The mathematical notion of infinity can be conceptualized in different ways. First, like counting by hundreds for the rest of our lives, an infinite quantity. It can also be seen as digging a whole into hell for eternity, negative infinity. The concept I am going to explore, however, concerns infinitesimally small quantities, through radioactive decay. Infinity is by definition an indefinitely large quantity. It is difficult to grasp the magnitude of such an idea. When we examine infinity in more detail by establishing one-to-one correspondences between sets, we notice some peculiarities. There are as many natural numbers as there are even numbers. We also see that there are as many natural numbers as 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. By transfinite arithmetic, we can see this as an 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 set of evens, in this case 0, it appears that the bottom set is larger, but when we move the bottom set on top, our original statement is true again.1 2 3 4 5 6 7 8 9 …0 2 4 6 8 10 12 14 16 …We have again reached a one-to-one correspondence with the top row, this proves that the cardinality of both is the same, i.e. o. This correspondence leads to the conclusion that o+1=o. When we add two infinite sets, we also get the sum of infinity; o+o=o. That being said, we can try to find larger sets of infinity. Cantor was able to show that some infinite sets actually have a cardinality greater than o, given 1. We need to compare the irrational numbers to the real numbers to get this result. to find another irrational number that has not been listed. Simply choose a digit different from the first digit of our first number. All it takes is that our second digit is different from the second digit of the second number, this can continue indefinitely. Our new number will always differ by one digit from the one already present on the list..