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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

Hive
5. scampi 6. Cantor s infinities 7. half a genotype 8. seventh Quidditch player 9. bridal or baby 10. minister, reverend, priest, etc. 11. 1sphere 12.
http://techhouse.brown.edu/puzzles/archive/puz/8/

Extractions: "I'll bet you didn't know Brown has a beekeeping department," quips an odd man standing by a buzzing hive on Lincoln Field. "Here, have a mask. I'm trying to figure out from what direction they're getting their food, but I'm not very good at reading their dancing." As he hands you the mesh mask you notice several bees flying between the hive and a patch of flowers several yards away, but rather than point this out you take a closer look. opportune; not late
[Phil-logic] Cantor's Axiom
Right. This is the sort of thing I see as a supporting consideration for Cantor s view. What are actual infinities ? What sense of actual is meant here?
http://philo.at/pipermail/phil-logic/2003-November/001959.html

Extractions: Wed Nov 26 10:55:17 CET 2003 Paul McNamara wrote: Apropos of this, those remarks I recently quoted from Cantor linking the-square-root-of-two to w (omega) seem helpful. Sloppifying Cantor, one way to look at w is that it is the limit of the natural numbers, just as the-square-root-of-two is the limit of a certain sequence. What are "actual infinities"? What sense of "actual" is meant here? Good question. I had in mind the former as a term of art in Phil of Math, so that it is in contrast to "potential infinity". On the latter view, the set of natural numbers would not be deemed to exist, on the former it would exist ("as a totality", as they used to say). Likewise for (say) the reals. I think it is more challenging to say what "potential infinity" means than to say what "actual infinity" means, although defending the truth of latter sort of claim is clearly challenging. -paul > Phil-logic mailing list Phil-logic at philo.at
[Phil-logic] Cantor's Axiom
PM Right. This is the sort of thing I see as a supporting consideration for Cantor s view. CS What are actual infinities ?
http://philo.at/pipermail/phil-logic/2003-November/001960.html

Extractions: Wed Nov 26 10:29:38 CET 2003 Right. This is the sort of thing I see as a supporting consideration Cantor's view. > What are "actual infinities"? What sense of > "actual" is meant here? Good question. I had in mind the former as a term of art in Phil of so that it is in contrast to "potential infinity". On the latter set of natural numbers would not be deemed to exist, on the former would exist ("as a totality", as they used to say). Likewise for reals. I think it is more challenging to say what "potential infinity" > means than to say what "actual infinity" means, although defending truth of latter sort of claim is clearly challenging. -paul Sure, I'm familiar with the adjectives "actual" and "potential" when applied to "infinity", but I'm wondering how this distinction really plays out. It seems okay to say that according to the view that there's only a potential infinity, "the set of natural numbers would not be deemed to exist." But, what is "would not be deemed to exist" supposed to mean in this context? (Where's the "deeming" coming from?) Previous message: [Phil-logic] Cantor's Axiom Next message: [Phil-logic] Cantor's Axiom Messages sorted by:
Cantor S Alpha One - Technology Services
I ask because cantor s proof seems a bit shaky to me, at least the way I ve I can t comment on the idea of how many infinities there are without knowing what
http://www.physicsforums.com/archive/t-23442
Media Center - Math Projects
The Abacus Index. Cantor s infinities Math Forum Infinite Sets Math Forum Cantor s Solution Denumerability. Computation Systems Computer
http://www.anderson2.k12.sc.us/schools/bhp/math_proj.htm
Certainty, Infinity, Impossibility
Key ideas Numbers, The Pythagoreans and the irrationality of 2 , orders of infinities, Cantor s diagonalization argument, R = R n ; n=2, 3
http://cs.wwc.edu/~aabyan/CII/book.html

Extractions: Syllabus Schedule Project Reports ... Course Rationale Instructors: Anthony Aaby Thomas Thompson These documents are written using XHTML1.1 and MathML. For those documents that include MathML, Amaya Opera , or Mozilla are appropriate browsers. (Social and cultural implications of mathematics and computer science.) Course Description: Reading, reflection, and discussion of the implications of the limits of reason and objective reality that underlie rational western civilization. The topics are an eclectic collection selected from 2,500 years of mathematics and logic. Prerequisites: Upper division standing, general studies mathematics, and strong curiosity about the limits of reason. Option: Upon request, graded S/NC. Distance learning: This course is avaliable on the internet as a distance learning course. The lecture outlines are provided online. In lieu of oral presentations, three peer evaluations for each essay are required. Goals: To paraphrase Bertrand Russell, "A course should have either intelligibility or correctness; to combine the two is impossible" thus our choice is to focus on intelligibility. We hope to avoid the situation described by Clifford Allen: "More intellectual `ticking off' from B.[ertrand] R.[ussell] at dinner because I used the word 'sentence' when I should have used `phrase'. I'm dead sick of it." Upon completion of this course you will be aware of and understand some of the philosophical implications of: The paradoxes of naive set theory
Ecclectica - Aleph
However, Cantor reasoned that Kronecker would only read the title and abstract, looking for some mention of Cantor s objectionable infinities.
http://www.ecclectica.ca/issues/2002/1/williams.asp

