Infinity is a number which really doesn't exist. There are an infinite number of numbers: The number of odd numbers is infinite: The number of even numbers is infinite: This means that the number of even numbers, or the number of odd numbers that exist, is the same as the number of all numbers! In other words:
Isn't that en extraordinary statement? Take the set of Natural numbers, and throw half of them away. You still have as many left as when you started!
We can show in a similar fashion that the number of numbers in the set of Natural numbers: is the same as the number of numbers in the set of Whole numbers: even though this second set clearly seems to have one more number in it. It does, but the size of both sets is still infinite. Both sets have the same number of elements! This implies that 'infinity + 1' equals 'infinity'. In fact, you can add as many numbers to an infinitely long set (or take as many out as you want) and the new set you get is still infinite in length. Infinity plus or minus anything still equals infinity. Moreover, if we take the set of Integers: Here's the proof:
We're not done yet! Consider the set of rational numbers: There are the same number of numbers here as in the simple set 0, 1, 2, 3, 4, ... Both sets have an infinite number of elements. Clearly, 'infinity + infinity + infinity + ....' still equals infinity. Now things start to get weird. A famous mathematician named Georg Cantor proposed and demonstrated that this infinity is actually the smallest infinity of three different infinities! He called the infinity that we've been looking at 'aleph zero', using the symbol aleph from the Hebrew alphabet: represents the number of numbers, of any kind. It's the infinity you're familiar with. But there are bigger infinities! A point, in mathematics, is considered to be an infinitely small dot; a position without length or width.
is greater than Now consider all the possible curved shapes you can draw. ALL the possible shapes: is greater than which is greater than Neither Cantor nor anyone since has been able to conceive of anything which would require a fourth infinity ... it seems that three infinities may be are all there are! |