on interval [0,1) denoted as set R01, thus with zero integer part and only decimals to the right of the decimal (or binary) point. Compare R01 with the non-negative integers set N. Then there is the simple 1-1 correspondence implied by symmetry about the decimal point, for instance 0.12345 in R01 Newsgrp: sci.math Subject: Re: Why are rationals countable? Date: Fri, 19 Apr 2002 From: Nico Benschop "Zdislav V. Kovarik" wrote: > > In article , > Agapito Martinez Author: benschop_nf Date: 20 nov 1998 Forum: sci.math - Re: Cantor's Diagonal Proof: FLAWED! - In article Date: Fri, 12 Mar 1999 From: Nico Benschop Newsgrp: sci.math In article , Mike Deeth Strict reasoning < c Re: 1/0 . . . (sci.math 10jan2001)

27. 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

Subject: Re: Allright, how many reals *are* there? Author: Nico Benschop Re: How diagonal is Cantor's diagonal? ...(sci.math 6jun98)

28. Cantor's Conjecture Cantor himself produced a lot of texts as to potential, actual, absolute, in concreto , in abstracto , metaphysical, theological, etc. infinities. http://www.ontologystream.com/beads/Cantor/Zenkin/bead1.htm

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.

29. Media RelationsNews Releases They are free and open to the public. The topics are Jan. 26. Cantor s infinities The lecture subject is Georg Cantor, a mathematician who has had a great http://www.northwestern.edu/univ-relations/media/news-releases/1999-00/*uwn/nemm

CONTACT: Charles Loebbaka at (847) 491-4887 or by e-mail at c-loebbaka@nwu.edu FOR RELEASE: Immediate Distinguished Mathematician John Conway Gives Nemmers Lectures EVANSTON, Ill. - John H. Conway will deliver his second talk in the Nemmers Prize Lecture series at Northwestern University Tuesday (Jan. 18 )on the topic "Archimedes and His World." Conway, the Frederic Esser Nemmers Professor of Mathematics at Northwestern University, is lecturing during the winter quarter on the theme "Thinking About Mathematics (and Many Other Things)." The talks are aimed at a lay audience with the goal of connecting people to the science of mathematics. One of the preeminent theorists in the study of finite groups (the mathematical abstraction of symmetry) and one of the world's foremost knot theorists, Conway is the author of more than 10 books and 130 journal articles on mathematical subjects. He has done pathbreaking work in number theory, game theory, coding theory, tiling and the creation of new number systems. Conway, who is the von Neumann Professor of Mathematics at Princeton University, was selected for the Nemmers honor in 1998. The lecture series is among his scholarly activities while in residence at Northwestern this quarter.

30. 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

home human ecology home mathematics Human ecology Bounded infinities 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, the mapping from S to T* is injective (as is the mapping from T to S) A' - A B' - B C' - C D' - D E' - E F' G' H' with the example before: the mapping from T* to S is surjective (as well as from T to S, and from S to T) Bijective: the mapping is both surjective and injective, i.e. from S to T, and from T to S) This is alright for finite sets. What happens if you have infinite sets? one - 1 two - 2 three - 3 four - 4 five - 5 ten - 10 eleven - 11 infinite - ?

31. Aristotle's Infinite And Cantor Aristotle s Potential and Actual Infinite and Cantor. Aristotle considering the Paradox due to the infinite small and large considered two types of infinities. http://www.mlahanas.de/Greeks/Infinite.htm

Aristotle's Potential and Actual Infinite and Cantor 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?

