What are we to make of this passage?
Among the triumphs of set theory are Gödel’s Incompleteness Theorems and Paul Cohen’s proof of the independence of the Continuum Hypothesis. Gödel’s theorems in particular had a dramatic effect on philosophical perceptions of mathematics, though now that it is understood that not every mathematical statement has a proof or disproof most mathematicians carry on much as before, since most statements they encounter do tend to be decidable. However, set theorists are a different breed. Since Gödel and Cohen, many further statements have been show to be undecidable, and many new axioms have been proposed that would make them decidable.
Well, we might complain that this is at least three ways misleading:
- Gödel’s Incompleteness Theorems are a triumph, but not a triumph of set theory.
- Gödel’s Incompleteness Theorems do not show that “not every mathematical statement has a proof or disproof”.
- Cohen’s and Gödel’s results are significantly different in type and shouldn’t be so swiftly bracketed together. While the Cohen proof leaves it open that we might yet find some new axiom for set theory which settles the Continuum Hypothesis (and other interesting propositions which can similarly be shown to be independent of ZFC), Gödel’s First Theorem — perhaps better called an Incompletability Theorem — tells us that adding new axioms won’t ever give as a negation-complete theory (unless we give up on recursive axiomatizability of our theory).
I’m slightly embarrassed to report, then, that the rather dodgy passage above is by a local Cambridge hero, Timothy Gowers, writing in the Princeton Companion (p. 6). Oops. I rather suspect he didn’t run this bit past his local friendly logicians …