Ordinal: Difference between revisions

* \(\alpha^{\beta+1} = \alpha^\beta \cdot \alpha\)

* If \(\beta\) is a limit ordinal, \(\alpha^\beta = \bigcup\{\alpha^\gamma: \gamma < \beta\}\)

 

There are helpful visual representations for these, namely with [[Matchstick diagram|matchstick diagrams]]. For example, \(\alpha + \beta\) can be visualized as (a diagram for) \(\alpha\), followed by a copy of (a diagram for) \(\beta\). Note that our definition gives \(1 + \omega = \bigcup\{1+n: n < \omega\} = \omega\), and this makes sense, since a single line, followed by infinitely many lines, is no more than just infinitely many lines, and they therefore have not only the same [[Cardinal|cardinality]] but the same order type. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formalized by the following statement, which is provable over [[ZFC]]: "if \(X\) and \(Y\) are well-ordered sets with order types \(\alpha\), \(\beta\), respectively, then \(X\), concatenated with a copy of \(Y\), has order type \(\alpha + \beta\)".