JohnF
Posts:
97
Registered:
5/27/08
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Re: Finitely discernible reals
Posted:
Jun 18, 2012 12:43 AM
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In sci.math Zuhair <zaljohar@gmail.com> wrote:
> I want to define two kinds of Reals.
Please see the book,
Computability in Analysis and Physics,
Marian B. Pour-El and Jonathtan I. Richards,
Springer-Verlag 1989, ISBN 0-387-50035-9
which discusses all this kind of stuff you've mentioned
in great detail, and many other ideas about computable reals
(and Banach spaces, etc) that you apparently haven't thought
of yet.
> Those that are finitely discernible to be irrationals, and those that
> are not.
>
> We will only consider reals that are represented by infinitely many
> decimals of the form 0.d1d2d3.......
>
> Now a real in the above sense is said to be finitely discernible to be
> irrational if for some i the segment 0.d1...di cannot be an initial
> decimal segment of an infinite decimal presentation of a rational
> number.
>
> Now to explain this, the infinite decimal presentation of a rational
> number always has a repeated moiety *starting* by some i_th decimal
> that cannot be broken after and so if we have an infinite decimal
> representation of some number where this repeated moiety is broken at
> some j_th decimal, then we can decide that this infinite decimal
> representation is of an irrational number. For example
> 0.998998998.....
> actually for any sequence of decimals d1d2....... the number
> 0.998d1d2..... is an irrational number.
>
> Obviousely we have uncountably many reals that are discernible
> finitely to be irrationals.
>
> But what about Reals that are not finitely discernible to be
> irrationals, those would be reals of the form that for every i the
> the segment 0.d1...di is part of the initial segment of some infinite
> decimal representation of a rational number.
>
> Now I have two questions.
>
> (1) how many reals do exist of the later type? are they countably
> many?
>
> (2) is the set of reals in (1) complete? i.e. the limit of every
> ascending sequence of them is one of them?
>
>
> Zuhair
>
--
John Forkosh ( mailto: j@f.com where j=john and f=forkosh )
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