The mathematical notion of infinity can be conceptualized in many different ways. First, counting in hundreds for the rest of our lives, an infinite amount. It can also be thought of as digging out an entire hell for eternity, negative infinity. The concept I will explore, however, is that of infinitely smaller quantities, through radioactive decay. Infinity is by definition an indefinitely large quantity. It is difficult to understand the significance of such an idea. When we further examine the infinite by establishing a one-to-one correspondence between the sets we see some peculiarities. There are as many natural numbers as there are even numbers. We also see that there are as many natural numbers as there are multiples of two. This raises the problem of designating the cardinality of natural numbers. The standard symbol for the cardinality of natural numbers is o. The set of even natural numbers has the same number of members as the set of natural numbers. Both have the same cardinality o. With transfinite arithmetic we can see this example.1 2 3 4 5 6 7 8 …0 2 4 6 8 10 12 14 16 …When we add a number to the even set, in this case 0 seems like the lower set is larger, but when we move the bottom set above our initial statement is true again.1 2 3 4 5 6 7 8 9 …0 2 4 6 8 10 12 14 16 …We have once again achieved a one-to-one correspondence with the top row , this shows that the cardinality of both is the same being o. This correspondence leads to the conclusion that o+1=o. When we add two infinite sets together, we also get the sum of infinity; o+o=o. Having said this we can try to find larger infinite sets. Cantor managed to prove that some infinite sets have cardinality greater than o, given 1. We must compare the irrational numbers with the real numbers to achieve this result.1 0.1426784352 0.2937587783 0.3839028924 0.563856365: :No matter what matching system we devise, we will always be able to find another irrational number that has not been listed. We just need to choose a digit other than the first digit of our first number. Our second digit just needs to be different from the second digit of the second number, this can continue forever. Our new number will always differ by one digit from the one already in the list.