Infinity Cantor was able to demonstrate that there are different sizes of infinity. Can YOU demonstrate that the infinities of points on a big circle http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Math ForumInfinite Sets Prime Numbers Finding Prime Numbers Famous Paradoxes Zeno's Paradox cantor's infinities cantor's infinities, Page 2 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Math Forum Cantor's Solution Denumerability Prime Numbers Finding Prime Numbers Famous Paradoxes Zeno's Paradox cantor's infinities cantor's infinities, Page 2 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Infinite Ink The Continuum Hypothesis By Nancy McGough and higher levels are called "uncountable infinities." The natural After Cantor showed that there are different levels of infinity, Cantor http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Sci.math Message The Math Forum sci.math Search. All Discussions sci.math Topic Message next Message Cantor's "infinities" cannot exist http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Sci.math Message sci.math Search. All Discussions sci.math Topic Message previous next Message Re Cantor's "infinities" cannot exist http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Cantor's Infinities cantor's infinities 32 in X 24 in.(any, same ratio) Digital painting. Home Page Next Page http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Ecclectica - Aleph However, Cantor reasoned that Kronecker would only read the title and abstract, looking for some mention of Cantor's objectionable infinities. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Math Forum:Infinite Sets Cantor s infinities · Cantor s infinities, Page 2 The Problem of Points · Pascal s Generalization · Summary and Problems · Solution, Problem 1 http://mathforum.org/isaac/problems/cantor1.html
Extractions: Prime Numbers ... Links Are there more integers or more even integers? Seems like a simple question, right? After all, every even integer is an integer but what about all the even integers? So there are more integers than there are even integers, right? But wait a second. How many even integers are there? An infinite number. And how many integers are there? An infinite number. Hmmmm.... "Infinity," says math student A, "is just a term... there's no way you can actually show me that there is the same number of each." "Okay, lets play..." says math student B. "Give me an integer, and I'll give you an even integer that corresponds to it. And if two of your integers are different, I guarantee that my two even integers will be different." Math Student A: Okay... 1 Math Student B: 2 A: 2 B: 4 A: 18 B: 36 A: -100 B: -200 A: n B: 2n A: I'm beginning to see what you mean. But let's consider some of the set theory we learned in math class. The set of even integers is contained in the set of integers, but is not equal to that set. So the two sets can't be the same size. (Who's right? What kind of sets did the teacher put on the board in class? How do these sets differ from those?)
Math Forum: Cantor's Solution: Denumerability Cantor s Solution Denumerability. A Math Forum Project Zeno s Paradox · Cantor s infinities · Cantor s infinities, Page 2 The Problem of Points http://mathforum.org/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.
Mathematics Revenge: How Numbers Don't Behave As They Should! Like with integers, normal mathematical operations could be applied to Cantor s infinities. Thus, w + 2 w + 1 because w + 2 = w + 1 + 1, where w + 1 is as 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
Cantor's Diagonal Proof Simplicio I m trying to understand the significance of Cantor s diagonal proof. proof starts out with the assumption that there are actual infinities, 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.
Science News Cantor s hierarchy of infinities was such a revolutionary concept that many of his contemporaries rejected it out of hand. Their derision, coupled with http://www.phschool.com/science/science_news/articles/infinite_wisdom.html
Extractions: I. Peterson 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. If that's the case, how much bigger is that infinity? This innocent-sounding question has stumped mathematicians from Cantor's time to the present. More than that, the question has exposed a gaping hole in the foundations of mathematics and has led mathematicians to reexamine the very nature of mathematical truth. Now, Hugh Woodin, a mathematician at the University of California, Berkeley, may finally have found a way to resolve the issue, long considered one of the most fundamental in mathematics.
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, http://www.ceptualinstitute.com/uiu_plus/necsij1send.htm
Extractions: 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.
Cantor's Diagonal Proof Here, we are attempting to build an infinity of infinitiesa new infinity of But now let us try one more trick for cramming infinities into infinities. http://home.ican.net/~arandall/abelard/math12/Cantor.html
Extractions: Cantors 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
Infinity - Wikipedia, The Free Encyclopedia In quantum field theory infinities arise which need to be interpreted in such a way as A concise introduction to Cantor s mathematics of infinite sets. http://en.wikipedia.org/wiki/Infinity
Extractions: Infinity is a term with very distinct, separate meanings which arise in theology philosophy mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from Latin is not ended In theology , for example in the work of theologians such as Duns Scotus , the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity (leading to the question, an unlimited quantity of what?). In philosophy , infinity can be attributed to space and time, as for instance in Kant 's first antinomy . In both theology and philosophy, infinity is explored in articles such as the Ultimate the Absolute , God, and Zeno's paradoxes In mathematics, infinity is relevant to or the subject matter of articles such as limit (mathematics) aleph number class (set theory) Dedekind infinite ... extended real number and absolute infinite In popular culture , we have Buzz Lightyear 's rallying cry, "To infinity â and beyond!", which may also be viewed as the rallying cry of
TalkCantor S Diagonal Argument - Wikipedia, The Free Encyclopedia If we accept the existence of uncountable infinities (and i guess you do if you Cantor s diagonal argument (was devised) to demonstrate that the real http://en.wikipedia.org/wiki/Talk:Cantor's_diagonal_argument
Ecclectica - Aleph Cantor s work with mathematical set theory, which would eventually lead to a revolution 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
