The following clip is taken from Chapter 4 - Cantor: Detour through Infinity (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ in the sequence) be, $\omega$-th or $(\omega + 1)$-th? Davis put the former, but I found the latter elsewhere on the Internet (e.g.). I am confused. Could you help?
Reference
- Davis, M. (2018). The universal computer: the road from Leibniz to Turing. CRC Press.
Ordinals probably shouldn't be used as an index to reference the individual elements of some well-ordered set, especially not $\omega$ which is a limit ordinal (as opposed to a successor ordinal or zero).
The order type of the even numbers as listed is $\omega$ and the order type of the even numbers followed by "1" as above is $\omega+1$.
One convention would be to index an element by the ordinal of the rest of the well-ordered set preceding it, which would give "1" the index $\omega$ but would then give the least element "2" the index $0$ instead of $1$.