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Tim Gowers has a very nice piece on his blog about functions, multivalued functions, relations and the like, called "Why aren't all functions well-defined?".
The intuitionists have it easy, of course: they can just waffle about extensional operations. (One is reminded of the trivial but fallacious intuitionistic proof of (certain form of) choice...)A standard illustration of the hazards of talking about well-definedness of functions is provided by the proof that recursive definitions (along some well-ordering, say) define unique functions. (At least I seem to recall this is a standard illustration.) Given a recursive definition we prove by induction, so one naturally surmises, that the function is well-defined. This is poppycock, as a moment's reflection reveals: we're trying to apply induction to show that an object we haven't proven to exist has a property that we haven't given any mathematical definition. And so it goes.
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