Ordinal: Difference between revisions

There are helpful visual representations for these, namely with sets of lines. For some basic intuition, [https://www.youtube.com/watch?v=SrU9YDoXE88 see this video]. For example, \(\alpha + \beta\) can be visualized as \(\alpha\), followed by a copy of \(\beta\). 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. Meanwhile, \(\omega + 1 = \omega \cup \{\omega\}\): you have infinitely many lines, followed by a single one after all of them. This intuition is formally provedformalized 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 \(\alpha + \beta\)".