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I have a question regarding proof by contradiction and the well ordering principle on integers.

Okay lets says for example an arbitrary value $p$ is a positive integer and which I assume is the smallest positive integer for some property.

Then lets say I find an arbitrary value $r$, also a positive integer, which I calculate to be less than $p$ for the same property. Then I have a contradiction.

So my question is how is this possible. Lets substitute $1$ for $p$ and it satisfies the property, then how can I find $r$ since there is no smaller positive integer? I'm not sure if $0$ is considered a positive integer but if it is then substitute $p$ for $0$.

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    $\begingroup$ Another term for this is proof by infinite descent. I don’t have time to write an answer but you may find something helpful here en.m.wikipedia.org/wiki/Proof_by_infinite_descent $\endgroup$ – spaceisdarkgreen Mar 1 at 20:18
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    $\begingroup$ It will also help to get more of a feel for the hypothetical nature of proofs by contradiction by studying other proofs by contradiction that might be easier to understand. $\endgroup$ – spaceisdarkgreen Mar 1 at 20:21
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The point of the proof by contradiction is that you can find no such $r$. That's the contradiction. You assume something you want to be false, then prove something that can't be possible. It may be helpful to give a specific example of a proof that's confusing you.

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  • $\begingroup$ mathoverflow.net/questions/107153/… $\endgroup$ – rert588 Mar 1 at 20:13
  • $\begingroup$ See the answer using the well ordering principle $\endgroup$ – rert588 Mar 1 at 20:13
  • $\begingroup$ from the link sent $\endgroup$ – rert588 Mar 1 at 20:14
  • $\begingroup$ Yeah, so in that proof you assume that $\sqrt{2}$ is irrational. Then using well ordering you can find a least $q$ such that $q\sqrt{2}$ is an integer. Then without assuming anything else, you can find $r < q$ a positive integer such that $r \sqrt{2}$ is a positive integer, which is a contradiction by minimality of $q$. $\endgroup$ – bitesizebo Mar 1 at 20:17
  • $\begingroup$ So the fact that an r can't actually be found is part of the contradiction. Am I correct? $\endgroup$ – rert588 Mar 1 at 20:19

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