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Cantor's Infinities:     more detail
1. Satan, Cantor and Infinity: Mind-Boggling Puzzles (Dover Books on Mathematical & Word Recreations) by Raymond M. Smullyan, 2009-03-26
2. Satan, Cantor and Infinity and Other Mind-boggling Puzzles by Raymond M. Smullyan, 1993-02-18
3. Set Theory and the Sizes of Infinity: An entry from Gale's <i>Science and Its Times</i> by Sherri Chasin Calvo, 2000
4. Infinity: Countable set, Cantor's diagonal argument, Surreal number, Continuum hypothesis, Hyperreal number, Extended real number line
5. Transfinite number: Transfinite Number, Cardinal Number, Ordinal Number, Finite Set, Absolute Infinite, Georg Cantor, Infinity, Cardinality
6. Satan, Cantor & Infinity by Raymond M. Smullyan, 1994-06-12

lists with details

1. Mathematics Revenge: How Numbers Don't Behave As They Should!
What I don't understand about Cantor Infinities. Cantor defines w as the number of numbers integers, normal mathematical operations could be applied to cantor's infinities. Thus, w
http://starship.python.net/crew/timehorse/new_math.html

Extractions: www Cantor defines w w elements. The reason this is called an Ordinal Infinity is because the set of numbers has a specific ordering: 2 always follows 1, etc. This set of Ordinal Numbers does not itself include Infinity ( Inf ) since Inf is not a number you can count. For more information on this please visit: Robert Munafo's Large Number Pages Section 4 Eric Weisstein's MathWorld page on Ordinal Numbers Following the logical definition of w , Cantor further devised the concept of even larger sets. If you imagine w w because w is Inf . Adding w to that set would produce a set 1 bigger than w , which Cantor denoted w + 1. It must be noted however that Cantor did not consider 1 + w to be the same as w w w w w but w w Cantor further defined even larger sets. Like with integers, normal mathematical operations could be applied to Cantor's infinities. Thus, w w + 1 because w w + 1 + 1, where w w w w w w w w w w - 1 is a place holder for the last integer in the w w w in the same way we do normal integers: w 2. By analogy, Cantor went further to define

2. The Repressed Content-Requirements Of Mathematics
HBT requires uncountable infinities of rotations. Cantor's interpacking of infinite decimals Wittgenstein's arguments against cantor's infinitiesin turn considered idiotic by most
http://www.henryflynt.org/studies_sci/reqmath.html

Extractions: The Repressed Content-Requirements of Mathematics Henry Flynt [started c. 1987; this draft 1994] (c) 1994 Henry A. Flynt, Jr. A. Mathematics, as it is conceived in the twentieth century, has presuppositions about perception, and about the comprehension of lived experience relative to the apprehension of apparitions, which are repressed in professional doctrine. It also has presuppositions of a supra-terrestrial import which are repressed. The latter concern abstractions whose reality-character is an incoherent composite of features of sensuous-concrete phenomena. In the twentieth century, mathematicians have been taught to say by rote, "We are beyond all that now. We are beyond psychology, and we are beyond independently subsisting abstractions." This recitation is a case of denial. Indeed, if this recitation were true, it would allow mathematics nowhere to live. Certainly social conventionsbeloved by the Vienna Circleare too unreliable to found the truths which mathematicians claim to possess. (Truths about the decimal value of [pi], or about different sizes of infinity, for example.) To uncover the repressed presuppositions, a combination of approaches is required. (One anthropologist has written about "the locus of mathematical reality"but, being an academic, he merely reproduces a stock answer outside his field, namely that the shape of mathematics is dictated by the physiology of the brain.)

3. Math Forum: Cantor's Solution: Denumerability
Zeno's Paradox ·. cantor's infinities ·. cantor's infinities, Page 2
http://mathforum.com/~isaac/problems/cantor2.html

Extractions: Now, Cantor made the following definition: Definition : Two sets are equal in magnitude (i.e. size) if their elements can be put into one-to-one correspondence with each other. This means that the natural numbers, the integers, and the even integers all have the 'same number' of elements. Cantor denoted the number of natural numbers by the transfinite number (pronounced aleph-nought or aleph-null). For ease of notation, we will call this number d, since the set of all natural numbers (and all sets of equal magnitude) are often called denumerable , a , a corresponds to the natural number 1, a to 2, and so on. Theorem: The set of rational numbers is denumerable, that is, it has cardinal number d.

4. Math & Science: More Points In A Line Or In A Plane
The end result is not all infinities are created equal sorry pushing the limits of my know-how of cantor's infinities - too many more questions and I will have run
http://www.experts-exchange.com/Miscellaneous/Q_20309516.html

Extractions: Seems like a plane should have an infinite number of points more than the infinite number of points in the line. This is a comparison of two like kinds/classes of infinity so they (the number of points in a line and the number of points in a plane) are really-really equal. If those two infinities are equal (equal means, "every point in a plane can be mapped to a single unique point in a line and the converse is true") than describe such a (isomorphic) mapping.

