Set theory is not easier than arithmetic! Zero is a finite number, and N+1 is a finite number if and only if N is.
Yes, that is a much better definition. I don’t know why this one occurred to me first.
Sewing Machine’s previous comment isn’t really a definition, but it leads to the following:
“n is a finite ordinal if and only if for all properties P such that P(0) and P(k) implies P(k+1), we have P(n).”
In other words, the finite numbers are “the smallest” collection of objects containing 0 and closed under successorship.
(If “properties” means predicates then our definition uses second-order logic. Or it may mean ‘sets’ in which case we’re using set theory.)
Set theory is not easier than arithmetic! Zero is a finite number, and N+1 is a finite number if and only if N is.
Yes, that is a much better definition. I don’t know why this one occurred to me first.
Sewing Machine’s previous comment isn’t really a definition, but it leads to the following:
“n is a finite ordinal if and only if for all properties P such that P(0) and P(k) implies P(k+1), we have P(n).”
In other words, the finite numbers are “the smallest” collection of objects containing 0 and closed under successorship.
(If “properties” means predicates then our definition uses second-order logic. Or it may mean ‘sets’ in which case we’re using set theory.)