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Slide 1

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. Mathematics for Computer Science MIT 6.042J/18.062J The Well Ordering Principle

Slide 2

Well Ordering principle Every nonempty set of nonnegative integers has a least element. Familiar? Now you mention it, Yes. Obvious? Yes. Trivial? Yes. But watch out:

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Every nonempty set of nonnegative integers has a least element. Well Ordering principle rationals NO!

Slide 4

Well Ordering principle Every nonempty set of nonnegative integers has a least element. NO!

Slide 5

What is the youngest age of MIT graduate? smallest # neurons in any animal? smallest #coins = $1.17?

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For rest of this talk, “number” means nonnegative integer N ::= nonnegative integers

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proof used Well Ordering …can always find such m, n>0 without common factors… why always ?

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Proof using Well Ordering and m/c < m

Slide 9

Proof using Well Ordering This contradiction implies m, n have no common factors. Find smallest number m s.t.

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