5. Untitled
in the building of stronger theoretical foundations, nonEuclidean geometries, and cantor's infinities we shall see in Cantor's work on infinities, geometry contains a problem
http://www.andrews.edu/~closserb/215_Logan.html

Extractions: Clara Logan During the 1800s, we find the theme of independence, or freedom from outside constraints, in the development of two different frontiers. We find it in the American West through Manifest Destiny, freedom from caste, and in the chance that homesteaders had to acquire virtually free land. We find independence in math through in the building of stronger theoretical foundations, non-Euclidean geometries, and Cantor's infinities. Independence involves breaking from the commonly accepted, traditional views in order to explore the new. It is not necessarily individual people working alone. We can see independence in a community of thought as well as in the work of a single person. The concept of Manifest Destiny is also seen in the actions of the common people. These common people included the family of my great grandmother's grandmother, Mary Rocell. In the 1880s, Hulda Rocell and her daughter Mary emigrated from Sweden to the United States. Abe Lincoln had just been shot. Mr. Rocell had to stay in Sweden because of his tuberculosis. Nevertheless, Mr. Rocell said, "Go to the United States. It is strong enough that Lincoln's assassination will not plummet the nation into chaos." Although he did not place this optimism under the title of Manifest Destiny, the idea that the United States is strong, and will continue despite opposition, is a part of this concept.

Zeno's Paradox ·. cantor's infinities ·. cantor's infinities, Page 2
http://mathforum.com/isaac/problems/zeno1.html

Extractions: Prime Numbers ... Links The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved. Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. Here we will present the first of his famous four paradoxes. Zeno's first paradox attacks the notion held by many philosophers of his day that space was infinitely divisible, and that motion was therefore continuous. Paradox 1: The Motionless Runner A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters. Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

7. [FOM] As To Strict Definitions Of Potential And Actual Infinities.
cantor's and any modern axiomatic infinities" (see FOMarchive at http//www.cs.nyu.edu/pipermail/fom/2002- December/006121.html) I have given a quite impressive list of cantor's
http://cs.nyu.edu/pipermail/fom/2003-January/006173.html

Extractions: Thu Jan 16 14:19:32 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:

8. Cantor's Diagonal Proof
cantor's Diagonal Proof. Simplicio I'm trying to understand the significance of cantor's diagonal proof. I find it especially confusing that the rational numbers are considered to be countable, but
http://www.mathpages.com/home/kmath371.htm

Extractions: Simplicio: I'm trying to understand the significance of Cantor's diagonal proof. I find it especially confusing that the rational numbers are considered to be countable, but the real numbers are not. It seems obvious to me that in any list of rational numbers more rational numbers can be constructed, using the same diagonal approach. Interfering With PI Simplicio: You said there is no upper bound on the size of natural numbers, and thus the least upper bound on the naturals is infinite, even though every natural number is finite. To me this implies that there can be numbers which do not have such a bound. Is that not so? Salviati: It sounds like you're trying to invent a kind of "number" that has infinitely many digits in the direction of geometrically increasing significance, somewhat analagous to the reals, which have infinitely many digits in the direction of geometrically decreasing significance. Number systems like what you are talking about have actually been developed, Simplicio, (see p-adic numbers) but the crucial difference is that the infinite sequence of digits is in the direction of increasing, not decreasing, significance, so the resulting implied "sum" does not converge to a value that behaves consistently like a magnitude. (The valuations are said to be "non- Archimedian".) There's nothing wrong with conceiving new forms of numbers like this, but we need to be clear about how they differ from other forms of numbers.

9. Cantor's Diagonal Proof
These numbers would, of course, also be infinite, but would be larger infinities than the standard infinity, ¥, that we are used to. Cantor called these
http://home.ican.net/~arandall/abelard/math12/Cantor.html

Extractions: Cantors Diagonal Proof Cantor introduced the idea of sets. A set was, for Cantor, a purely intuitive concept, not formally defined. A set is a kind of collection of objects that we can imagine intuitively in our minds. We can speak, for instance, of the set of all red things. Or a set can contain other sets, such as the set of all sets of coloured things. . However, Cantor asked us to consider that there could be larger numbers than infinity. These numbers would, of course, also be infinite, but would be larger infinities than the standard infinity, , that we are used to. Cantor called these numbers "transfinite" numbers. After , we get ( + 2), and so on. So what is the cardinality, or size, of ( . But, in fact, I can place all the elements of the first set into a one-to-one correspondence with the second, like so: Cantor defined the size of his sets so that any two sets that could be matched up one-to-one like this were considered to be the same size. The same holds true for any finite number of new elements I insert into the set. Note that it does not really matter that I placed the new element at the beginning, since the one-to-one match-up would still hold no matter where I inserted the new element. So it really makes no sense to say that one of these sets is larger than the other. In other words, + 1, and we do not seem to have created a larger number with

10. Set Theory
7, 1873, the date of Cantor s letter to Dedekind informing him of his discovery.) Until then, no one envisioned the possibility that infinities come in
http://plato.stanford.edu/entries/set-theory/

Extractions: JUL 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 B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members the only objects one needs for the theory are sets. See the supplement Basic Set Theory for further discussion.