Extractions: The book, or movie, A Beautiful Mind, has recently entranced us all with a view into the troubled thoughts of Princeton mathematician and Nobel Laureate John Nash. He is often compared with the painter, Vincent Van Gogh, another unhappy spirit who spent much time in and out of mental institutions but never wavered from his quest. Many of us have a weakness, even envy, for these people, driven to madness by their single-minded search for truth and their determination to declare it. Georg Cantor is another example. He was born in Russia in 1845, but lived most of his life in Germany. Although he trained with some of the foremost mathematicians of the day, and obtained his doctorate in 1869, he was unable to secure a position at any of the prestigious research universities. Disappointed, he accepted a post at Friedrich's University in the small industrial town of Halle, almost midway between the two great university cities of Gottingen and Berlin. Time and again, he applied for positions in both of these places, always to be refused. Cantor's work with mathematical set theory, which would eventually lead to a revolution in our understanding of the infinite, began with the question: How do I count the number of elements (members) in a set? The answer, he concluded, was to associate each element in turn with the so called
Mathematics Emperor's New Clothes.
Not until the late twentieth century did Cohen prove the relationship between Cantor s infinities and the Axiom of Choice. So what s the big deal?
http://www.netautopsy.org/jharempr.htm

Extractions: One of the luxuries of being an amateur mathematician, who does not earn his living at mathematics, is that one may ask questions that might get a salaried mathematician fired from his job. That is, the questions are so outrageous or apparently simple-minded, that one is fired either for blasphemy or for gross incompetence. So what does all this have to do with mathematics? Somewhere late in my graduate school training in biomathematics, it dawned on me that there are about a dozen central ideas in mathematics, all of them basically fairly simple once understood, from which one may derive all the important theories of mathematics. The amazing thing is that such simple things took such a long time to internalize in our culture. For example, why was the Greco-Latin culture so resistant to the idea of ZERO, discovered one thousand years B.C. by the Babylonians (as a place-holder on the abacus)? The idea was banished from ancient Greece, and not really embraced in Europe until the sixteenth century, by merchants not mathematicians, who could do their accounting far more easily with Arabic numerals (with zero) than with Roman numerals (without zero). A few more examples: Pythagoras's proof [Singh]; infinity; infinitesimals (Calculus; Newton/Leibniz; Weierstrass; Robinson's calculus); open/closed sets (Heine-Borel theorem); computational complexity (NP complete problem; why the Sieve of Eratosthenes takes so long to solve); symbolic logic [Boole; Lewis and Langford]; Goedel's proof [Casti and DePauli]; Hilbert's Tenth Problem [Davis]; Fermat's Last Theorem [Singh]; Riemann hypothesis [Davis]; fractals [Lauwerier], etc.
[math-ph/9909033] Infinities In Physics And Transfinite Numbers In Mathematics
Upon examining these examples in the context of infinities from Cantor s theory of transfinite numbers, the only known mathematical theory of infinities, we
http://arxiv.org/abs/math-ph/9909033

Extractions: Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known mathematical theory of infinities, we conclude that apparent inconsistencies in physics are a result of unfamiliar and unusual rules obeyed by mathematical infinities. We show that a re-examination of some familiar limiting results in physics leads to surprising and unfamiliar conclusions. It is not the purpose of this work to resolve the problem of infinities but the intent of this analysis is to point out that the study of real infinities in mathematics may be the first step towards delineating and understanding the problem of infinities in physics. References and citations for this submission:
[math/0305310] Infinite Sets Are Non-denumerable
Finally it is shown that Cantor s second diagonalization method is inapplicable at all remains no evidence for the existence of different infinities denoted by
http://arxiv.org/abs/math.GM/0305310

Extractions: Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no evidence for the existence of transfinite cardinal numbers. References and citations for this submission:
The Power Set
This multiplicity of infinities deserves a name. Cantor gave it the name aleph2. Each time you layer in a new bag of infinities you get a new cardinal number
http://descmath.com/diag/power.html

