Here's a couple of observations in rapid fire. Have any philosophers argued that these, and other, related observations mean that the Godel incompleteness theorems aren't as devastating as they're often made out to be?
Observation one. There's no 'incompleteness theorem for Turing-completeness'. In particular, there's plenty of Turing-complete computational languages.
Observation two. Define that the 'diagram of true arithmetic' is the set of all true sentences of the form
- not 3+2=6
More precisely, I guess I should be writing SSS(0)+SS(0)=SSSSS(0). Anyway, the point is, the diagram of true arithmetic is recursively ennumerable. Indeed, it is recursive, a more stringent condition. For example, we can decide whether 3+2=5 or not.
This means that, for example, that if we axiomatized arithmetic in a highly defective (but internally consistent) manner - say, for example, we ended up proving 438+21=555, or something like that - well we can demonstrate the our axiomatization is defective by noting that 'not 438+21=555' is an element of the diagram of true arithmetic.