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 there are, 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: 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:which continues on to infinity in both directions, it has the same number of numbers in it as all the previous sets! Here's the proof:
We're not done yet! Consider the set of rational numbers: between each and every pair of Integers. An infinity of infinities, if you like.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 , using the symbol 'aleph zero'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. Consider all the possible points on a line. Or all the possible points on a flat surface. Or all the possible points inside a region of space:Cantor proposed and proved that this infinity is bigger than the previous one. In other words, the number of points on a line is greater than the number of numbers.He called this infinity 'aleph one': is greater than
Now consider all the possible curved shapes you can draw. ALL the possible shapes: more of them than there are points on a shape, or numbers. He called this infinity 'aleph two':
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,
aleph three, to count. It seems that three infinities are all there are! |