Well, what's a computation? Is y=x a computation, where x is a number, including an irrational? I thought we were talking about mathematical numbers?
Proof of my other assertions: an interval (x,y) (or [x,y) or (x,y] or [x,y]) has a 1-1 mapping onto (0,1), and in (0,1) there are a countable number of rationals and an uncountable number of irrationals. The mapping is a linear mapping and the parameters are easy to figure out. On Sat, Mar 30, 2019 at 6:22 PM William Tanksley, Jr <[email protected]> wrote: > Roger Hui <[email protected]> wrote: > > I assume you are talking about mathematical numbers rather than > > finite-precision floating point representations of same. In that case, > the > > first claim (countable number of irrationals can appear as the limit of > > computations) is false. > > There are only countably many computations -- Godel's correspondence. > So there are only countably many irrationals which can be the limit of > a computation (that is, a non-terminating computation, such as > Chatin's Omega). Chatin proved this result, on the way to proving that > the only irrationals which are uncountable are the ones we cannot work > with in any way. > > > The second claim (some pairs of rationals have no > > irrationals between them) is false. > > Good proof, I'm persuaded. > > -Wm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
