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Mathematics
General Math
Writing proofs that are more or less formal
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[QUOTE="andrewkirk, post: 6005818, member: 265790"] The principle of recursive definition is actually a theorem, which has to be proven rigorously. The claim of the theorem is set out in the above link. Essentially it says that [I]there exists a unique function[/I] that has the properties given in a recursive definition. The proof of the theorem typically uses the induction axiom of Peano arithmetic. There are two long proofs of the theorem given [URL='https://proofwiki.org/wiki/Principle_of_Recursive_Definition']here[/URL], as well as a short, fallacious one. I had a quick look at the first proof and it looks sound, except that in the last line of the theorem statement ##g(f(m))## should be ##g(f(n))##. But I did not check it step by step. On reflection I realize that a more precise way of invoking the theorem in my above post is to say "the principle of recursive definition tells us that there exists a unique function" rather than "use the principle of recursive definition to define a function". I will make the change above. [/QUOTE]
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Writing proofs that are more or less formal
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