We can get a good start on quantity with tennis balls...
If you have a empty tennis ball can, the set of balls in the can is the empty set. The number of balls in the empty set is zero.
Axiom 1: Zero is a valid number of balls to find in a can.
Consider the act of adding a tennis ball to the can. Call this the "successor" function and use S() to denote it. [Assume, hypothetically that there is no limit to the size of a can].
Axiom 2: If n is a valid number of balls in a can then S(n) is also a valid number of balls.
Consider that if you add a tennis ball to the can, the result will always have more than zero balls.
Axiom 3: There is no number n such that S(n) = 0.
Consider that if you have two cans, added a ball to each can and found that they both now have the same number of balls in them. Then it follows that the number of balls they had to start with were equal.
Axiom 4: If S(a) = S(b) then a=b
Consider a set of numbers for saying how many balls are in cans. If zero is in the set and if for every number in the set the successor of that number is also in the set then every possible number of tennis balls in cans is in the set.
Axiom 5: This one is called the induction axiom.
These are the
Peano Axioms.
