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”
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”