What Viliam is looking at here doesn’t (I think) have much to do with surreal numbers.
I don’t understand the question “why you don’t stop at the previous −1 ones?”. I think there may be a false assumption built into it. You don’t “count up to ω from 0″, in the surreal numbers. You do do that, in some sense, in the ordinals, but there there isn’t a ω−1.
Perhaps there might be a variant of p-adic numbers where you allow a digit for each ordinal position rather than each non-negative integer position. But that seems like it might be problematic because there are too many ordinals; for instance, in ZF set theory with the usual conventions there are no functions from the ordinals to {0,1,...,p-1} to be these numbers. (You could consider partial functions, with the convention that anything not specified thereby is zero. But then e.g. you no longer have a representation for −1.)
There are only countably many p-adic numbers with periodic digit sequence. There are uncountably many once you fill in the gaps and allow non-periodic sequences. (Just like with ordinary decimals.)
It might be a little informal in my head but I liken that you get the ordinary finite integers from a successor function and the finite integers get their birthdays by finite induction by being constructed from the previous birthday. So each of that steps seem like “+1”. Then when you do the first transfinite induction it feels like “+1″ “real hard”. And when you have calculations like ω∗1=ω that can seem like it correspond to the operation of “+1, omega times”
What Viliam is looking at here doesn’t (I think) have much to do with surreal numbers.
I don’t understand the question “why you don’t stop at the previous −1 ones?”. I think there may be a false assumption built into it. You don’t “count up to ω from 0″, in the surreal numbers. You do do that, in some sense, in the ordinals, but there there isn’t a ω−1.
Perhaps there might be a variant of p-adic numbers where you allow a digit for each ordinal position rather than each non-negative integer position. But that seems like it might be problematic because there are too many ordinals; for instance, in ZF set theory with the usual conventions there are no functions from the ordinals to {0,1,...,p-1} to be these numbers. (You could consider partial functions, with the convention that anything not specified thereby is zero. But then e.g. you no longer have a representation for −1.)
There are only countably many p-adic numbers with periodic digit sequence. There are uncountably many once you fill in the gaps and allow non-periodic sequences. (Just like with ordinary decimals.)
ω has birthday ω and ω−1 has a birthday of ω+1.
It might be a little informal in my head but I liken that you get the ordinary finite integers from a successor function and the finite integers get their birthdays by finite induction by being constructed from the previous birthday. So each of that steps seem like “+1”. Then when you do the first transfinite induction it feels like “+1″ “real hard”. And when you have calculations like ω∗1=ω that can seem like it correspond to the operation of “+1, omega times”