univerz == pavel kropitz (bb6 record holder)

]]>Hmm, if I understand the discussion correctly, the problem is that once you get to transfinite ordinals the method you use to code up the ordinal can contain lots of information, so you can do some prestidigitation to be able to prove every statement. I guess one might then ask how much one can prove if one uses “natural” representations of ordinals, and don’t use any “tricks to code up extra information”. But then the problem is that the terms in quotes are not well-defined.

]]>Cool! Thanks for the update! Like you, I hope someone publishes this work, because it’s a bit risky to have results scattered around like this, a bit hard to assemble, relying on us to trust various pseudonymous authors.

I’m not sure I’d say we’re “extremely close” to solving BB(5). It all depends on how hard the remaining 17 math problems are. If they’re as difficult as the Collatz conjecture, it could take centuries to solve them.

]]>We discussed some further results on Scott Aaronson’s blog. You can find Daniel Briggs’ website on web.archive.org:

https://web.archive.org/web/20121026023118/http://web.mit.edu/~dbriggs/www/

In particular he lists which of Skelet’s 42 HNR machines he has resolved:

https://web.archive.org/web/20121109222914/http://web.mit.edu/~dbriggs/www/Turing/proofs/record.txt

Apparently he resolved 23 machines with the help of someone named “univerz”

Another person who worked on Skelet’s HNR machines was Cloudy176 from the Googology Wiki, resolving 14 of the 42 machines using a computer program:

http://googology.wikia.com/wiki/Forum:Sigma_project#Confession_.28and_proofs_for_14_HNRs.29

There’s a lot of overlap, but combined they appear to have resolved 26 of the 42 and leaving 16 unresolved. However, Briggs notes that one of the machines labelled “BL_2” should also be classified as an HNR machine, bringing the total back up to 17.

That is, if you trust Skelet, Briggs, univerz, and Cloudy176. Probably some extra verification will be necessary.

It does seem like we are extremely close to solving BB(5)!

]]>http://www.randform.org/blog/?p=6184

The application itself (programmed by Tim) is however intended to be also used with other sensory input (the most banal is of course again the keyboard, or buttons and mouse, and in fact those were of course used during prototyping the application). The experiment is summarized in a blog post:

http://www.randform.org/blog/?p=6184

Some example of “dance gestures” are also given there. ]]>

I’m just saying that if there’s a derivation of F = T in the predicate calculus, we can probably derive almost anything in any system of logic, so we can’t trust the arguments usually given to prove the consistency of mathematical systems.

]]>The following paper, due to appear in the December issue of *Cognitive Systems Research*, gives a finitary proof of consistency for the first-order Peano Arithmetic PA as sought by Hilbert in the second of his twenty three Millenium ICM-1900 problems:

Ergo, the first-order predicate calculus FOL too is consistent in Hilbert’s sense, which is how I believe the term ‘consistency’ is usually understood when referring to a formal mathematical system.

If the propositional calculus is ‘inconsistent’, it may be in some sense of the word ‘consistent’ other than that which I believe Hilbert had in mind.

]]>I don’t know any papers about it. There should be some philosophy and logic papers about the concept of “logical certainty”, and the question of when we can trust a consistency proof: they are well-known issues that have been discussed for a long time. But I don’t know any papers raising doubts about the usual consistency proofs for the propositional calculus.

Personally I think the consistency of very fundamental principles like the propositional calculus, the predicate calculus and Peano arithmetic is widely believed for the following reasons: 1) they haven’t yet led to any contradictions, and 2) we can prove certain subsets of these principles won’t lead to contradictions if we assume certain other subsets won’t lead to contradictions. So, everyone has decided that these principles are consistent. And I think that’s fine — *as long as we notice what we are doing!*

John, the statement that the propositional calculus could be inconsistent is very interesting (almost disturbing :-)). Is there anything more to be said about it, aka are there any papers about it?

]]>I am thinking that there are many Turing machine that have the same BB(N). For example a machine that make the same operation to the right instead of the left, or if there is an initial tape full of 1s (an arbitrary redefinition of starting tape and there is a busy beaver game that have the same dynamic with opposite symbols), or a Turing machine with permutation of state symbols (or a combination of all these conditions); but if this is a dynamic, so is possible increase the BB(N) with an arbitrary initial condition, an initial string with some 1s and 0s, of finite lenght? If the initial string is not a string that the Turing machine write on the tape starting from the empty tape, then it is possible to increase the length of BB(N).

If the Turing machine does not halt, then there is a periodicity in the written string with written states (another tape).