Infinity is by definition an indefinitely large quantity. It is hard to grasp the magnitude of such an idea. When we examine infinity further by setting up one-to-one correspondence’s between sets we see a few peculiarities. There are as many natural numbers as even numbers. We also see there are as many natural numbers as multiples of two. This poses the problem of designating the cardinality of the natural numbers. The standard symbol for the cardinality of the natural numbers is o. The set of even natural numbers has the same number of members as the set of natural numbers. The both have the same cardinality o. By transfinite arithmetic we can see this exemplified.

1 2 3 4 5 6 7 8 …

0 2 4 6 8 10 12 14 16 …

When we add one number to the set of eve

1 2 3 4 5 6 7 8 9 …

: :

We again have achieved a one-to-one correspondence with the top row, this proves 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.

Infinity has many theories and applications. It is a concept that defies regular arithmetic and has puzzled mathematicians and theologians alike. We are left only to try to grasp the sheer magnitude of such a concept.

Georg Cantor did not come up with the concept of infinity, but he was the first to give it more than a cursory glance. Many mathematicians viewed infinity as unbounded growth rather than an attained quantity like Cantor. The traditional view of infinity was something “increasing above all bounds, but always remaining finite.” Galileo (1564-1642) noticed the peculiarity that any part of a set could contain as many elements as the whole set. Berhard Bolzano (1781-1848) made great advancements in the theory of sets. Bolzano expanded on Galileo’s findings and provided more examples of this theme. One of the most respected mathematicians of all time is Karl Friedrich Gauss. Gauss gave this insight on infinity:

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 need only to choose a digit different than the first digit of our first number. Our second digit needs only to be different than the second digit of the second number, this can continue infinitely. Our new number will always differ than one already on the list by one digit. 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. We now have a set, the irrationals, with a greater cardinality, hence its designation as 1.

Some topics in this essay:
, Gauss Gauss, Georg Cantor, Berhard Bolzano, Mathematics Introduction, digit digit, cardinality o, Publishing Company, Prentice Hall, Mathematics Citizens, Brown Publisher, 5 6 7, … 0 2, 6 7 8, one-to-one correspondence, 4 6, 6 8, 8 10, 12 14 16, cardinality natural, 10 12 14, 12 14, bottom set, 1 2,