Question I have heard that Gödel Proved that Arithmetic cannot be reduced to logic or formal logic. Although I have read explanations which basically state that arithmetic is not complete and thus not definitional like in formal logic, I cannot get my head around how 1+1=2 is NOT reducible to formal logic. This seems like an obvious analytic statement in which "one and one" is the same as saying "two". Can anyone shed light on this?

Accepted:
September 24, 2009

Comments

Well, there is a logical truth in the vicinity of 1 + 1 = 2. Or perhaps better, a whole family of logical truths. Fix on a pair of properties F and G. Then it is a theorem of first-order logic that if exactly one thing is F and one thing is G and nothing is both F and G, then are exactly two things are either-F-or-G. Here the numerical quantifiers 'exactly one thing is' and 'exactly two things are' can be defined in standard ways from the ordinary first-order quantifiers and identity. And the theorem holds whatever pair of properties we choose. This elementary logical result probably captures what is driving your intuition that in some sense 1 + 1 = 2 is "reducible to formal logic". (For a bit more on this sort of thing, see my Intro to Formal Logic §33.3 -- or any other standard logic text!)

But all that is quite compatible with Gödel's first incompleteness theorem. For Gödel's theorem isn't about some limitation or incompleteness in our ability to prove (or disprove) simple equations relating particular numbers like 1 + 1 =2. Rather it is, roughly speaking, about the incompleteness of any theory rich enough to prove quantified claims about numbers. For any given rich enough axiomatized theory, there will some quantified truth about numbers it can't prove. So that means we can't wrap up all the truths of arithmetic (including all the quantified ones) into a single axiomatized theory . Hence, a fortiori we can't wrap them up into a single axiomatized theory that might be thought of as part of logic. (For a bit more on this sort of thing, see my Intro to Gödel's Theorems -- or any other standard text which covers incompleteness results!)