141–146 of 146 posts
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Excuse me?
I’m saying that all you’ve proven is that if you know a priori whether ZFC is consistent, you can construct a Turing machine that returns this value. If you consider that to be window dressing I don’t know what else I can tell you.
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His point is if you know the value of BB(748) then you don't have to wait forever, just BB(748) steps, as after that the Turing machine is guaranteed not to halt. The problem with his argument is that we don't know the value of BB(748). Not only that, it is incomputable, which resolves the contradiction.
It is categorically false that BB(748) is not computable. On the contrary any particular BB(n) can be computed by some Turing Machine even though there is no Turing Machine that can compute BB(n) for every n.
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It is categorically false that BB(748) is not computable. On the contrary any particular BB(n) can be computed by some Turing Machine even though there is no Turing Machine that can compute BB(n) for every n.
So then what's stopping you from running the BB(748) machine, getting the number, then running the ZFC machine and proving ZFC consistent or not?
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> I think the point here is that it was not known whether all 3,3 halting problems are reducible to trivial Collatz-like problems. True, but it still isn't known either way. Nothing changed in that regard.
'trivial' isn't a mathematical concept. If no one has shown why this problem is trivial, it isn't trivial.
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Yes. My reasoning applies to a function n=> BB(n) Isn't that what "the function is not computable" is about? Or is the thesis that the value of BB(748) can't be computed?
BB(n) for any particular n is always computable, no exceptions. There simply is no single computable function that can compute BB(n) for every n, but for any particular n there absolutely is a TM that computes it.
The same applies to more limited settings too, e.g. boolean questions like whether the Collatz conjecture holds: I know that one of the following programs calculates the right answer, but I don't know which:
- PRINT "true"; HALT
- PRINT "false"; HALT
(Perhaps we should also allow `PRINT "independent of the given axioms"; HALT` as well!)
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'trivial' isn't a mathematical concept. If no one has shown why this problem is trivial, it isn't trivial.
This problem isn't trivial, but neither are the ones we could already reduce BB(3, 3) to (via Conway's construction).