111–120 of 232 posts
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The number itself is not independent of ZFC. (Every integer can be expressed in ZFC.) What's independent of ZFC is the process of computing BB(748).
I think the more correct statement is that there are different models of ZFC in which BB(748) are different numbers. People find that weird because they don't think about non-standard models, as arguably they shouldn't.
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
No individual number is uncomputable. There’s no pair of a number and proof in ZFC that [that number] is the value of BB(748). And, so, there’s no program which ZFC proves to output the value of BB(748). There is a program that outputs BB(748) though, just like for any other number.
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Couldn't you make the same argument for sqrt(2), or better yet for zero [0]? [0] https://en.wikipedia.org/wiki/Zero:_The_Biography_of_a_Dange...
For sqrt(2) I can tell you the order of magnitude and output as many digits as you want. I think that's plenty specific for this use case. For zero I can not only do that, I can also count to it if you let me count both up and down, which seems like a very simple ask.
But as someone in this generation, I see a good argument for rejecting the big busy beaver numbers, which are provably outside of the realm of calculating with all the resources of our universe's runtime, from being fully accepted as numbers, any more than the first uninteresting number [0].
[0] https://en.wikipedia.org/wiki/Interesting_number_paradox
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Not an expert, but I've read about this a bit because it bothered me also and I think this is the answer: Most of these 'uncomputable' problems are uncomputable in the sense of the halting problem: you can write down an algorithm that should compute them, but it might never halt. That's the sense in which BB(x) is uncomputable: you won't know if you're done ever, because you can't distinguish a machine that never hal…
I am also not an expert, but this does not sound right to me. Godel's incompleteness theorem shows that there are certain things that cannot be proven. Being independent of ZFC means that something is such a case. So BB(643) being independent of ZFC means that we cannot prove or disprove that a certain number is BB(643). Aka we don't have the math to know for certain.
ZF & ZFC are as important as they are because they're the weakest set theories capable of working as the foundations of mathematics that we've found. We can always add axioms, but taking axioms away & still having a usable theory on which to base mathematics is much more difficult.
Earlier quoted context omitted.
No individual number is uncomputable. There’s no pair of a number and proof in ZFC that [that number] is the value of BB(748). And, so, there’s no program which ZFC proves to output the value of BB(748). There is a program that outputs BB(748) though, just like for any other number.
Individual numbers can be uncomputable! For example, take your favorite enumeration of Turing machines, (T1, T2...) and write down a real number in binary where the first bit is 0 if T1 halts and 1 otherwise, second bit is 0 if T2 halts... clearly this number is real and between 0 and 1, but it cannot be computed in finite time.
Earlier quoted context omitted.
No individual number is uncomputable. There’s no pair of a number and proof in ZFC that [that number] is the value of BB(748). And, so, there’s no program which ZFC proves to output the value of BB(748). There is a program that outputs BB(748) though, just like for any other number.
Individual numbers can be uncomputable! For example, take your favorite enumeration of Turing machines, (T1, T2...) and write down a real number in binary where the first bit is 0 if T1 halts and 1 otherwise, second bit is 0 if T2 halts... clearly this number is real and between 0 and 1, but it cannot be computed in finite time.
Earlier quoted context omitted.
The number itself is not independent of ZFC. (Every integer can be expressed in ZFC.) What's independent of ZFC is the process of computing BB(748).
I think the more correct statement is that there are different models of ZFC in which BB(748) are different numbers. People find that weird because they don't think about non-standard models, as arguably they shouldn't.
Suppose model A proves BB(748) = X and model B proves BB(748) = Y > X. But presumably the models can interpret running all size 748 Turing machines for Y steps. Either one of the machines halts at step Y (forming a proof within A that BB(748) >= Y contradicting the assumed proof within A that BB(748) = X I'm guessing the only way this could ever work would be some kind of nastiness like X and Y aren't nailed down integers, so you can't tell if you've reached them or not, and somehow also there's a proof they aren't equal.
It boggles my mind that a number (an uncomputable number, granted) like BB(748) can be "independent of ZFC". It feels like a category error or something.
I wonder if the visible universe is large enough to write down the exact value of BB(6).
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What makes BB(748) independent of ZFC is not its value, but the fact that one of the 748-state machines (call it TM_ZFC_INC) looks for an inconsistency (proof of FALSE) in ZFC and only halts upon finding one. Thus, any proof that BB(748) = N must either show that TM_ZF_INC halts within N steps or never halts. By Gödel's famous results, neither of those cases is possible if ZFC is assumed to be consistent.
I think what's most unintuitive is that most (all?) "paradoxes" or "unknowables" in mathematics involve infinities. When limiting ourselves to finite whole numbers, paradoxes necessarily disappear. BB(748) is by definition a finite number, and it has some value - we just don't know what it is. If an oracle told us the number, and we ran TM_ZFC_INC that many steps we would know for sure whether ZFC was consistent or n…
We already have more powerful systems, but what causes the inability to self-reason is exactly that power: only first order logic can prove its own consistency. Once you get powerful enough to model arithmetic, you can build statements with self-referential weirdness.
I don’t see it as a paradox, but as growth: a sufficiently rich system can pose questions which suggest a richer system — and thereby scaffold up the universe hierarchy.