An -number is an ordinal number, or more generally a surreal number, , such that , where is the first infinite ordinal. In other words, it is a fixed point of the exponential map .
The ordinal -numbers are unbounded: for every ordinal there is an -number greater than , namely the limit of the sequence . Thus, there are a proper class of ordinal -numbers, and their order type? is the same as that of the ordinals themselves. Hence, we can label them by writing for the -number, where is any ordinal.
In particular, is the first ordinal -number, namely the limit of the sequence . It may be tempting to write as
but in fact every -number could be written in this way, so this notation does not specify a unique number.
Similarly, the surreal -numbers have the same order type as the surreals, and can be notated as where is a surreal.
Last revised on March 30, 2024 at 12:40:41. See the history of this page for a list of all contributions to it.