There’s Something about Gödel, Ch. 12 (concluded)

I’d hoped to have time, before setting off for a “term” (i.e. half a semester) in New Zealand, to comment in a bit of detail on Berto’s discussion of the dialetheist riff on Gödel’s theorem. But that’s not going to happen. And since I’m not going to pack Berto’s book, I’m afraid — if I ever get round to commenting at length — it won’t be for nearly three months.

The trouble is, however, — and no doubt this is the reason I’ve not got round to doing the job before — I do find it jolly difficult to take dialetheism here seriously.

This isn’t to dismiss dialetheism out of hand, across the board. Perhaps there’s a just-so story to be told about how, when we add a minimalist truth-like predicate to a language without prior explicit semantic apparatus, the smoothest thing to do — all things considered — allows some extraordinary sentences containing this new predicate to then come out both “true” and “false”. So be it. But the dialetheist line on Gödel incompleteness, when the wraps are off, is committed to saying that there’s a number (an ordinary, common-or-garden, natural number) which both does and does not satisfy some primitive recursive condition (a complicated condition, to be sure, but still primitive recursive in an entirely straightforward way). Here’s a sketch of why, in my words.

Recall: the Routley/Priest suggestion is that our overall informal mathematics — the body of assumptions and deductive processes that mathematicians take take to lead to proofs that establish mathematical truth — should be susceptible to being regimented as a recursively axiomatized theory T (recursively, because negotiable by us limited humans). But T is consistent (because a body of truths) and includes enough arithmetic for Gödel’s theorem to apply. So, fixing on a scheme of Gödel-numbering, there is a Gödel sentence G, true if and only if unprovable-in-T, which is indeed unprovable-in-T, and hence true. In principle, we could spell out that informal reasoning for the truth of G in our all-embracing theory T which, by hypothesis, includes all informal mathematics. So there’s a T-proof of G, which will have Gödel-number g. But, as is familiar G (truly) “says” that no number numbers a T-proof of G. So g is also not the number of a T-proof of G. But numbering a T-proof of G is a primitive recursive property.

That conclusion — that there’s a number  which both does and does not satisfy some primitive recursive condition — I, for one, just find incomprehensible.

“But an incredulous stare is not an argument!” Indeed. But I’m not incredulous in the sense of understanding what is being said to hold, but then treating the suggestion as beyond belief (“Another concrete world, as real as this one,  in which there are talking donkeys? Come off it, David, pull the other one!”). My trouble, to repeat, is that I just don’t understand what it would be for a perfectly ordinary number both to satisfy a primitive recursive condition and not to satisfy it. Not so much incredulous stare as incomprehending boggle.

Now I don’t pretend that that‘s the end of the matter. But it is, so to speak, the beginning of the matter. And I guess my main complaint about Berto in this chapter is that — although he cheerfully takes the dialetheist to be committed in the way I’ve described — he doesn’t get far enough past the beginning and  explain what that commitment could mean. True, Berto says something briefly about strict finitism — but that‘s not going to help, unless we have a proof that, whatever the system of Gödel numbering in question, the relevant number g will have to beyond a sensible finitist’s ken (so can be treated as an “inconsistent number”). And I don’t see any reason to suppose that’s true. So I’m left boggling! As will, surely, be most of Berto’s readers.

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