We keep going back to the Greeks because the paradoxes of the Eleatics (such as Zeno) and the Skeptics have never been satisfactorily addressed; they apply as well to the modern formal systems you laud as to the old syllogisms. Thinking in that vein may sharpen & formalize the paradoxes in such forms as Godel’s theorems, but they won’t dissolve them; we need different approaches to resolving the many Skeptical arguments about, say, circularity, like this metacircular approach.
Right! The Axiom of Choice is just one example of something like a ‘sharpened’ paradox, or really something ‘sharp’ that implies other paradoxical conclusions.
We keep going back to the Greeks because the paradoxes of the Eleatics (such as Zeno) and the Skeptics have never been satisfactorily addressed; they apply as well to the modern formal systems you laud as to the old syllogisms. Thinking in that vein may sharpen & formalize the paradoxes in such forms as Godel’s theorems, but they won’t dissolve them; we need different approaches to resolving the many Skeptical arguments about, say, circularity, like this metacircular approach.
Right! The Axiom of Choice is just one example of something like a ‘sharpened’ paradox, or really something ‘sharp’ that implies other paradoxical conclusions.