On Fri, May 27, 2011 at 12:21 PM, Jason Van Patten <[email protected]> wrote: > i think what Dan is trying to say is that there is not more numbers > between 0-1 and 0-2. Infinity = infinity regardless of the rate at which > it is reached but if you were to take the sum of all numbers from 1 to > 1/(infinity^2) you would end up with a discrete value. In fact any power > of the denominator 2 greater than the numerator always will. For example > the sum of the limit 1/(n^2) as n-> infinity = (pie^2)/6 however the sum > of the limit of 1/n as n-> infinity = infinity albeit extremely slowly. > in both cases we have an infinite number of values being added together, > but the final results are radically different.
No. I meant to say exactly what I said - that there are in fact "more" numbers between 0-2 than there are between 0-1. That is, the cardinality of the set of all numbers between zero and one is of a lower order than the cardinality of the set of all numbers between zero and two. However, since neither of these sets are countable, the concept or "more" or "less" (which applies just fine to finite sets - iirc, all finite sets are countable) breaks down when talking about infinite sets, especially ones that are uncountable. I *believe* that I am using the phrase "the cardinality of (the set) is greater" correctly in the above paragraph, but it's been a few years; I trust that Levi will correct me if I'm using the words incorrectly, or if I'm just plain wrong =) On Fri, May 27, 2011 at 12:49 PM, Lonnie Olson <[email protected]> wrote: > In your example, your use of the term infinite is in relation to the > count of numbers, not the sums or values. > This count is in fact infinite, no upper bound. There are no more > numbers between 0 and 1 than there are between 0 and 2. This upper > bound is limitless. You cannot multiply infinity by two, since > infinity is already fully encompassing that result. There is no value > greater than infinity. There *is* no count of the set of all numbers between X and Y, because that set is always uncountably infinite. You can't multiply infinity by two, because infinity is not a number or a value, it's a concept. However, not all infinities are equal, so you can't correctly say that "infinity is already fully encompassing that result." To use our earlier example, the set of numbers between 0-1 is infinite but it definitely does not contain the number 1.1, 15, or 42. However, there are other sets (also infinite) that do contain these numbers. Further reading: http://en.wikipedia.org/wiki/Cardinality - Cardinality of sets, covers both finite and infinite http://en.wikipedia.org/wiki/Bijection http://en.wikipedia.org/wiki/Injective_function http://en.wikipedia.org/wiki/Surjection Bijection, injection and surjection, which are types of functions which can be used to describe or think about sets, including infinite sets. http://plato.stanford.edu/entries/set-theory/ - A good site on set theory. -Dan /* PLUG: http://plug.org, #utah on irc.freenode.net Unsubscribe: http://plug.org/mailman/options/plug Don't fear the penguin. */
