Virgil
Posts:
9,012
Registered:
1/6/11


Re: Finitely definable reals.
Posted:
Jan 14, 2013 5:39 PM


In article <2b3f683c0b794c6ea25c6e2a1f9c970b@bx10g2000vbb.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> 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. > But no one ha ever claimed that infinite strings need be "meaningful", just that certain infinite strings can exist. 

