1 thought on “Functions and gunctions”

  1. Aatu Koskensilta

    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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