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From: ellis@spar.UUCP (Michael Ellis)
Newsgroups: net.philosophy,net.math
Subject: Ineffable numbers
Message-ID: <667@spar.UUCP>
Date: Thu, 21-Nov-85 09:17:01 EST
Article-I.D.: spar.667
Posted: Thu Nov 21 09:17:01 1985
Date-Received: Sat, 23-Nov-85 06:15:16 EST
References: <2467@sjuvax.UUCP> <11700014@inmet.UUCP>
Reply-To: ellis@spar.UUCP (Michael Ellis)
Organization: Schlumberger Palo Alto Research, CA
Lines: 53
Xref: watmath net.philosophy:3194 net.math:2553
Summary: 

{
    For newcomers, `ineffable' numbers are those which cannot be
    named. The naming convention may include any of the common methods
    used in math, including a convention for specifying infinite
    summations, taking the root(s) of arbitrary polynomials, 
    evaluating the solutions to integrals or differential equations
    or whatever you might wish!

    The assertion is that the vast majority of real numbers are
    ineffable, since there are only denumerably many names but
    an uncountable number of real numbers.
}
>Here is a more elegant  and  direct  treatment  of  Todd  Moody's
>"ineffable  number" model than in my previous response.  The con-
>struction is by Bill Homer, of Intermetrics, Inc.   Whatever  the
>language  for  number  naming,  it  produces  a unique unnameable
>("ineffable") number, thus refuting Todd's thesis that an example
>of an individual ineffable number cannot be given. (If the naming
>language is powerful enough to  include  the  construction  itself,
>this is a paradox).

>The construction is a straightforward application of Cantor's diagonal
>method: Consider "nameable" numbers in the interval (0,1).  Since the set is
>countable, assign each number a positive integer. This can be done in a
>constructive way, say, by ordering the *names* lexicographically and in the
>order of increasing length, and discarding extra names for the same number.
>Now consider the decimal notation of each number.

>Construct the number X such that the i'th decimal digit of i equals the i'th
>decimal digit of the i'th number plus 1 mod 10. Then X cannot be an element
>of the set of nameable numbers and is, therefore, ineffable.  (There is a
>tiny snag with numbers that have two notations, e.g, 0.1000... and 0.0999...
>; but it is so easy to fix that I won't spell it out). -Jan Wasilewsky

    You've just proven that any set of effables E(0) can be expanded into a
    larger set of effables E(1), by a complex Cantorian diagonal process
    you mentioned which is equivalent to expanding the naming convention.

    Indeed, you could continue this process infinitely, producing E(2),
    E(3)... But the size of your list would still always be denumerably
    infinite, and, as the order of numbers in the interval (0,1) has
    the power of the continuum (which is a vastly larger infinity)
    there would always be an uncountable infinity of numbers in (0,1) 
    that are not included in any E(n).

    In other words, the particular set of `effables' is determined by
    the naming convention used. This set of effables can always
    be expanded by incorporating improvements into the naming convention,
    but there always remain ineffables that have not been included.

    Thus, I still believe in Todd Moody's ineffables.

-michael