Suppose you have encountered Zeno’s Achilles paradoxfor the first time. Zeno offers a rigorous (looking) proof that, having once given the tortoise a head start, Achilles can never overtake it. Would you regard this as
# A startling new discovery in athletics;
# A demonstration of the transcendent capacity of the human spirit – although the laws of logic forbid it, Achilles does in fact catch and overtake the tortoise; or
# A warning about how not to take limits?
In this case, I assume nearly all readers will go for option 3. But things aren’t always so easy. The Einstein-Podolsky-Rosen Paradox was supposed to be a type-3 paradox demonstrating the incompleteness of quantum mechanics. But on most modern views, it is really a type-1 paradox, predicting various highly counter-intuitive consequences of quantum mechanics that nonetheless turn out to be be empirically valid.
Although I don’t accept that there are any good examples of type-2 paradoxes, plenty of others would offer this solution in relation to both Godel’s theorem and Schrodinger’s cat.
With these three possibilities in mind, how should we think about paradoxes involving probability measures over infinite sets that are finitely, but not countably additive? The two-envelopes problem we’ve been discussing here falls into this class and so, with a little bit of tweaking, do St Petersburg and related paradoxes. I’ll leave this question hanging and offer my own answer in a later post.