This very short second chapter ‘The First Theorem, two versions’ makes a key distinction between the semantic and syntactic flavours of the incompleteness theorem. That distinction is already there in Gödel 1931; but if I recall rightly, it was Andrzej Mostowski in his short book twenty-one years later who first brings it out really clearly and explicitly. The chapter also makes another key point: we should really talk, not of an incompleteness theorem (after all, mere incompleteness in itself might be boringly repairable) but of an incompletability theorem.
5 thoughts on “Gödel Without Tears, slowly, 2”
I was startled by the word ‘express’ there too, but I quite like it being there because it made me stop and think about what it might mean. I took it to be referring to the second part of the hypothesis in the theorem; not so much the ‘soundness’ part but the ‘language of basic arithmetic’ part.
I think the point is that that the theory must have all the expressive resources mentioned in def. 11 in the language. If it doesn’t, for example if the language doesn’t have quantifiers, then the theorem doesn’t apply.
This point is maybe worth emphasising. You *can* have a negation complete effectively axiomatised theory of a body of arithmetical truths if you don’t have quantifiers for example, or if you have just the addition symbol and not multiplication. And this a point about ‘what the theory can express’, in the sense of what is available in the language. Whether this is the right place to make this point I don’t know …
Yes, ‘express’ is naturally understood as being about the ‘language of basic arithmetic’ part. The problem in the final paragraph on p 11 is that the ‘language of basic arithmetic’ part is syntactic, and
is treating ‘express’ as semantic. The semantic part is soundness.
However, your post has made me think, and I’m wondering whether the 2nd version of the incompleteness theory (the syntactic version — Theorem 2) also needs a theory “whose language contains the language of basic arithmetic”. Is that brought in by it being able to “prove a certain modest amount of arithmetic”, or might it be able to prove enough of arithmetic even without having the full language of basic arithmetic? I think that’s something that should be explained at this point in GWT.
I agree — there is an (unintended) disconnect between the ways that Theorems 1 and 2 are stated which certainly needs tidying!
p 10, Re the question ‘are set theory natural numbers the genuine article?’, it occurs to me that there’s a similar issue about the way Godel does logic as arithmetic, with numbers as surrogates for formulas.
p 11, par just before Theorem 2: style: I don’t know how you feel about such repetitions, but there’s But But and So So each in successive sentences in that paragraph.
p 11, final par, “express”: I don’t think it’s clear (at least at this point) why / how the semantic assumption in the first version of the theorem is about what the theory can express. Neither this chapter nor the previous explains soundness in terms of what can be expressed, and when “can express” is used in Defn 11 on p 10, it seems to be syntactic. In light of Defn 11, the “can express” at the bottom of p 12 seems syntactic as well.
p 12, incompletability – As a reader, I find myself wondering at this point how GU relates to the original Godel sentence, GT. It must be different, but what causes the difference? (That becomes clear enough, eventually; but is there an informal discussion, even in IGT?) It’s also presumably quite similar, if the same method for constructing a theory’s Godel sentence is used. So the incompletability seems to be of a particular and quite circumscribed sort.
Of course, you can add more interesting axioms and those systems will also be incomplete, but once you’ve added the ones you care about, the further incompletability is back to adding Godel sentence after Godel sentence.
How does such incompletability get wider significance (if it has any)?
I can’t remember whether this is discussed in IGT.
Thanks for these! Yes, to the first comment. And yes again to the last one — though I think the place to say more about this is when we actually get to prove the theorem in Ch. 10.
And yes, most immediately importantly, yes to the comment about “express” on p. 11: that does indeed look like a thinko, and not what I should have said, now I re-read it! So particular thanks for that.