Historical context

Preliminaries

Standard representation of ordinal numbers

The definition of ordinal numbers offers little insight into their nature. In situations like this pure mathematicians create representations of the objects they wish to study. Such representations are built from familiar mathematical constructions and are equivalent to the obstruce objects. By manipulating the familiar objects in the representation, the pure mathematician may thus investigate the structure of the mysterious abstract entities.

The most common representation of the ordinal numbers, due to Von Neumann, is as follows. The ordinal 0 is defined to be the empty set ${\displaystyle \emptyset }$; the ordinal 1 is defined to be the set ${\displaystyle \{0\}}$, which is of course equal to ${\displaystyle \{\emptyset \}}$. Similarly, the ordinal 2 is the set ${\displaystyle \{0,1\}}$; the ordinal 3 is the set ${\displaystyle \{0,1,2\}}$; the ordinal 4 is the set ${\displaystyle \{0,1,2,3\}}$. Any finite ordinal ${\displaystyle n}$ is defined to be the set ${\displaystyle \{0,1,2,\ldots ,n-1\}}$ (or, in a rigorous notation, the successor of an ordinal α is defined as the set ${\displaystyle s(\alpha )=\alpha \cup \{\alpha \}\,}$).

In fact this definition extends naturally to transfinite ordinals. The ordinal ${\displaystyle \omega }$ is the set consisting of every finite ordinal ${\displaystyle \{0,1,2,3,\ldots \}}$. Again, ${\displaystyle \omega +1}$ is the set ${\displaystyle \{0,1,2,3,\ldots ,\omega \}}$; ${\displaystyle \omega +2}$ is the set ${\displaystyle \{0,1,2,3,\ldots ,\omega ,\omega +1\}}$; and so on.

The ordinal ${\displaystyle \omega +\omega }$ (or ${\displaystyle \omega *2}$) is the set consisting of all finite ordinals and ordinals of the form ${\displaystyle \omega +n}$, where ${\displaystyle n}$ is a finite ordinal. Thus ${\displaystyle \omega +\omega =\{0,1,2,\ldots ;\omega ,\omega +1,\omega +2,\ldots \}}$.

The ordinal ${\displaystyle \omega _{1}}$ is the first uncountable ordinal, and is the set of all countable ordinals.

Ordinal arithmetic