32. 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

Proof That Not All Infinities Are The Same Size The proof is as follows: we count by matching the natural numbers to some set. For example, the set: 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

33. Deterministic Chaos: Mandelbrot's Fractal Dimensions Are Not Special Species Of next 10 Cantor s schizophrenic infinities A reminder of the way for generating numbers according to set theory. A reminder of http://pro.wanadoo.fr/quatuor/english/mathematics.htm

home french version contents Mathematics new: Quatuor site has its own domain name: quatuor.org - Then, the page you are looking at, is available by: www.quatuor.org/english/mathematics.htm . If you have made a link to the site, thanks, and please, update this link where is the order of deterministic chaos hiding From the dimensions with coordinates to the dimensions of deformation start what is a dimension ? The evolution of the notion of dimension, from antiquity to fractal dimensions of Mandelbrot. [next] the measurement of the deformation of a contrast How we can measure a phenomenon without any notion of coordinates on an axis. The fundamental difference between coordinate dimensions and deformation dimensions. [next] in theory the dimension of a contrast can be self-similar The problem of dimensions which vary with the scale of the measurement. [next] the trap in the vectorial representation of the forces Usually a force is summarized by a vector which applies on a point. This representation leads to a serious abnormality when we want to calculate the interference of several forces. We suggest another type of representation that correctly uses the effects of their interference in all directions of space: this representation requires the measurement of infinity of vectors in every point.

34. Fractal Dimensions Of Mandelbrot And Chaos Theory - 5 - Self-similarity As A For begin with a detour through infinity. Then we go. Cantor s schizophrenic infinities. Ask a particles physicist about the infinitesimal http://pro.wanadoo.fr/quatuor/english/fractal_10.htm

home contents Mathematics previous : the trap in the vectorial representation of the forces next : let's go back from zero direct to all other pages of the section Counting in a different way (why most of the time .5 is anything but between and 1) Counting in a different way? What does it mean? That 2 and 2 would not make 4? Yes they do, but 2 + .1 will not make 2.1 every time. For we will have to reconsider the relation between whole numbers and their decimal digits. And this will enable us to see fractional dimensions of Mandelbrot in a more pertinent way, for precisely these dimensions are dimensions with decimal values. But before seeing these dimensions, we have to see first, why 0.5 for example, has nothing to do with the path which goes from to 1. Why most of the time, 0.5 cannot possibly be between and 1. Strangely enough, the simplest way to understand how to travel from to 1 without ever reaching 0.5, is to begin with a detour through infinity. Then we go. Cantor's schizophrenic infinities Ask a particles physicist about the infinitesimal, he will tell you how very strange things happen there, which are unthinkable at our scale. Such as that a particle can be a well-located corpuscle, and a wave of infinite size endlessly expanding in space, both at the same time.

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

Infinity Back to 5 Numbers More about Infinity INFINITY You can read more about the number below, and there are links to other infinity sites and you can hear the programme infinity Infinity Given the old maxim about an infinite number of monkeys and typewriters, one can assume that said simian digits will type up the following line from Hamlet an infinite number of times. "I could confine myself to a nutshell and declare myself king of infinity". 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, 4Â .). 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:

36. Re: Infinities There is a greater immediacy of completed infinities in Cantor s application for the uncountability of the real numbers; but natural number arithmeticas, for http://hhobel.phl.univie.ac.at/phlo/199707/msg00192.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Infinities To phil-logic@bucknell.edu Subject : Re: Infinities From n_nelson@ix.netcom.com Date : Thu, 31 Jul 1997 09:48:50 -0500 (CDT) Prev by Date: Re: Infinities Next by Date: Re: Infinities Prev by thread: Re: Infinities Next by thread: Re: Infinities Index(es): Date Thread

37. Re:Sameness And Self-Identity and symbolic logic the influence of such as Ramón Lull and Kabbalah on Leibnitz and Newton to the present day (from Cantor s infinities through Spinoza s http://hhobel.phl.univie.ac.at/phlo/199701/msg00014.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re:Sameness and Self-Identity To phil-logic@bucknell.edu Subject : Re:Sameness and Self-Identity From : "David A.Hernandez Purnell" < David@metatheology.org Date : Fri, 3 Jan 1997 13:45:43 -0000 Prev by Date: Re: Let's Not Forget Set Theory Next by Date: Re: Message to Adriano (Private) Prev by thread: Re: "Let's Not Forget Set Theory" Next by thread: Re: Sameness and Self-Identity Index(es): Date Thread

