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.

]]>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. ]]>

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.

]]>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!*

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