Recent content by Wenti

So then don't we have m \in S \Rightarrow Sm \in S \Leftrightarrow Sm \notin S \Rightarrow m \notin S? Or is that not correct? If it's not, that's probably exactly where my problem is.

That makes sense. ... However, I'm still a little confused, because I thought that conclusion was based on the second hypothesis of PMI, not on PMI itself. Isn't it analogous to say "how do you know that m+1 exists?"; the second hypothesis of PMI asserts that existence. So shouldn't the...

m-1 = \underbrace{1 + 1 + \cdots + 1}_{m-1 times}. I'm not sure why the statement needs to be proved or how to prove it statement. A little stab in the dark: are you saying that proving that statement requires induction itself, so I'm using my problem in my proof?

"Proving" PMI from... nothing? Help find the error! Homework Statement Prove the Principle of Mathematical Induction from the Well-Ordering Principle. Homework Equations Principle of Mathematical Induction (PMI): If A \subset \mathbb{N} satisfies (i) 1 \in A, (ii) if k \in A, then...