11. Cantor's Diagonal Argument
The great granddaddy of diagonal arguments is Cantor s, which proved that some infinities are bigger than others. Cantor s Diagonal Argument.
http://www.chaos.org.uk/~eddy/math/diagonal.html

Extractions: Various arguments prove extremely strong results by phrasing the result in such a way that some `two-parameter' thing can be used to supply a `diagonal', the special case where the two-parameter thing has both parameters equal, analysis of which suffices to prove some property of the two-parameter thing which suffices to imply the desired strong result. The great grand-daddy of diagonal arguments is Cantor's, which proved that some infinities are bigger than others. can can enumerate any set of subsets of the naturals as long as there's an upper (finite) bound on the size of the subsets. One may use each natural in turn as choice of upper bound and construct an enumeration of all the subsets of the naturals smaller than this. It might seem that `in the limit at infinity' this process should get us at least an enumeration of all finite sub-sets of the naturals; and one might hope that some analogous trickery might enable one to get all sub-sets of the naturals. In any case the diagonal argument shows only one The diagonal argument can be read as an algorithm for converting a function, from a set to its power set, into a function from the set's successor to

12. [FOM] As To Strict Definitions Of Potential And Actual Infinities.
well as modern nonnaive ) set theory is based on Cantor s theorem on previous message As to strict definitions of potential and actual infinities (see FOM
http://www.cs.nyu.edu/pipermail/fom/2003-January/006173.html

Extractions: Thu Jan 16 14:19:32 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:

13. [FOM] As To Strict Definitions Of Potential And Actual Infinities.
FOM As to strict definitions of potential and actual infinities. Dean Buckner Dean.Buckner at btopenworld.com Thu Jan 16 210320 EST 2003 CANTOR S AXIOM.
http://www.cs.nyu.edu/pipermail/fom/2003-January/006174.html

Extractions: Thu Jan 16 21:03:20 EST 2003 I believe that all the axiomatic set theories break down the classical logic and the classical mathematics in the following point. We have [...] the following two opposed axiomatic statements. ARISTOTLE'S AXIOM. All infinite sets are potential. CANTOR'S AXIOM. All sets are actual (since all finite sets are More information about the FOM mailing list

14. Untitled
The ideals and beliefs of Rome started to catch on again. Art, literature, politics, and many other things began to change. The social classed became less pronounced. them. All others cannot. Failing to contemplate these infinities, men have recklessly taken it on themselves to which lies between the two infinities which enclose and flee from it
http://www.acceleratedschools.com/s9/ehcombwp.txt

15. 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

16. Teacher Training HomePage
Failing. to contemplate these infinities, men have recklessly finite which lies between the two infinities which enclose and
http://www.acceleratedschools.com/oldt9/intro_ehcomb.htm

17. 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, neural
http://users.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.

18. Infinity: You Can't Get There From Here -- Platonic Realms MiniText
Cantor completely contradicted the Aristotelian doctrine proscribing actual, completed infinities, and for his boldness he was rewarded with a lifetime

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.

19. Re: Limitations Of C*algebras
functions. To clear up some apparent confusion, it was I who dismissed Cantor s infinities and the Reals and Rationals; not Charles. The
http://www.lns.cornell.edu/spr/2000-01/msg0020769.html

Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index OKw6EIAgV4c4EwYE@clef.demon.co.uk ">news: OKw6EIAgV4c4EwYE@clef.demon.co.uk Follow-Ups Re: limitations of C*algebras From: References Re: limitations of C*algebras From: Prev by Date: Photon Tunnelling Next by Date: Re: Who put the 8PI in the bomp de bomp bomp Prev by thread: Re: limitations of C*algebras Next by thread: Re: limitations of C*algebras Index(es): Date Thread

20. Re: Limitations Of C*algebras
To clear up some apparent confusion, it was I who dismissed Cantor s infinities and the Reals and Rationals; not Charles.
http://www.lns.cornell.edu/spr/2000-01/msg0020819.html

Extractions: Date Prev Date Next Thread Prev Thread Next ... Thread Index lc_c4.9104\$oJ5.19011@newsfeeds.bigpond.com OKw6EIAgV4c4EwYE@clef.demon.co.uk ">news: OKw6EIAgV4c4EwYE@clef.demon.co.uk Re: limitations of C*algebras From: "Terry Padden - news" <TCCG@bigpond.com> Prev by Date: Re: Measurement and Observers Next by Date: Re: Cosmic Background Radiation (Not Very Speculative, Honestly) Prev by thread: Re: limitations of C*algebras Next by thread: Re: limitations of C*algebras Index(es): Date Thread

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