On 5/27/2011 2:52 PM, 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 =) > There are only more numbers if you are looking at a discrete non zero interval between each number in a set. Unfortunately this doesn't exists if you are considering an infinite number of values in a set. Let me show you what i mean. To determine the number of values in a set with a discrete non zero value we take the set and divide it by the interval. Hence if my discrete interval is .5 then 1/.5=2 and 2/.5 = 4. So far it looks like you are right the set of 0-1 has 2 values and the of 0-2 has 4 at a .5 interval, but as the interval approaches zero this changes. As the size of the interval approaches zero and our number of values in each set approaches infinity the size of each interval will become 0. if we take the size of the sets and divide both by the interval of zero, which is the value of the interval at infinity when either of them are equal to infinity, we get 1/0 and 2/0. which is best understood as the limit(1/n) as n->0+ which = limit(2/n) as n->0+ which = infinity. This can also be written as limit(1/|n|) as n->0 which = limit(2/|n|) as n->0 which = infinity for those of you who aren't that familiar with limit notation. I'm sorry but there is an equal number of values in the sets of 0-1 and 0-2. > 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. > */ >
/* PLUG: http://plug.org, #utah on irc.freenode.net Unsubscribe: http://plug.org/mailman/options/plug Don't fear the penguin. */
