Topics Mathematics

Question Hello Philosophers Is there any axiomatized theory of arithmetic that is much stronger to be afflicted by Gödel theorems? I've read that there are axiomatized theories that are weaker than the theorem's criteria, i.e not expressive enough, and their consistency is proved within the same theory. I wondered if there would be something like that, which is stronger than the Gödel theorem's criteria for a axiomatized theory.

Accepted:
March 17, 2011

Comments

Gödel's first theorem, with later improvements by Rosser and others, tells us that any theory of arithmetic T that is (i) consistent, (ii) decidably axiomatized (i.e. you can mechanically check that a purported proof obeys the rules of the arithmetic) and (iii) contains Robinson Arithmetic (a very weak fragment of arithmetic) is incomplete. Strengthen T by adding more axioms and the theory will still be incomplete (unless you throw in so much it becomes inconsistent or stops being decidably axiomatized). In sum, you can't "outrun" the reach of incompleteness theorem by going to a stronger theory which is still consistent and properly axiomatized. This is explained in any standard introduction to Gödel's theorems (e.g. in the first chapter of mine).