From: mpd@netcom.com (Mike Duvos)
Date: Sun, 19 Jan 1997 23:35:25 -0800 (PST)
To: cypherpunks@toad.com
Subject: Re: [Math Noise] (fwd)
In-Reply-To: <199701200456.UAA00404@toad.com>
Message-ID: <199701200735.XAA17096@netcom19.netcom.com>
MIME-Version: 1.0
Content-Type: text/plain


Jim Choate <ravage@EINSTEIN.ssz.com> writes:

 > An arbitrary Real can be constructed from the Rationals. If
 > we accept the proposition, as posed apparently by you and
 > others, of uncountable Reals then your 'assumption' fails,
 > otherwise the 'uncountable' members would be countable.

We can construct the Reals from the Rationals without having to
speak of each specific Real while doing so.  A formal system,
having only a countable number of strings of symbols from its
alphabet, can speak of "The Real Numbers" even though it cannot
speak of "The Real Number X" for every single X in the Reals.

 > A 'limit point' is not the same as 'equal to'. Arbitrarily
 > close is not equivalent, inherent in the definition of a
 > limit point is the concept of 'little o' and 'big o', or
 > worded differently our axiomatic definition of infinity.

In standard analysis, the limit of a sequence A[n] is a value x
such that given any positive epsilon, no matter how small, we can
find a point in the sequence such that all its members after that
point are within epsilon of x.

Such a limit, if it exists, is unique and exactly defined.

"Little o" and "big o" are concepts from complexity theory and I
am not precisely sure why you feel they need to be mentioned.

 > I can, but there is no fundamental rule in mathematics
 > that requires me to ignore the small but distinct
 > difference between the element in the original set and the
 > sequences used to approximate it.

In the general case, the limit point will not be a member of the
sequence which approximates it.  Although every member of the
sequence is a finite distance away from the limit, the limit
itself is, as I previously mentioned, exactly known without any
ambiguity.

 > It is something that must be agreed upon by accepting a
 > particular axiomatic definition of infinity.

An "infinity" is simply the property of being able to be put in
1-1 correspondence with a proper subset of oneself.

 > The number of cuts are 1-to-1 with the Reals, they are not
 > the Reals. There is no way I can make a cut which is
 > 3.1527, only 1-to-1 with the number(s). Important
 > distinction.

Mathematical objects are sets with structure.  We generally
consider two mathematical objects equivalent if there exists a
1-1 correspondence between the respective sets which is structure
preserving. What the actual members of the set are, and how they
were constructed, is usually unimportant.  For all practical
purposes we may refer to any mathematical object isomorphic to
the Reals as "The Reals", without any confusion.

--
     Mike Duvos         $    PGP 2.6 Public Key available     $
     mpd@netcom.com     $    via Finger.                      $