I just read this in the first chapter of the text we use for our senior class Advanced Algebra. The book is Introduction to Advanced Algebra by W. Keith Nicholson. I like it, so I will pass it on.
If you study math, you have probably heard of the Well-Ordering Axiom, a property of the integers which is equivalent to the Principle of Induction. The Axiom states: Every non-empty set of non-negative integers has a smallest member.
Intuitively clear, no? Here's how it works in practice:
I claim that every positive integer is interesting. To show this, let's assume it is false (this is a proof technique known as proof by contradiction, in which one assumes the claim is false, then works by deduction until something absurd follows, a clear contradiction. If the logic is solid, only the assumption can be flawed. Thus your original statement must be true. )
Since the statement is assumed false, there must be a non-empty set of uninteresting positive numbers. But by the Well-Ordering Principle, there then must be a smallest uninteresting number. But an extreme element of any ordered set is automatically interesting (in that it is special)! Hence we arrive at the contradiction, making our assumption false and the original statement true.
Silly, yes?
3 comments:
Unfortunately this central theorem isn't introduced early enough in our educational system.
In my opinion, Dummit & Foote is a better undergraduate algebra textbook.
Absolutely, in my opinion, we are not teaching the logic and playful beauty of pure mathematical thought early enough in our educational system. I have thrown stuff like this at kids as young as 10 and they can certainly see and appreciate the ideas. Even Lewis Carroll understood that.
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