Good sources for a project on the Continuum Hypothesis (specifically UNCOUNTABLE and COUNTABLE infinities?
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There's an early chapter on the continuum hypothesis in Leonard Wapner's "The Pea and the Sun," which is about the Banach-Tarski Theorem.  The wonderful thing about that book is that by the end I actually understood the theorem.\n\nAlso, when I was quite young I read Lilian Lieber's book "Infinity."  She was a mathematician and took it far beyond a high school student's level, yet it was a fascinating read.  Understood why Cantor went nuts.
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The earliest known documented knowledge of infinity was presented in ancient India in the Yajur Veda (c. 1200–900 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".\n\nThe Indian Jaina mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. Each of these was further subdivided into three orders:\n\n    * Enumerable: lowest, intermediate and highest.\n    * Innumerable: nearly innumerable, truly innumerable and innumerably innumerable.\n    * Infinite: nearly infinite, truly infinite, infinitely infinite.\n\nThe Jains were the first to discard the idea that all infinites were the same or equal. They recognized five different types of infinity: infinite in one and two directions (one dimension), infinite in area (two dimensions), infinite everywhere (three dimensions), and infinite perpetually (infinite number of dimensions).\n\nAccording to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null &#92;aleph_0 (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest.\n\nIn the Jaina work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asmkhyata and ananata, between rigidly bounded and loosely bounded infinities.
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Don't know yet, but lots of wild theories and mathematical approaches being tried (some, via\nhttp://myweb2.search.yahoo.com/search?p=4%208%2015%2016%2023%2042):\n\nhttp://www.d.kth.se/~dicander/lost.html\n\nhttp://www.4815162342.com/forum/viewforum.php?f=3
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I am working on a project (a long one, no answers yet) that involves INFINITY and requires a good understanding of it.\n\nI found one infinity website so far (months) that was not very good (had international contributors, but I already zapped the url).\n\nThere are mathematical considerations of INFINITY, best found in math reference section at university or NYC public library, or equivalent.\n\nBUT, the BEST GLIMPSE, to use your word, I have ever found in many years of looking is THE URANTIA BOOK.\n\nYou can find urls for this on the right hand side of my blog at http://spaces.msn.com/up-escalator\n\nI would recommend reading the whole book, and that will take some time, in the months and years range.  You read this stuff with mind engaged, not like a Bond novel.\n\nThe project I am working on is to explain, in a manner assimilable by swift teenagers, the Seven Absolutes of Infinity.\n\nIn just o few sentences:  Infinity means there is A LOT MORE than what we are now seeing.  More than three dimensions plus time.  And more variety than we can possibly imagine.\n\nMuch of this is invisible to us, or even to our minds, unless we make the effort.  Francis Bacon was one person who really thought about things invisible.\n\nSince the subject is ENDLESS, we will only get a glimpse as you say, no matter how far we go.\n\nBut the attempt is a journey well worth the effort.\n\nDon't be in a hurry, you picked a biggie.
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Good Question, Mandelbrot's book and Barnsley's book will provide you with many examples.
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Theres a poster in my maths class that says "a trillion steps towards an infinitae pi" So i assume the best so far is a trillion number after the decimal point. WOW!
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Check out Stewart's Calculus(refer to the first link). If you are looking for something  more rigorous refer to Apostol's Calculus(refer to second link).
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I did a paper on the History of 0,1, and infinity for a grad school paper.  It worked out really well.  There is a lot of history of how different societies developed mathematics and accepted fractions, algebra.  Good luck!
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many have tried:\n\nConvergence in science includes a plethora of unification theories. In Consilience, Edward O. Wilson argues that all knowledge is unified around a small number of natural, interlocking laws. These fundamental laws, which we are just beginning to comprehend, govern disciplines as diverse as physics and the humanities. In A New Kind of Science Stephen Wolfram uses simple cellular automata (265 programs) to produce many results found in nature such as: tree branches, streams dynamics, and even the possibility of a truly fundamental theory of physics. In The Elegant Universe, Brian Greene describes superstring theory: a theory of everything.
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I don't feel like doing the review, but I can offer something.  It turns of that there are functions can are LeBesgue (sp?) integrable but no Riemann.  The f(x) = 0 x rational, =1 x irrational is an example.  You can cook up a set that lands on just the rationals so that it has a limit at a given rational, and hence every rational even though both the rationals and irrationals are infinitely dense in any small neighborhood of any point.  I don't think that you can do the same with irrationals.  If I'm correct then the rationals would be continuous and the irrationals not.  But I don't think that I'm correct.\n\nYou are essentially talking about what's known as point set topology, a part of an undergraduate, or graduate, real analysis course.\n\nI've found Wikipedia to be very helpful for math concepts.\n\nI just found something that said\n\nf(x) = 0, x irrational\nf(x) = x, x rational\n\nis continuous at one point, x = 0.
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Algorithms,\nFormal Methods,\nComputational Geometry,\nComplexity theory,\nTemporal Logic, and ...
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how do you count to infinity?
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NO!\nThe cardinality of the integers the same as that for the rational numbers for instance.  Denoted aleph (א)-null, this "infinity" is "countable", i.e., it can be put in a 1-1 correspondence (bijection) with the natural numbers.\n\nThe cardinality of the real numbers is higher.( א-one *) the number of real numbers cannot be put into a 1-1 correspondence with the natural numbers.\n\nThe number of possible curves in a plane is even higher (א)-two\nETC.......(ad infinitum - no pun intended)\n\nWhat is strange here is that the number of odd natural numbers is the same as the number of all natural numbers, even though the former is a subset of the latter!  That is because a bijection exists between the two sets.\nFor example, let every natural number x be associated with the number 2x-1.   The pairings are:\n1 - 1\n2 - 3\n3 - 5\n4 - 7\n5 - 9\netc.\nEvery natural number is paired with every odd natural number.  Thus the sets have the same cardinality.\nNo bijection exists between these countable sets and say, the real numbers however.  In fact, if you pick ANY two numbers, no matter HOW CLOSE TOGETHER, there are still more real numbers between them than all rational numbers combined. The proof of this is rather cumbersome for here, I suggest you read up on Georg Cantor or infinite sets elsewhere\n\n*It is my "choice" to accept as convention (one side or another of) the continuum hypothesis - it doesnt really matter, but as one answerer pointed out the real numbers' cardinality usually represented as 'c' (Although I never liked that designation-too easily misinterpreted as something else.  I can try to sway supporters, cant I?)
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Maybe.  But the question is hypothetical -- infinity is a mathematical concept, and does not exist in the real world.
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that's right. cause infinit is indefinit.
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Consider a discrete infinite set, like the natural numbers (also known as the counting numbers), or the set of rational numbers. These sets are called countable by mathematicians because they can be associated with the counting numbers in a manner such that each element is uniquely labelled with a single natural number. \nThe set of real numbers and the set of all rotations in the plane cannot be associated with the natural numbers in this manner, and we call such sets uncountable. The first proof of this fact was given by Cantor.\nIn normal conversation, countless simply refers to any infinite set, or even just a very large set of unknown size, ie the number of fishes in the sea or stars in the sky.
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yes, there are infinte, the universe is infinite and besides that, there are infinite spacial-temporal or just temporal realities, not just one, not ten, not a million but infinite. Just imagine ! You should read The Fabric of Reality by David Deutsch. a great scientist in physics and , he is English, proffesor at Oxford I think. If you wanna talk more contact me on messenger Y, on axelbarsan or theocritos82