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Are there any theorems that can only be proved by induction? Induction seems to be proof by technicality.

N. F. Taussig
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Jimmy360
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    "Induction seems to be proof by technicality." What were you expecting? Proof by intuition? Proof by committee? The technicality is what makes proof by induction so strong and desirable, like a supermodel who chops wood. – Jon Mar 31 '15 at 03:10
  • Induction is based on two properties of the natural numbers. They are the Well-Ordering Principle (each nonempty subset of the natural numbers has a least element) and the fact that each element of the natural numbers has a successor. – N. F. Taussig Mar 31 '15 at 03:12
  • @Jon that is the greatest analogy that I have ever heard – Jimmy360 Mar 31 '15 at 03:12
  • What I mean is, for example proving the Consecutive Gauss Formula. The inverted addition proof shows how it works, while the inductive proof just lets us know that somehow it does. – Jimmy360 Mar 31 '15 at 03:14
  • It's a good question, are all inductive proofs replaceable by some other kind of proof? I really don't know, but I do know that the theorems we prove by induction we do so because it's the easiest way that anybody has found to prove them. So why would we want to work harder? It's not like proof by induction is not mathematically valid. – Gregory Grant Mar 31 '15 at 03:22
  • I wish I could find a similar question that came up not long ago, someone gave a good definitive answer. But I feel similarly, often: Proofs by induction feel less enlightening than alternatives. – pjs36 Mar 31 '15 at 03:24
  • Here we go: link. Not exactly the same, but quite similar. The answer is quite good, but way beyond me. – pjs36 Mar 31 '15 at 03:29
  • This is a duplicate of [this question], to the point that the question asker almost (possibly unintentionally) quoted that question verbatim. – Jack M Mar 31 '15 at 03:32
  • @JackM wow that is like.. the exact same thing – Jimmy360 Mar 31 '15 at 03:33

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What set of axioms are you using? If you use the Peano Axioms, you can't prove much without induction. In particular, the commutative and associative properties of addition and multiplication require induction. The inverted addition proof that you mention needs a lot of work to justify if you go all the way back to the axioms. If we don't go back to the axioms we rely on a lot of intuitive understanding of the naturals.

Ross Millikan
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