Ordinal and Cardinal Numbers
“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” - Jon von Neumann
When I begun to write this post I knew next to nothing about the ordinal or cardinal numbers. They seemed to keep popping up on me, and I never quite understood the meaning of \(\omega\) and \(\aleph\). This post is intended to fix that!
Informally
Ordinal numbers tell us an item's position in a sequence, for example, first, second, third, fourth, ... Each natural number is an ordinal number.
Cardinal numbers tells us the size of a sequence, it is a measure of magnitude, for example, the list [11, 43, 8] is 3 elements long. Each natural number is also a cardinal number. They are different creatures though!
Formally
Ordinals
Now things start to get a little messy. There seem to be many definitions of ordinal numbers.
It seems to be the case that the now standard definition of the ordinals is due to Von Neumann (at the age of 19!!). It is as follows:
Ordinal Number
A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and every element of S is also a subset of S.
Remembering the construction of the natural numbers according to Peano's axioms, we have that each of the rows below satisfies the definition to be an ordinal number.
Colloquially "each ordinal is the well-ordered set of all smaller ordinals."
Cardinal numbers
There's not as much confusion here, however there is a little controversy depending on whether or not you're willing to accept the Axiom of Choice. If the axiom of choice is not assumed, then a different approach is needed.
Cardinal Number
Assuming the axiom of choice, the cardinality of a set \(X\) is the least ordinal number \(\alpha\) such that there is a bijection between \(X\) and \(\alpha\).
The above definition is again due to von Neumann, and is known as the von Neumann cardinal assignment.
Each ordinal is associated with one cardinal, called its cardinality
Some Famous Known Cardinal and Ordinal Numbers
After all the natural numbers comes the first infinite ordinal, \(\omega\). This corresponds to the cardinal \(\aleph_0\), the "size" of the natural numbers
Next comes \(\omega + 1, \omega + 2, \omega + 3, \cdots , 2\omega = \omega + \omega, \cdots \)
After all of these comes the first uncountable ordinal, \(\omega_1\). It's associated cardinal is \(\aleph_1\), the "size" of the real numbers.
The continuum hypothesis asks whether there is a cardinal between \(\aleph_0\) and \(\aleph_1\). It turns out this statement is independent of the typical formulation of mathematics (the Zermelo-Fraenkel Axioms of set theory).
This means that either the answer to the continuum hypothesis can be added as an axiom without ruining the consistency of Zermelo-Fraenkel.
The ordinal \(\epsilon_0\) is the limit of the sequence \(0, 1, \omega, \omega^\omega, \omega^{\omega^\omega}, \cdots\).
Conclusion
I have barely scratched the surface here. There seems to have been a lot of work done in this area of mathematics in the past, but not so much anymore.
I'm not really sure what is happening here today, I remember reading in John H. Conway's biography that it was hypothesised that major results would follow when the theory of ordinals and infinitesimals where combined (as was done by Conway's Surreal Number system). To my VERY limited knowledge, this hasn't happened.