Gleason's Theorem, proven in 1957, says that the Born rule is the only one that is unitary, the only one where all the probabilities add up to exactly 1. So if you want probabilities its got to be proportional to the square of the magnitude of the particle's wave-function and not the cube or...
The key is what "collapse of the wave function" means physically, I think it means there is something special about one particular state of that function. The Many World's Interpretation says no state is special, nothing collapses, and everything that can happen does happen. This is how I think...
If ZFC+Con(ZFC) can figure out what BB(7918) is why can't ZFC alone figure out what BB(5) is? It is after all VASTLY smaller. No matter how many times you say "ZFC is consistent" it won't help you determine if a given Turing Machine will halt, and that's what you need to do to figure out what...
If all you want is a proof there is a far far easier way to get one than adding an infinite number of iterated axioms, you can get a very simple proof by just adding one axiom, “BB(7918) = 42”. But I tend to think truth is more important than proof.
John K Clark
You can't get something from nothing and when you add Con(ZFC) to ZFC you're really adding nothing, all you're doing is saying ZFC is consistent without any evidence to back it up . A critic could simply say "oh yeah, says you!". I don't see the point of having a proof if it can't be trusted...
I don’t see what additional power adding Con(ZFC) would bring other than being able to prove the consistency of ZFC, and if it solved that problem it would immediately create another one that is too obvious to point out; it certainly wouldn’t help in figuring out what large but finite number...
I agree that if we add some stronger axiom to ZFC the question "what is BB(7918)?” can be answered, for example it can be answered in the system ZFC+X if X is “BB(7918) = 42”. But I don’t just want answers, I want correct answers.
John K Clark
An axiom should be as simple and self evidently true as possible and Con(ZFC) is certainly not simple, and although I think its probably true it would be pushing it to say the truth of it was self evident. Con(ZFC) is just the claim that ZFC is consistent, sticking that in as an axiom does not...
That was an interesting link, thanks. But if a "much weaker system” than ZFC is good enough for "virtually all mainstream mathematics” and if ZFC can’t determine what BB(7918) is then a much weaker system certainly can’t either. That just gives me more confidence that Aaronson was correct when...
I agree that with stronger axioms the question could be answered, but could it be correctly answered? Most people are pretty confident that ZFC is consistent, me too, but if more axioms are added my confidence drops fast. If a Turing Machine is connected to a bomb that will go off if it ever...
Regardless of what reasoning system is used a given Turing Machine will either halt or it won’t, so the Busy Beaver number always exists, the question is if we can ever know what it is. We know for sure that BB(5) is at least 47,176,870 because one 5 state Turing Machine has been found that...
The 7918th Busy Beaver number is huge but it’s finite and a natural number, but Scot Aaronson proved it is not computable, no algorithm can find it. I wanted to know if its possible to find the smallest non-computable Busy Beaver number or is that non-computable too. I wouldn’t be very surprise...
Why wouldn’t it be meaningful to say a Real Number is not computable if no finite algorithm exists that can approximate it with arbitrary precision?
John K Clark
We know for sure the first four Busy Beaver numbers exist are finite and are computable because they have in fact been computed, they are 1,6,21 and 107. And we know for sure that the 7918th Busy Beaver number exists and is finite but is NOT computable thanks to the work of Scot Aaronson, but...
How did Einstein manage to get by with just 10? Did he cut corners or are some of those spacetime curvatures unphysical and so are of interest to a mathematician but not a physicist?
John K Clark