On 27 May 2011, at 14:52, Daniel C. wrote: > 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. >
My definition of infinity: more numbers than I have the ability or desire to count to. Using this definition, my 2 year old would probably consider 20 to reside in the definition of infinity. /* PLUG: http://plug.org, #utah on irc.freenode.net Unsubscribe: http://plug.org/mailman/options/plug Don't fear the penguin. */
