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"Interesting" numbers and the Hangman paradox in proof theory?
First off, wikipedia gives a *decent* discussion of the hangman paradox. If you don't know it, you can find it here:
http://en.wikipedia.org/wiki/Unexpected_hanging_pa...
Now, normally, we don't seem to care much about this paradox in mathematics, since there isn't even consensus on where the paradox originates from. However, it occurs to me that a standard proof that appears in mathematical literature may be an instance of this same paradox.
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Proof that all natural numbers are interesting:
First, let N be the set of natural numbers and X the set of interesting natural numbers. We want to show that N is a subset of X.
Suppose otherwise, that there is an element n of N that isn't interesting, then the set N\X is nonempty.
Since N is well-ordered, N\X has a least element - call it y. Well, the fact that y is the least natural number that isn't interesting makes it interesting. Thus y is interesting and y is not interesting. Contradiction.
Therefore every element of N is interesting. Q.E.D.
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Now, this proof may just be included in books in a tongue-and-cheek manner, but it seems like a disservice to mention it but not mention its connection to such a famous paradox. After all, when we finish the proof, any single element is just one element in a sea of other "interesting" elements, making it rather boring (which to me shows the connection to the paradox).
Does anyone know enough about proof theory to confirm or deny that these situations are identical?
2 Answers
- JBLv 71 decade agoFavorite Answer
Long ago I thought I was interested in logic and philosophy. Now it puzzles me that I once felt that way.
That was an interesting proof you gave (which I had seen before), but it reminded me that there are theorems about the length of proofs, so maybe there should be theorems about the interestingness (or not) of proofs.
Well, I know nothing about proof theory, so can't possibly answer your question properly. All I can do is make a few comments.
It seems to me the similarity to the interesting numbers proof may draw from the fact that there are multiple days in a week. If there are only two days in the week, then at least superficially it seems more like the Russell paradox, or the heterological paradox. And if you have only one day in a week, the paradox (apparently) cannot even be made at all.
In any case, the link below may be of interest to you, although I suspect you are already familiar with it. Perhaps, the author of that paper is who you should ask.
