00:01
Hi, in this question we are asked to prove that the well -othering property of positive integers.
00:09
Like we have to prove it given that we have mad induction as an axiom.
00:16
So usually it's the other way around, but this time we have that mad induction is true.
00:23
And we want to prove the well -ordering properties, which state that any, not -earned induction is true.
00:31
Empty subset of positive integers has the least element.
00:41
So it can be infinite set, it can be finite set, it must have the least element.
00:47
To prove this, i will assume the opposite and show contradiction.
00:53
So let's suppose that there really is a non -empty subset, call it s, such that it has no list element and we will find a contradiction from this.
01:10
So first let p of n be a statement that integer n, positive integer n, is not inside this set, this subset for n in positive integer.
01:25
And we will use induction to prove that p of n is true for any end and that will lead to a contradiction.
01:32
But first, a basic step is clear.
01:38
So, number one, cannot be inside this subset s.
01:43
Why? because if it does, it will be the least element...