User talk:Lord Farin/Backup/Principle of Finite Induction

"As $$S$$ is well-ordered, and $$T' \subseteq S$$, it follows that $$T'$$ has a minimal element. Call this minimal element $$x$$." Should this just be based on the well ordering principle? --cynic 23:19, 30 September 2008 (UTC)

Not the way I see it, as the Well-Ordering Principle specifically refers to the set $$\mathbb{N}$$ of natural numbers. The Principle of Finite Induction is demonstrated on the naturally ordered semigroup before we've gone anywhere near defining what the natural numbers are, and in fact is used in the construction of the proofs that establish that $$\mathbb{N}$$ is a naturally ordered semigroup in the first place. To assume the well-ordering principle would make the argument circular.

The fact that the natural numbers are mentioned at the bottom is an explanatory amplification which is not germane to the proof itself but follows a lot later down the line after the set of natural numbers has been defined and proven to be a naturally ordered semigroup. --Matt Westwood 06:09, 1 October 2008 (UTC)