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

Extractions: The standard introduction to transfinite theory is fraught with paradox and muddled dialectics. In this article, I suggest a different direction for approaching the interesting issue of sets, continuity, large sets and infinity. I've titled the approach "rich theory". The main gist of rich theory is that the behavior of large sets is a fertile and worthwhile area of investigation. There are many different ways to approach the issue. Rich theory is in keeping with the goals of Descriptive Mathematics in that the primary goal of the theory is to introduce students to the language used to describe large sets. For example, rather than trying to explain the nature of continuity, the teachers of rich theory simply try to introduce the concept of continuity, and the problems we have in trying to master this illusive topic. The primary goal of this approach is to build the student's vocabulary and basic understanding of the subject. The goal is not to establish the teacher as a guru, or to serve as a foundation of a theory of everything. There is nothing startling or new about rich theory. I chose the name simply to indicate that mathematics is a rich subject, and that there are many different ways to approach complex subjects. Too much of mathematical and philosophical writings are dedicated to trying to find fault in the works of others. The truth of the matter is that there is not a simple single way of thinking.
MIND Exchange
) It would seem to me that you would have to have a meta-mathematics , a framework larger than Cantor s set of infinities in order to prove what DS is saying
http://www.kurzweilai.net/mindx/show_thread.php?rootID=22349
When You Get To The End, Keep Going Csmonitor.com
While trying to parse Cantor s diagonal proof establishing that some infinities are larger than others, you ll also be attempting to parse Wallace s
http://www.csmonitor.com/2003/1104/p14s02-bogn.htm
DC Comics Message Boards » DC Comics » Superman
Given Cantor s infinities are not equal to each other, we know some infinities are larger/stronger than others; to us, Aleph nunber is about as unattainable as
http://dcboards.warnerbros.com/web/messages.jsp?topic=49758698&board=superman
The Infinite And The Unknowable
Cantor invented a nomenclature to keep track of all the infinities that come about by taking unions and power sets, but we shall not attempt to explain it here
http://spicerack.sr.unh.edu/~dvf/Pathways/inf
Everything And More: A Compact History Of Infinity (Great Discoveries) - Book Re
Cantor s counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown
http://www.bookfinder.us/review9/0393003388.html
The Mystery Of The Aleph - By Aczel, Amir D.
But (as he also showed) some infinities are bigger than others. To distinguish them, Cantor used the Hebrew letter aleph the number of whole numbers is aleph
http://www.bookfinder.us/review6/156858105X.html

Extractions: The search for infinity, that sublime and barely comprehensible mystery, has exercised both mathematicians and theologians over many generations. Jewish mystics, in particular, labored with elaborate numerological schema to imagine the pure nothingness of infinity, while scientists such as Galileo, the great astronomer, and Georg Cantor, the inventor of modern set theory (as well as a gifted Shakespearean scholar), brought their training to bear on the unimaginable infinitude of numbers and of space, seeking the key to the universe. In this sometimes technical but always accessible narrative, Amir Aczel, author of the spirited study Fermat's Last Theorem, contemplates such matters as the Greek philosopher Zeno's several paradoxes; the curious careers of defrocked priests, (literal) mad scientists, and sober scholars whose work helped untangle some of those paradoxes; and the conundrums that modern mathematics has substituted for the puzzles of yore. To negotiate some of those enigmas requires a belief not unlike faith, Aczel hints, noting, "We may find it hard to believe that an elegant and seemingly very simple system of numbers and operations such as addition and multiplicationelements so intuitive that children learn them in schoolshould be fraught with holes and logical hurdles." Hard to believe, indeed. Aczel's book makes for a fine and fun exercise in brain-stretching, while providing a learned survey of the regions where science and religion meet. Gregory McNamee
Salon.com Books | What's Bigger Than A Kazillion?
Using reasoning that I ve always thought of as his Squiggly Argument, Cantor proved that But don t jump to the conclusion that all infinities are the same size
http://www.salon.com/books/review/2003/11/12/infinity/

Extractions: By Polly Shulman The greatest thrill I remember from my girlhood better than my first kiss, first airplane flight, first taste of mango, first circuit around the ice rink without clinging to a grown-up's sleeve was the heart-lifting moment when I first understood Georg Cantor's Diagonal Proof of the nondenumerability of the real numbers. This proof, the Mona Lisa of set theory (to my mind, the most satisfying branch of mathematics), changed the way mathematicians thought about infinity. Yet Cantor's diagonal argument, in its essence, is so beautifully simple that even someone who hasn't yet entirely mastered trigonometry can understand it. I know, because I did, and so can you, whoever you are. I've often wanted to share the thrill with my intelligent but mathematically innocent friends and family English teachers, textile designers, photo editors, Internet journalists, soccer moms, wedding guests and I've succeeded, too, whenever I can get them to stay put. Being stuck in an elevator together helps. But elevators don't stall that often, so I was delighted to learn that the gloriously articulate novelist and essayist
Date Sun, 27 Feb 2000 141136
But the arguments of Cantor and others showed, to my satisfaction, anyhow, that none of the traditional objections to actual infinities holds water.
http://personal.bgsu.edu/~roberth/triad.html

Extractions: Date: Sun, 27 Feb 2000 14:11:36 Subject: An Inconsistent Triad There are three propositions that many Objectivists believe and that Rand herself almost certainly believed that form an inconsistent triad. That is, they form a set that has the property that any two are consistent, but the addition of the third makes the set contradictory. The first rarely gets stated, because it is taken for granted by almost everybody. It may well be that it is so thoroughly taken for granted, that it is not even recognized as a belief to which there might be some alternative. Objectivists take it for granted, but are hardly ever pushed to defend it, because the people they engage in discussion also take it for granted. It is the assumption that time is non-cyclical. There is a definite future and past, so it is not the case that, say, a future event is the very same event as some past event. In fact, if time were cyclical, every event would be in both the future and past of every other (and even in its own future and past). There could be, say, three events, A, B and C, with A preceding B, B preceding C and C preceding A. (This is not to say that the cycle would

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