It just MIGHT be relevant to the ZF. Writing down every real from the interval (0,1) onto the Euclidian plane using some (arbitrarily resizable) Lucida console or whatever font, (in an abstract way, of course), would be enough to have the consistency crisis in the ZF.
You can do this in a finite volume amount of 3D space, using flat 2D font. But that’s okay, regarding ZF, because most numbers written this way, covers no rational point when in 3D space.
You don’t need to actually write them down in a font, you can instead represent every real from the interval (0,1) with just some small circle or square, or with whatever shape with an area, and they also should not overlap. And with all that, the ZF is done.
But you don’t need to use every real from (0,1), just say aleph-middle of them.
What is aleph-middle? It’s some Cohen’s cardinality between aleph-zero and aleph-one, you can always postulate.
This MIGHT be feasible. If it is, it’s enough. I don’t know, if it’s feasible, let alone how exactly.
Then, you don’t even need finite area shapes, you need shapes with at least one rational point.
Then, those shapes MAY overlap. Just not in that rational point, but everywhere else.
I wouldn’t be too much surprised, if someone comes with such a construction. With all the necessary rigor, of course.
If we had provided contradictory solutions to any of your previous puzzles then that would also doom ZF, no?
Yes, but I don’t find those likely to have two opposite simple solutions. At least not less than extremely complicated. Which is practically useless. We could never have agreed about something that complicated.
I think all this means is that you find this proof less obvious than some other proofs. That’s fair enough, but finding something difficult to grasp doesn’t mean it’s likely to be wrong.
The way it looks to me: no, it’s not feasible, it’s plainly not feasible, for exactly the reason cousin_it gives; you might as well be asking for three positive integers with x^3+y^3=z^3. (Actually, even more so; I find the cardinality argument here clear at a glance, but Euler’s infinite-descent argument intricate and requiring sustained concentration. But, again, the fact that I can’t just look at it and immediately see why there are no solutions in no way calls into question the proof that there are no solutions.)
It just MIGHT be relevant to the ZF. Writing down every real from the interval (0,1) onto the Euclidian plane using some (arbitrarily resizable) Lucida console or whatever font, (in an abstract way, of course), would be enough to have the consistency crisis in the ZF.
You can do this in a finite volume amount of 3D space, using flat 2D font. But that’s okay, regarding ZF, because most numbers written this way, covers no rational point when in 3D space.
You don’t need to actually write them down in a font, you can instead represent every real from the interval (0,1) with just some small circle or square, or with whatever shape with an area, and they also should not overlap. And with all that, the ZF is done.
But you don’t need to use every real from (0,1), just say aleph-middle of them.
What is aleph-middle? It’s some Cohen’s cardinality between aleph-zero and aleph-one, you can always postulate.
This MIGHT be feasible. If it is, it’s enough. I don’t know, if it’s feasible, let alone how exactly.
Then, you don’t even need finite area shapes, you need shapes with at least one rational point.
Then, those shapes MAY overlap. Just not in that rational point, but everywhere else.
I wouldn’t be too much surprised, if someone comes with such a construction. With all the necessary rigor, of course.
Yes, but I don’t find those likely to have two opposite simple solutions. At least not less than extremely complicated. Which is practically useless. We could never have agreed about something that complicated.
I think all this means is that you find this proof less obvious than some other proofs. That’s fair enough, but finding something difficult to grasp doesn’t mean it’s likely to be wrong.
The way it looks to me: no, it’s not feasible, it’s plainly not feasible, for exactly the reason cousin_it gives; you might as well be asking for three positive integers with x^3+y^3=z^3. (Actually, even more so; I find the cardinality argument here clear at a glance, but Euler’s infinite-descent argument intricate and requiring sustained concentration. But, again, the fact that I can’t just look at it and immediately see why there are no solutions in no way calls into question the proof that there are no solutions.)