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> There are no reals in between 0.001 and 0.002, unless you generate them. You seem to be thinking of a real as a string of digits, or a block of bits, which i

> There are no reals in between 0.001 and 0.002, unless you generate them. You seem to be thinking of a real as a string of digits, or a block of bits, which is just the representation of a real. Like, a number doesn't exist until you think of it. That view doesn't help much in thinking about these matters. > So then what Cantors proof is saying is that if you could somehow count to infinity and get to a result, some infinities would have more elements than others. But that is a contradiction, because you can’t finish counting to infinity. So you're saying Cantor's "proof" embodies a contradiction. You must be a very great mathematician. I'm not a mathematician, so I'd better just bow out now. Anyway, I'm way too old to start thinking of numbers as some kind of generative process.

nico 4y ago
Why would someone need to be a mathematician to think and understand? That’s just gate keeping through labels. And probably the reason why so many old ideas go completely unchecked and are taken as gospel for so long with no one questioning them. You can clearly reason about this stuff, even if you are not a mathematician. But maybe it scares you to think that something that you thought was “true”, is just someone else’s opinion about what they thought it was true.
- denton-scratch 4y ago
  Unfortunately, a lot of results (proofs) about infinities are counter-intuitive, and they can't be disproven simply by showing how they go against (your) intuition. A proof is not "someone else's opinion"; it is formal reasoning that stands on its own, apart from whatever opinion may have been held by the creator of the proof. In fact many proofs went against the opinion of the proof's creator. So, I'm not "scared"; I'm prefectly prepared to take a stand against unproven orthodoxies. But I have enough humility to realize that if I think I've found a contradiction in a proof by a great mathematician such as Cantor, and nobody else has found it, then it must be me that's wrong.

nico 4y ago

Why would someone need to be a mathematician to think and understand? That’s just gate keeping through labels. And probably the reason why so many old ideas go completely unchecked and are taken as gospel for so long with no one questioning them. You can clearly reason about this stuff, even if you are not a mathematician. But maybe it scares you to think that something that you thought was “true”, is just someone else’s opinion about what they thought it was true.

denton-scratch 4y ago
Unfortunately, a lot of results (proofs) about infinities are counter-intuitive, and they can't be disproven simply by showing how they go against (your) intuition. A proof is not "someone else's opinion"; it is formal reasoning that stands on its own, apart from whatever opinion may have been held by the creator of the proof. In fact many proofs went against the opinion of the proof's creator. So, I'm not "scared"; I'm prefectly prepared to take a stand against unproven orthodoxies. But I have enough humility to realize that if I think I've found a contradiction in a proof by a great mathematician such as Cantor, and nobody else has found it, then it must be me that's wrong.