
Re: Finitely definable reals.
Posted:
Jan 14, 2013 1:34 PM


On 14 Jan., 14:47, Dick <DBatche...@aol.com> wrote: > On Sunday, January 13, 2013 4:47:29 PM UTC5, WM wrote: > > On 13 Jan., 22:13, Dick <DBatche...@aol.com> wrote: > On Friday, January 11, 2013 4:16:39 AM UTC5, zuhair wrote: > > Lets say that a real r is finitely definable iff there is a predicate P that is describable by a Finitary formula that is uniquely satisfied by r. Formally speaking: r is finitely definable > > I think this would be more helpful if "finitely definable" were defined more carefully. That is simple. A finitely definable item has a finite definition. A finite definition is a definition consisting of a natural number of characters of a finite alphabet of your choice or even of your construction in a language of your choice or even of your construction. No computers or Turing machines required. Everybody can understand the definition. Regards, WM > > This is not true. Given an arbitrary string of characters it is impossible to determine whether it is a meaningful statement or not.
But given a meaningful statement it is simple to determine that it is a finite string. Infinite strings cannot be read.
> The language of Turinh Machines  or abacus machines  or logic statements is a way to bring order to this. It remains (recursively) undecidable whather one of these constructions is meaningful or not. However, it allows one to determine that some are meaningful. More, by introducing the ides of an oracle (even though an oracle is admittedly impossible) it allows on to move forward into the Kleene hierarchy. > Dick
You cannot determine whether an infinite string is meaningful, because you cannot read it to the end. After everything you have read a joke or negation or complete nonsense may follow.
You are right, not all finite strings are meaningful. But all meaningful strings are a subset of all finite strings. That is sufficient to know that all meaningful strings are countable.
Regards, WM

