Georg Cantor did not come up with the concept of infinity, but he was the first to give it more than a cursory glance. We now have a set, the irrationals, with a greater cardinality, hence its designation as ГЂ1. This being true we cannot put the natural and irrational numbers in a one-to-one correspondence like we could with the naturals and evens. Our new number will always differ than one already on the list by one digit. Our second digit needs only to be different than the second digit of the second number, this can continue infinitely. We need only to choose a digit different than the first digit of our first number.

No mater which matching system we devise we will always be able to come up with another irrational number that has not been listed. We must compare the irrational numbers to the real numbers to achieve this result. Cantor was able to show that some infinite sets do have cardinality greater than ГЂo, given ГЂ1. This being said we can try to find larger sets of infinity. When we add two infinite sets together, we also get the sum of infinity ГЂo+ГЂo=ГЂo. This correspondence leads to the conclusion that ГЂo+1=ГЂo. We again have achieved a one-to-one correspondence with the top row, this proves that the cardinality of both is the same being ГЂo. When we add one number to the set of evens, in this case 0 it appears that the bottom set is larger, but when we shift the bottom set over our initial statement is true again. By transfinite arithmetic we can see this exemplified. The set of even natural numbers has the same number of members as the set of natural numbers. The standard symbol for the cardinality of the natural numbers is ГЂo. This poses the problem of designating the cardinality of the natural numbers. We also see there are as many natural numbers as multiples of two. There are as many natural numbers as even numbers.

When we examine infinity further by setting up one-to-one correspondence?s between sets we see a few peculiarities. It is hard to grasp the magnitude of such an idea. Infinity is by definition an indefinitely large quantity. The concept I will explore, however, is infinitely smaller quantities, through radioactive decay It can also be thought of as digging a whole in hell for eternity, negative infinity. First, as counting by hundreds for the rest of our lives, an endless quantity. The mathematical notion of infinity can be conceptualized in many different ways.

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