perhaps the problem is none of you have an accurate concept of what infinity means example : let N be the collection of all positive whole numbers clearly N contains infinitely many elements (since there are infinitely many positive whole numbers) it can be shown that the number of elements in (cardinality of) N is the same as the cardinality of all whole numbers, is the same as the cardinality of all fractions,... etc all these collections contain an infinite of elements now consider the collection of numbers between 0 and 1, and they can be any numbers, for example 1/Pi is in the collection, because 1/Pi is between 0 and 1. this collection has strictly more elements than the collections discussed earlier (even though both are infinite). so we see there are different types of infinity. now observe a result proved by euler in a less decadent age : 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + .... = Pi^2/6 Said in plain english "The sum of all squared reciprocals is equal to Pi^2/6" by the ... i mean etc, added on forever and ever. it turns out that although this series has infinitely many members, it has finite value, namely Pi^2/6 http://en.wikipedia.org/wiki/Infinity http://en.wikipedia.org/wiki/Countable [these were the collections or "sets" i discussed earlier, like the set of all whole numbers, we call "countably infinite"] it can be shown that the interval i considered earlier, the collection of all numbers between 0 and 1, is uncountable, and we say "uncountably infinite"