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It seems straightforward to encode a wff in PA as a number. I can't see how to encode a proof. Could someone please tell me how this is usally done? Thanks.

Rando McRandom
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  • Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos Dec 16 '21 at 06:41
  • "It seems straightforward to encode a wff in PA as a number:" See What does a Godel sentence actually look like? – Mauro ALLEGRANZA Dec 16 '21 at 08:42
  • "how to encode a proof?" As a sequence of formulas. – Mauro ALLEGRANZA Dec 16 '21 at 08:42
  • If you know how to encode wffs as numbers, then you already know the key trick: how to code (finite) sequences of numbers as numbers. But this is a trick we can iterate: using it twice, you can code sequences of sequences of numbers as numbers, and so on. In particular, we can code sequences of wffs by numbers. Now a bit of care is needed to identify which sequences of wffs correspond to proofs (as opposed to just being random sequences of wffs), but this is no different than the situation with wffs themselves (we need to distinguish between arbitrary and well-formed sequences of symbols). – Noah Schweber Dec 16 '21 at 15:27
  • Given that a proof consists of a sequence of well-formed formulas in which every step is either an axiom or follows from previous step(s) in the proof preceding that step using the given rule(s) of inference, we would replace "well-formed formulas" with "Goedel numbers". This also presumes that Goedel numbers are well-formed formulas also. – Doug Spoonwood Dec 16 '21 at 16:28
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    Once you know how to encode sequences, you can just fix an encoding of proofs-as-sequences. – Mark Saving Dec 17 '21 at 18:47

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