James R. Newman (1907–1966)

What Gödel's Theorem really says is this: In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be INDEPENDENT OF the set of axioms. They are neither false nor true in the system. They are INDEPENDENT (cannot stress this enough). We want axioms to be independent of each other, for instance. That's because if an axiom is dependent on the other axioms, it can then be safely removed from the set and it'll be deduced as a theorem. The theory is THE SAME without it. Now, the continuum hypothesis, for instance, is INDEPENDENT of the Zermelo-Fraenkel axioms of the set theory (this was proved by Cohen). Therefore, it's OK to have two different set theories and they will be on an equal footing: the one with the hypothesis attached and the one with its contradiction. There'll be no contradictions in either of the theories precisely because the hypothesis is INDEPENDENT of the other axioms. Another example of such an unprovable Gödelian sentence is the 5. axiom of geometry about the parallel lines. Because of its INDEPENDENCE of the other axioms, we have 3 types of geometry: hyperbolic, parabolic and Euclidean. And this is the real core of The Gödel Incompleteness Theorem. By the way... What's even more puzzling and interesting is the fact that the physical world is not Euclidean on a large scale, as Einstein demonstrated in his Theory of Relativity. At least partially thanks to the works of Gödel we know that there are other geometries/worlds/mathematics possible and they would be consistent.

Without a clear and explicit reference to the concept of a formal system all that is said regarding Gödel's theorems is highly inaccurate, if not altogether wrong. For instance, if we say that Gödel's statement is true, after saying that Gödel's Theorem states that it can't be proved either true or false. Without adding "formally", that doesn't really make much sense. We'd only be only talking about axioms, which are only a part of a formal system, and totally neglecting talking about rules of inference, which are what the theorems really deal with.

By independent I mean 'logically independent', that is only a consequence of Gödel's theorem in first order languages, whose logic is complete. In second order arithmetic, the Peano axioms entail all arithmetical truths (they characterize up to isomorphism the naturals), so that no arithmetical sentence is logically independent of such axioms. It occurs, however, that second order logic is incomplete and there is no way to add to the axioms a set of inference rules able to recursively derive from the axioms all of their logical consequences. This is why Gödel's theorems holds in higher order languages too. In fact, this is how the incompleteness of higher order logic follows from Gödel's theorems.

What prompt me to re-read this so-called seminal book? I needed something to revive my memory because of Goldstein's book on Gödel lefting me wanting for more...I bet you were expecting Hofstadter’s book, right? Nah...Both Nagel’s & Newman’s along with Hofstadter’s are failed attempts at “modernising” what can’t be modernised from a mathematical point of view.

Read at your own peril.… (more)