Infinity (∞)

Infinity is a quantity that is bigger than all "finite" numbers. What does "finite" mean? We can define it recursively like this.

  1. \(0\) is finite
  2. If \(n\) is finite, then \(n+1\) is finite

Together, we can define an infinite quantity as one which cannot be reached by application of 1. and 2. above. We will need a new abstraction to really make this precise, and that is a set. The Axiom of Infinity states that there exists a set \(S\) that contains \(0\) and all the applications of 2., and this set \(S = \mathbb{N}\), the set of natural numbers.

At the foundation of the sciences you will find mathematics, and at the foundation of mathematics, you will find faith in a set of axioms.

Why some axioms and not others? This is the correct question. Once you commit to a set of axioms, you must accept all the conclusions that follow from them. For example, accepting the Axiom of Infinity means that you can prove that there are multiple sizes of infinity. This is the work of a brilliant man named Cantor.

Types of Transfinite Numbers

Cardinal numbers (Sizes of Infinity)

Countable \(\aleph_0\)

Continuum \(c = |\mathbb{R}|\)

\(2^c\)

\(2^{2^c}\)

…

Ordinal numbers (Positions of Infinity)

\[1, 2, 3, 4, ..., \omega, \omega + 1, ..., 2\omega, ... \]