[Math Noise] (fwd)

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.

> Er, no.  But you can create a number which is not representable
> as a fraction as a limit point of very many fractions.

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.

> If we then consider equivalence classes of those Cauchy sequences
> which converge to the same limit, and consider an element of the
> original set to correspond to the class containing the sequence
> all of whose members are that element, we can consider the
> classes to form a "completion" of the original set by addition of
> all its limit points.

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

> 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.


                                                  Jim Choate
                                                  CyberTects
                                                  ravage@ssz.com


"The best lack all conviction, while the worst are full of passionate
 intensity."

                                                     Yeats

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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Jim Choate allegedly said:
> 
[...]
> 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.

They are 1-1 with the Reals, you can define all the operations on 
them that are defined for reals, for every representable real (like 
3.1527) there exists a DC.  You have some idea, perhaps that there 
are "real" reals, as opposed to the various constructs for defining 
them?  Perhaps you could define what a "real" real would be?

-- 
Kent Crispin				"No reason to get excited",
kent@songbird.com,kc@llnl.gov		the thief he kindly spoke...
PGP fingerprint:   5A 16 DA 04 31 33 40 1E  87 DA 29 02 97 A3 46 2F

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.                      $