Talk:Equivalence of Well-Ordering Principle and Induction

It's a little silly to talk about equivalence between tautologies - it would seem more meaningful (and useful) if this statement were generalized to induction on well-founded or well-ordered sets. --asalmon


 * Why? --GFauxPas 09:58, 29 November 2011 (CST)


 * If any two statements are tautologies, then they are automatically equivalent. -Andrew Salmon 15:02, 29 November 2011 (CST)
 * That strong and weak induction are logically equivalent holds even if they're not tautologies, they can all be false as well, or all be axiomatic (and actually I learned them as being axiomatic, not as being theorems). No? --GFauxPas 15:08, 29 November 2011 (CST)

Missing justification in the PCI implies WOP part.
In the line:

Now if $k + 1 \in S$ it follows that $k + 1$ would then be the minimal element of $S$.

How do we know that there is no number between k and k + 1 ? --TonyH 08:05, 9 May 2012 (EDT)
 * From the structure of $\N$ as derived either by the Peano axioms or the Naturally Ordered Semigroup technique. --prime mover 08:31, 9 May 2012 (EDT)
 * Well, the same could be said about every fact we know about math. Isn't the purpose of this site to give explicit proofs?--TonyH 09:02, 9 May 2012 (EDT)
 * Once we assume that natural numbers exist, we assume the peano axioms exist and that the natural numbers satisfy them. One of the peano axioms is: "$\forall m, n \in \N: s \left({m}\right) = s \left({n}\right) \implies m = n$, where $s$ is a Definition:Successor Mapping. Would it help to look at the expression $n + 1$ not as "the sum of $n$ and $1$", but as the "successor of $n$", i.e., the "immediate next number after $n$"? The definition of $s$ then guarantees that $s\left({n}\right)$ is unique for every $n$, and you don't need to prove definitions. --GFauxPas 09:37, 9 May 2012 (EDT)
 * As GFP says, the $+1$ function has already been established as a pre-existing definition, upon which this proof rests. Yes okay, a link is needed to it. This will be accomplished in due course. --prime mover 09:50, 9 May 2012 (EDT)
 * ... happy now? --prime mover 13:01, 9 May 2012 (EDT)

One could argue that it would be better were the three abbreviations used in the proof links to the appropriate pages. However, I would like someone backing me here, before I put myself through this exercise. --Lord_Farin 13:26, 9 May 2012 (EDT)
 * No reason why not. Don't know why I didn't do it like that in the first place. --prime mover 14:15, 9 May 2012 (EDT)