Re: Dedekind Cuts, Fundamental Sequences: why?

On Wed, 06 Jun 2007 13:54:33 -0400, Hatto von Aquitanien wrote:
Dave Seaman wrote:

We don't prove definitions.

Read Weyl yourself.

I doubt that Weyl claims to be able to prove definitions. If he does,
then he is seriously mistaken.

The fact that the reals are a complete ordered field can be derived
strictly from concepts related to rational numbers. What other
properties of the real numbers do you have in mind?

Define Pi strictly in terms of rational numbers.

Well, if you insist, pi can be expressed in many ways as the limit of a
Cauchy sequence of rational numbers.

But you are misreading what I said. I specifically did not say that
every real number can be specified, either in terms of the rationals or
by any other means. After all, there are uncountably many reals and only
countably many specifications to go around. What I said is that the set
of real numbers can be defined (and proved to be a complete ordered
field), using the rationals as a starting point. Defining a set does not
mean specifically defining every member of the set.

The set of Cauchy sequences of rationals, with elementwise addition and
multiplication, forms a ring. There is a maximal ideal in this ring,
consisting of all those sequences that converge to 0. The quotient ring
is therefore a field, and it is provably a complete ordered field. This
is what I meant by the statement above about proving the properties of
the reals strictly from the rationals.

"It is probably not necessary to repeat that it would be meaningless to
include among these principles an assertion such as the following: If A
is a property of properties, then one may form that property P_A which
belongs to an object x if and only if there is a property constructed by
means of
these principles and itself possesses the property A. That would be
blatant circulus vitiosis; yet our current version of analysis commits
this error and I consider this ground for censure."

Can you be more specific? Exactly where do you see circularity in the
construction of ther real numbers from the rationals?

I don't believe Weyl is going to respond. He's fairly dead.

I was asking you, not Weyl. You raised the claim, whether it was
original with you or not. Defining a new concept strictly in terms of
some previously existing concept is not circular.

Why not just extent the natural numbers immediately to the positive real
numbers using infinite decimal expression, introduce 0 and negation and
be done with it?

Because it is clumsy and inelegant,

Well, I don't know what the ballance of the development looks like, but the
definition of the set of reals using that approach was fairly clear up to
the point where the translator got the transcription wrong.

and because the integers and the rationals are worthy of study in their
own right.

Indeed. It's a shame the original definition of module was corrupted to
what it has become today.

I am assured that development of the algebra of real numbers is
excessively unwieldy when approached in this way, but I have not seen
that demonstrated.

The definition of addition is quite unwieldy. There is no upper bound on
the length of a carry that might occur. Multiplication is even worse.

I should probably try to find a source which goes through this development
before I comment much further on this, but just thinking through the
process in my head, it doesn't seem that outrageously difficult. But my
approach might not satisfy many mathematicians because I would treat the
problem in much the same way that arbitrary precision math operations are
handled in computer science, and then extend the definitions through
induction.

[snip]

The objective in this case is to achieve completeness. In fact, the
process involving Cauchy sequences is called "completion of the
rationals". Even if you don't fully understand the reason that something
was done, I don't see how that makes it a "loophole".

What I mean by loophole is that we are doing the following, if you will
forgive the informal language: After constructing the rational numbers we
conduct a few experiments and find out that there are expressions which
look as though they should represent (or at least imply) rational numbers,
but they don't. The paradigm is {x:x^2=2}. So someone comes up with the
very impressive demonstration that the set of all rational numbers bounded
from above by x has no supremum. That's all very good and well, except
that I don't have any formal way to talk about the place of x in the
ordering of \Q. Sure, there is an intuitively obvious meaning to the
notion, but it is formally undefined (in the developments I am aware of).

I have already addressed that point. You don't need to know what a real
number is in order to know that the set {x in Q: x^2 > 2} has no glb in
Q. Given any rational lower bound, we can find a larger one.

Once we have finished constructing the reals, we have a perfectly good order
relation on R, which is what we need to talk about the place of x = sqrt(2) on
the real line. If x and y are reals, represented as Dedekind cuts, then by
definition we take x < y to mean that the left set of x is strictly contained
in the left set of y.

So I am intuitively persuaded that there is a "hole" in the rational
numbers, and that the sets of rational numbers bounded by x are "open" at
x. So Dedekind comes along and says: "Hey, why not define the real numbers
as pairs of such disjoint sets?" Mathematicians look at the idea and
think: "Wow! That looks like it fills all the holes". They then play
around with the behavior of closed bounded sets and convince themselves
that such sets of rational numbers follow the same algebraic rules as the
rational numbers. "Problem solved!"

They don't just "convince themselves" that the reals form a complete
ordered field; they *prove* it.

But the problem we were trying to solve was the case where the set of
rational numbers is _not_ closed. "Shshshshs... Don't tell anybody. Just
stick that sqrt(2) thing in the hole and pretend like it belongs there...."

No, that's not what was done. Given either the Dedekind cuts or the
Cauchy sequence construction, we explicitly construct the real number x >
0 such that x^2 = 2. There is nothing up anyone's sleeve. It is no more
mysterious than constructing a rational x such that x + x = 1.


--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
.

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