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Personally (as someone who did several pure math courses as an undergraduate but is not by any means a pure mathematician) I think Cantor’s set theory and every
by jacobolus 8y ago
Personally (as someone who did several pure math courses as an undergraduate but is not by any means a pure mathematician) I think Cantor’s set theory and everything that flows from it is a bunch of hokum.
But if mathematicians’ convention is that they will refer to ZFC when writing their proofs about something else I am interested in, I am willing to humor them, and keep my caveats to myself.
I’m reasonably well convinced that most models mathematicians can make in the context of ZFC which have any real-world implications (e.g. accurately simulate some physical phenomenon) could be alternately proved under some more restrictive set of axioms, just sometimes with a lot more hassle.
- antidesitter 8y ago> I think Cantor’s set theory and everything that flows from it is a bunch of hokum. Why?
- jacobolus 8y agoIn my opinion it is entirely speculative, built on largely unconvincing foundations (involving a lot of hand waving and circular reasoning when you start poking at them) whose implications are often absurd. But it doesn’t really matter what I think. This is not a fight I care about. It’s like arguing with Buddhists about the nature of suffering or something. YMMV.
- bordercases 8y agoAre you more comfortable with a different set of foundations? Intuitionistic? Finitistic or non-completable infinities?
- antidesitter 8y agoCan a statement like Goodstein's theorem (which is about finite objects) be proven by finitist mathematics? It cannot be proven by Peano arithmetic, which is bi-interpretable with ZFC with the axiom of infinity replaced by its negation [1], or any weaker theory. [1] https://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa https://math.stackexchange.com/questions/315399/how-does-zfc...
- throwawaymath 8y agoCould you expand on this a little? Which reasoning do you find to be circular and which implications to you find to be absurd? If you explain what you an issue with others may be able to help you understand it more clearly.
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- antidesitter 8y agoWhat does "speculative" and "unconvincing" mean, in the context of a mathematical theory? What do you find hand-wavy or circular about it? Can you specify which absurd implications you're referring to? Not trying to argue, just genuinely curious.
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