Where have we got to in this slow introduction to the incompleteness theorems? Theorem 8 from Chapter 4 tells us that if a consistent, effectively axiomatized, theory is “sufficiently strong”, then it must be negation incomplete. And we *announced* that even the very weak arithmetic called Robinson Arithmetic is sufficiently strong (in which case, stronger consistent, effectively axiomatized, theories must also meet the condition, and so must be incomplete). But we didn’t say anything at all about what this weak theory of arithmetic looks like! Today’s Chapter 5 explains.

So here are revised versions of Chapters 1 to 4 together with the new Chapter 5 on ‘Two weak arithmetics’. (Many thanks to those who have commented on the earlier chapters here or by email: there are quite a few small improvements as a result.)

Rowsety MoidChapter 5

p 28: what exactly does ‘the likes of τ’ include? If it’s Greek letters generally, or some subset of them, why not say that?

p 28, footnote 1: What does “much odds” amount to?

Compare footnote 2, p 29 “it in fact wouldn’t make any difference to the strength of our theory”

p 29, “And so on and so forth” can just be “And so on”.

p 30: “Leibniz’s Law, which allows us to substitute terms which are equal.” — which have been shown to be equal earlier in the proof?

“(‘LL’ of course indicates ..)” — better without “of course” and without the parens, since both (though “of course” more strongly) suggest that the reader should already know what “LL” indicates.

p 31, might be worth having a footnote explaining what “MP” and “RAA” mean.

(I think they’re easier for more people to work out than “LL”, so that a footnote would do.)

p 32, Theorem 11 and its proof: I’m not sure why it’s “equivalent to”, rather than “is”.

p 33, “(so this is the least ambitious language which ‘contains the language of basic arithmetic’ in the sense of Defn. 11)” — I’m not convinced that it’s “so”, and it looks odd to have such a clause attached in parens anyway, so I think it would be better to drop the “so” and the parens and have a separate sentence “This is the least …”

p 33, Q: “And as for its non-logical axioms” — can just be “And for …”

p 35’s nonstandard model: so Axiom 3 dosn’t rule out strays after all?

Peter SmithMany thanks for these once more — all helpful!

Rowsety MoidA couple of other things, looking back after reading chapter 6:

Re the formal proofs — now that I’ve looked at chapter 6 and it’s much friendlier terminology — “by the identity laws” and “by Conditional Proof”, rather than “by LL” and “by RAA” — I feel more strongly opposed to “LL” and “RAA” and think Ch 6’s terminology should be used here too.

p 30, right after the 1st poof: “Exactly similarly” — why not just “Similarly”?

Sam ButchartI found just one typo, p. 39 middle of para. 2, ‘Let ‘0’ still to refer to zero’ should be ‘Let ‘0’ still refer to zero’.

I agree with Rowsety that using names rather than abbreviations for rules in proof is friendlier.

Sam ButchartThis is just a comment rather than a suggestion to change anything: I understand this is not meant to be a philosophical text.

bottom page 32 ‘that’s all very straightforward but also pretty unexciting.’

Maybe baby arithmetic is a little bit exciting if you’re a hopeful logicist. Kant thought that numerical equations like 7 + 5 = 12 could not be analytic, but must be synthetic, so we need some a priori intuition to justify them. But BA shows that we can derive all true numerical equations (and inequations!) from basic axioms governing +, x, successor and zero. If not exactly definitions, you might think of those axioms as partly explicating the meaning of those symbols. So we get some initial encouragement for logicism from this result…

Peter SmithThat’s a nice point … mathematically unexciting, maybe, but a little bit philosophically exciting. I’ll add a footnote!