User blog:Edwin Shade/Understanding The Infinite

Among the many concepts of Googology, infinity plays a key role. This blog post seeks to teach anyone who finds themselves in the situation I was in when I was first introduced to Googology, when I had many questions, (and still do !), but had, (and still have !), difficulty in understanding the notations used because they were not broken down step by step, in the simplest way possible.

Introduction to Cardinals and Ordinals
Before we get into the nature of infinity, it would be good to distinguish between two types of numbers known as cardinals, and ordinals. These will be important.

Cardinals are numbers which denote size or quantity, and are the type of numbers we are most familiar working with. For example, if you have 5 apple pies, the number 5 is a cardinal, since it refers to the amount of apple pies you have.

Ordinals are numbers which denote order, or where in a list a given object is. This is why centuries are always referred to by a number that is one higher than you might expect, (e.g. calling 2000 the "21st century", and the 1600s the "17th century"). It is because you are counting in what century you are in, rather than by how many centuries have elapsed.

A question: Suppose you are in a race and you pass the 2nd runner, which place are you ?

If you said 1st, you would actually be wrong ! Surprisingly, the answer is 2nd, because in order to pass the 2nd runner you would have to have been the 3rd runner in the race, and consequently, passing the 2nd would only make you the 2nd runner. The tendency when you read that question was probably that since you were getting one closer to the front of the race, you could subtract 1 from 2 and get 1st place. This is because you tried to solve a problem using ordinals, (position), with cardinals, (quantity). You can now see why is it vital that you make a clear distinction between cardinals and ordinals.

Cardinal and Ordinal Infinities
Just as there are two types of numbers, there are two main types of infinity, cardinal infinites, and ordinal infinities. These, like their finite counterparts, are distinguished by size and order.

Cardinal infinities are infinities which denotes the size of a set, or a collection of objects. As a brief primer on sets, all you need to know for now is that a set is any collection of objects enclosed in curly brackets, separated by comas. $$\{1, 2, 3, 4, ...\}$$ is a set for example, and one that is familiar as the set of counting numbers. The size of a set has a special name, called cardinality, which refers to how many objects are in that set. The cardinality of the counting numbers, or $$\{1, 2, 3, 4, ...\}$$ is given a special name, aleph-null, and is represented by the Hebrew letter Aleph with the subscript 0, or $$\aleph_0$$. It was proven to be the smallest cardinal infinity by George Cantor, and consequently it is the first of the aleph-numbers, or cardinal infinities.

[TO BE CONTINUED]