Re: [Math Noise] (fwd)

[Paul Foley <mycroft@actrix.gen.nz>]

On Sun, 19 Jan 1997 22:34:05 -0600 (CST), Jim Choate wrote:

   Forwarded message:

   > Yes, the Reals can be constructed from the Rationals.  No, the
   > Reals are not a subset of the Rationals.

   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.

There are no "uncountable" numbers -- uncountability is a property of
the set, not an individual member of the set.  The set of reals cannot
be placed in one-to-one correspondence with the integers (strictly,
positive integers, but it amounts to the same thing).

   > Dedekind Cuts are a simple abstraction, often used to construct
   > the Reals from the Rationals in undergraduate calculus courses.
   > Conceptually, one makes a single "cut" in the set of Rationals,
   > dividing it into two parts, all of the members of one part being
   > greater than all of the members of the other.  The number of ways
   > of doing this correspond to the Reals.

   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.

Cut the rational numbers into two sets, A containing all the negative
rationals and all those that have squares less than 2, and B
containing all the positive rationals that have squares greater than
2.  There you have a cut which is 1.41421... (i.e., sqrt(2)).  So you
can define irrational numbers from the rationals (an irrational number
is a cut such that the first set (A) has no largest member and the
second set (B) has no smallest member).

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Paul Foley <mycroft@actrix.gen.nz>       ---         PGPmail preferred

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