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Here was shown by induction that the order of parentheses is irrelevant when associativity is verified.

Question: Would this be a metatheorem about the formal language (say, of ZF) where the metalanguage is English and where the Induction Principle used is only an intuitive (informal) one that we accept on purely logical/intuitive grounds?

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Guest
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Yes.

Consider for example propositional logic; in the calculus you can derive e.g. :

$\vdash ((p_1 \lor p_2) \lor p_3) \leftrightarrow (p_1 \lor (p_2 \lor p_3))$

but there is no way to derive in the calculus a "one shot" formula expressing the fact that associativity holds with $n$ whatever.

To prove it, we have to use induction in the meta-theory.

Mauro ALLEGRANZA
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  • Would the argument given in the meta-theory, which in this case is English and is not encoded in a formal system, be considered less rigorous than an argument in a formal system? – Guest Dec 17 '15 at 01:31
  • @Guest - not at all... mathematical proofs are (quite) always " in the meta-theory" and not encoded into the formal system. We can use this terminology : a proof is a "standard" mathematical argument written semi-formally; this happens also in logic : see e.g. completeness theorem for propositional logic. A derivation is a sequence of formulae in a formal system according to the rules of the calculus (modus ponens, etc.) – Mauro ALLEGRANZA Dec 17 '15 at 07:21