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Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: Finitely definable reals.
Posted: Jan 14, 2013 5:39 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 14 Jan., 14:47, Dick <DBatche...@aol.com> wrote:
> > On Sunday, January 13, 2013 4:47:29 PM UTC-5, WM wrote:
> > > On 13 Jan., 22:13, Dick <DBatche...@aol.com> wrote: > On Friday, January
> > > 11, 2013 4:16:39 AM UTC-5, 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.
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