> On 5 Mar 2019, at 19:05, John Clark <johnkcl...@gmail.com> wrote: > > It's easy to prove that the Busy Beaver Function grows faster than ANY > computable function because if there were such a faster growing function you > could use it to solve the Halting Problem. So if you're ever in a contest to > see who can name the largest integer in less than 5 seconds just write > BB(9000) and you'll probably win.
Usually, when asked to name a big number, we mean to provide a number that e can compute in a finite time (no matter how long). BB(8000) will be rejected, because it is not a definite description, or name, because BB is not computable. How to name a big number? You can start with a sequence of growing function, like addition, multiplication, expoenntation, iteration, quintation, hexaxion, etc. (each one is just the iteration of the preceding one). Then, calling those functions F_0, F_1, F_2, … F_i, … you can get a “limit” by diagonalising them, which gives a growing total function, as all F_i are total, and it grow more quickly: G_0(n) = F_n(n) + 1, And then a full new sequence G_0, G_1, G_2, … And you can diagonalise on that sequence too, and again and again. This will work on the whole range of the constructive (aka recursive) ordinals. So, if you need to write the description of a big number, the usual method will be to name a big constructive infinite ordinal, like epsilon_zer, for example, although there are much bigger one … and write: F_epsilon-0 (999). See my old post on this published here a (long) time ago, for a more detailed account, going far above epsilon_zero. It is a cute problem we can ask six years old children. Usually some write 9+9+9+9+… +9, on the whole paper. Some write 9*9*9*9*9¨… *9. Later they get better ideas, and can discuss this all along their secondary school. To get the transfinite constructive original, you need the second recursion theorem of Kleene, which is con course much more advanced. O course, if a non computable number number is asked, and still want to win, you can do the same statring from BB(8000), like F_epsilon-0(BB(8000)). But again, that is usually not accepted, because the “name” is not constructive, and usually a name must be a constructive definite description. Bruno > > John K Clark > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