38. Newsletter On Proof the properties of the prime numbers and the difficulty of finding primes Famous Paradoxes Zeno s Paradox and Cantor s infinities - The Problem of Points http://www.lettredelapreuve.it/Newsletter/981112.html

Herbst P. G. What works as proof in the mathematics class . Ph.D. Dissertation, The University of Georgia, Athens GA. USA Lopes A. J. Raccah P.-Y. (1998) L'argumentation sans la preuve : prendre son biais dans la langue. Interaction et cognitions . II(1/2) 237-264. Arzarello F. Micheletti C. Olivero F. Robutti O. (1998) A model for analysing the transition to formal proofs in geometry. (Volume 2, pp.24-31) Arzarello F. Micheletti C. Olivero F. Robutti O. (1998) Dragging in Cabri and modalities transition from conjectures to proofs in geometry. (Volume 2, pp. 32-39) Baldino R. (1998) Dialectical proof: Should we teach it to physics students. (Volume 2, pp. 48-55) Furinghetti F. Paola D. (1998) Context influence on mathematical reasoning. (Volume 2, pp. 313-320) Gardiner J. Hudson B. (1998) The evolution of pupils' ideas of construction and proof using hand-held dynamic geometry technology. (Volume 2, pp. 337-344) Garuti R.

39. Infinity And Infinities Infinity and infinities. It was not until the 19th Century that mathematicians discovered that infinity comes in different sizes. Georg Cantor (1845 to 1918 http://www.gap-system.org/~john/analysis/Lectures/L4.html

MT2002 Analysis Previous page (Functions) Contents Next page (Axioms for the Real numbers) Infinity and infinities It was not until the 19th Century that mathematicians discovered that infinity comes in different sizes. Georg Cantor (1845 to 1918) defined the following. Definition Any set which can be put into one-one correspondence with N is called countable Remarks Given such a set, one can count off its elements: 1st, 2nd, 3rd, .. and will eventually reach any element of the set. We will see later that many infinite sets are countable but that some are not. Some versions of the above definition include finite sets among the countable ones, but we will (mostly) not do so. Examples of some countable sets The set Z of positive, zero and negative integers is countable. Proof Here is a counting. That is, we list the elements N Z You can (if you wish) write down a formula: n if n is even, n - 1) if n is odd. The set N N is countable. Proof Count the points with integer coefficients in the positive quadrant as shown. (The formula is now rather tricky to write down.) The set Q of all rationals is countable.

40. Philosophy For The Short Attention Span Further reading http//www.chinesefortunecalendar.com/yinyang.htm; Cantor s infinities In 1873 the mathematician Georg Cantor discovered the hierarchy of http://quantumlab.net/cgi-bin/blog/blog.pl/2003/09/

Philosophy for the Short Attention Span September Sun Mon Tue Wed Thu Fri Sat Deep Thoughts Matt blog@quantumlab.net Mon, 29 Sep 2003 The Dual Nature of Reality For my next blog I will sing some songs for the dance couples below. To appreciate the song and dance, try to look for the patterns. Yin/Yang According to legend, the Chinese Emperor Fu Hsi discovered the Yin/Yang symbol at around 4,000 BC. These dancers are Yin and Yang . Their special dance routine is the way they do not reach equilibrium, but rather, continue to dance around each other. Their interplay represents many things in life, possibly including all of existence. Further reading: http://www.chinesefortunecalendar.com/yinyang.htm Cantor's Infinities In 1873 the mathematician Georg Cantor discovered the hierarchy of infinite quantities. Geometrically, this means that there are more points on a continuous line than in a discrete set of points evenly spaced from each other, even though both sets are infinite. These dancers are the continuous real numbers and the discrete whole numbers . Their special dance move is a fake-out, where the discrete set begins to enumerate the continuous set, but the continuous set lets a point slip out that the discrete set cannot enumerate no matter how hard he tries. The result is a deeper glimpse of the structure of infinity.

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