Re: Dedekind Cuts, Fundamental Sequences: why?

Hatto von Aquitanien wrote:

quasi wrote:

On Sun, 03 Jun 2007 03:26:43 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

quasi wrote:

On Sat, 02 Jun 2007 19:34:51 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> wrote:

quasi wrote:

Or Pi.

With only rationals, you can get below Pi or above Pi but not exact.
If you want the area of a circle, you need to patch the hole. Might
as well patch all such holes simultaneously. Dedekind cuts is one way
to detect missing numbers. Cauchy sequences is another. Either
strategy, starting with the rationals, yields the reals.

I someone were to ask me in casual conversation if Pi is a real number
I
could say yes. But we are working tabular rasa here. Why should I
want to
admit Pi into the set of things I call numbers? Why should I think Pi
isn't already covered? Before I admit Pi as a number, what is Pi?


Given a circle, you can inscribe a regular n-gon. You can also
circumscribe a regular n-gon. When n is power of 2, the perimeter of
such regular n-gons can be found exactly using only rational numbers
and square roots. Square roots can be bounded, to arbitrary precision,
by rationals. Thus, you can bound the circumference of a circle, to
arbitrary precision, by rationals.

One can show that all circles are similar, hence the ratio of the
circumference of a circle to its diameter is independent of the choice
of circle.

If one then defines Pi as the ratio of the circumference of a circle
to its diameter, we have the concept of Pi as a number (a ratio of two
lengths).

Using Calculus, one can prove that Pi is irrational. Thus, starting
with just the rationals, Pi is a missing number.

quasi

So your argument that Pi is a number has to do with a concept of length.
It
seems to entail a necessary concept of continuity. Not only is there a
concept of linear continuity, but there is a concept of algorithmic
continuity in the sense that the algorithm can be executed continuously
to
an arbitrary refinement. You might argue that the latter is already
assumed from Peano, and therefore not a newly introduced assumption. I'm
not convinced that the assumption of continuity even in the recursive
sense doesn't presuppose what some authors assert they have proved.

I'm not sure if I understand the intent of your objections.

The reals were designed to model physical measurements such as
lengths.

I'm not sure all mathematicians will agree with that.

If you assume that every length can be represented by a number, then
sqrt(2) exists by the Pythagorean Theorem. The number Pi exists, by
definition, as the ratio of the circumference of a circle to its
diameter.

The numbers and their axioms came _after_ the underlying physical
concepts, and with the goal of modeling those concepts.

quasi

The way I see it, the attempts to exactly define the reals arose from the
desire to exactly define what we mean by multiplying two real numbers.
Pickert and Görke assert that the reals could be defined in terms of
infinite decimal sequences. They then assert that such a definition would
make proving the theorems necessary to show the reals properly extend the
rational numbers prohibitively difficult. I guess I'm having some trouble
understanding why that would be necessarily difficult, but I haven't
attempted to write out rigorous proofs either.

There is a convergence of two distinct ideas in this problem. We have the
idea of a geometric continuum, and the idea of discrete numbers. Cuttable
things, and countable things. What I really mean by cuttable
is "measurable" in the sense of pre-cision. Numbers are not cuttable.
When placed in the context of physical continuum, they represent
unextended locations on the continuum.

Part of why this has really come to bother me is because of what Weyl did
with vector spaces. Rather than extending vector addition to become
scalar multiplication, he chose to proclaim axioms:

"Gemäß dem Prinzip der Stetigkeit nehmen wir sie auch für beliebige reelle
Zahlen in Anspruch, formulieren sie aber ausdrücklich als Axiome, da sie
sich in dieser Allgemeinheit rein logisch nicht aus den Additionsaxiomen
herleiten lassen. Indem wir darauf verzichten, die Multiplikation auf die
Addition Zurückzufüren, setzen sie uns in den Stand, aus dem logischen
Aufbau der Geometrie die schwer zu greifende Stetigkeit ganz su
verbannen."

"In accordance with the principle of continuity we shall also make use of
them for arbitrary real numbers, but we purposely formulate them as
separate axioms because they cannot be derived in general form from the
axioms of addition by logical reasoning alone. By refraining from reducing
multiplication to addition we are enabled through these axioms to banish
continuity, which is so difficult to fix precisely, from the logical
structure of geometry."

Stetigkeit -> continuity.

Ironically, my online German-English dictionary translates Stetigkeit
as "consistency".

"By refraining from reducing multiplication to addition we are enabled
through these axioms to banish consistency, which is so difficult to fix
precisely, from the logical structure of geometry."

I'll have to finish this later.


<quote>
Two vectors \a and b uniquely determine a vector \a + b as their sum. A
number lambda and a vector \a uniquely define a vector lambda \a, which
is "lambda times \a" (multiplication). These operations are subject to the
following laws:-
(alpha) Addition-
(1) a + b = b + a (Commutative Law).
(2) (a + b) + c = a + (b + c) (Associative Law).
(3) If \a and c are two vectors , then there is one and only one value of x
for which the equation a + x = c holds. It is called the difference
between c and \a and signifies c - a (Possibility of Subtraction).

(beta) Multiplication-
(1) (lambda+mu) a = (lambda a) + (mu a) (First Distributive Law).
(2) lambda (mu a) = (lambda mu) a (Associative Law).
(3) 1 a = a.
(4) lambda (a + b) = (lambda a) + (lambda b) (Second Distributive Law)

For rational multipliers lambda, mu, the laws (beta) follow from addition if
multiplication by such factors be *defined* from addition. In accordance
with the principle of continuity we shall also make use of them for
arbitrary real numbers, but we purposely formulate them as separate axioms
because they cannot be derived in general form from the axioms of addition
by logical reasoning alone. By refraining from reducing multiplication to
addition we are enabled through these axioms to banish continuity, which is
so difficult to fix precisely, from the logical structure of geometry. The
law (beta) 4 comprises the theorems of similarity.

(gama) The "Axiom of Dimensionality", which occupies the next place in the
system will be formulated later.
</quote>
--
http://www.dailymotion.com/video/x1ek5w_wtc7-the-smoking-gun-of-911-updated
http://911research.wtc7.net
http://vehme.blogspot.com
Virtus Tutissima Cassis
.

Relevant Pages

  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
  • Re: counter example in analysis
    ... Cantor's interpretation of reals like more than infinitely many numbers ... I asked myself whether or not the singular point can have any meaning if ... at least unless EB can provide some definition of continuity of his own ... are restricted to only the rationals. ...
    (sci.math)
  • Re: Does the Calculus rest on Euclid?
    ... axioms I have assumed at the outset. ... Now you define the reals as ... A real number is a set S of rationals with the following properties: ... To say that you can order things implies a concept of a continuum. ...
    (sci.math)
  • Re: counter example in analysis
    ... Cantor's interpretation of reals like more than infinitely many numbers ... I asked myself whether or not the singular point can have any meaning if ... at least unless EB can provide some definition of continuity of his own ... are restricted to only the rationals. ...
    (sci.math)