On Sunday, October 1, 2017 at 1:32:32 PM UTC-6, John Gabriel wrote:
> Peano's Crapaxiom 5 is the induction axiom:
> If a set S of numbers contains zero and also the successor of every > number in S, then every number is in S.
Well, that's clearly wrong. 1/2 would not be in S, and we're all agreed that 1/2 is a number.
However, I think this is really what Peano was saying:
If S contains zero, and if it is also true that if S contains x then it contains the successor of x
(where the successor of x is what we would normally call x+1, but Peano hadn't gotten around to defining addition yet)
then S will contain 0, 1, 2, 3, 4, 5... and indeed any non-negative integer.
For example, it contains 5 because
S contained zero and because if it contains x, it contains x+1, it contains 1 because it contains 0 it contains 2 because it contains 1 it contains 3 because it contains 2 it contains 4 because it contains 3 it contains 5 because it contains 4
Clearly, this will work for any non-negative integer, however large, even if one doesn't bother to type that much.