Re: Peano curve is wrong

On Thu, 10 Apr 2008 04:01:25 -0700 (PDT), qiuzhihong
<qiuzhihong@xxxxxxxxx> wrote:

Peano Curve And Uncountability Of The Real
Number System


1 Cantor's definition of real number

Russell's paradox points out that a set which is a member of itself
will lead
to some paradox.

The idea that this has something to do with your
problems understanding how the Peano curve works
is fascinating. Let's see:

nalyse the definition of real number given by Cantor
[1]:

"The first section of Cantor's paper of 1872 was devoted to the real
numbers,
in particular to the irrationals. In a paper published somewhat later,
he stressed explicitly the objections he had to previous attempts to
define
irrational numbers in terms of infinite series.

Thus Cantor set himself the task of developing a satisfactory theory
of the
irrationals which in no way presupposed their existence. He followed
Weierstrass
in beginning with the rational numbers. The set of all rationals (A),
including zero, thus provided a foundation for all further concepts of
number.

Cantor then was explicit that whenever he spoke of a numerical
quantity in
any further sense he did so above all in terms of infinite sequences
of rational
numbers {an}:

Definition: The infinite sequence a1, a2, ..., an, ... is said to be a
fundamental
sequence if there exists an integer N such that for any positive,
rational valuve
of e, |an+m ? an| < e, for whatever m and for all n > N.

No, that is _not_ the definition! And I'm not just talking
about the notation "|an+m ? an| < e" being garbled
(this should be |a{n+m} -an| < e). The definition is this:

Definition: The infinite sequence a1, a2, ..., an, ... is said to be a
fundamental sequence if for any positive, rational value
of e, there exists an integer N such that |an+m ? an| < e, for
whatever m and for all n > N.

The difference in the order of the words makes a huge difference
in the meaning - the fact that you wrote the _definition_ wrong
simply shows that you're not reasoning about things nearly
carefully enough.

If a sequence {an} satisfied this condition, then Cantor said that
"the sequence
has a definite limit b." This had a very special meaning for Cantor
and must
be taken as a convention to express, not that the sequence {an}
actually had
the limit b, or that b was presupposed as the limit, but merely that
with each
such sequence {an} a definite symbol a was associated with it. At this
point
in his exposition Cantor was explicit in using the word
"symbol" (Zeichen) to
describe the status of b.

This may have been an axiom or assumption for Cantor.
Since then there have been explicit _constructions_ of
the reals (Dedekind cuts, for example) so that one does
not need to assume that any "fundamental sequence"
has a limit, one can _prove_ this fact.

. . .

A rational number a (where a could be defined by the constant sequence
{a},
which was clearly a fundamental sequence and for which the a was
actually
the value reached by the sequence in the limit).

. . .

The definitions to which Cantor referred were those for which
arithmetic operations
might be extended from the domain of rational numbers to the new
numbers b (Cantor now called them numbers instead of symbols).

. . .

Consequently an element b 2 B should be considered as a number only
for the
sake of convenience. Ultimately it only represented a fundamental
sequence. "

Obviously, Cantor used a rational number set to define a real number.
According
the definition, a real number is equal to a rational number sequence,
that is:

(1). In the real number field, a rational number a is equal to a
sequence {a};

(2). In the real number field, a irrational number b is equal to a
sequence
{b1, b2, . . . , bn, . . .}.

According the set theory, a number sequence is correspond to a number
set
rather than a number member. Therefore, the definition of rational
number in
the real number field, a rational number a is equal to a sequence {a}.
It will
construct a set that containts itself: number a is equal to a set, the
set has an
element that a itself. It will lead to the Russell's paradox.

Oh, so that's the connection.

You're simply confused by slightly informal terminology.
When people say that the rational a is equal to the sequence
{a, a, a, ...} they don't mean that literally. Speaking more
carefully, if a is a rational then {a, a, ...} is the corresponding
real number.

(People who know some math: Yes, I understand that that's
not quite correct either...)

Though Cantor's definition of irrational number didn't contain such
paradox,

Ah, that's a relief.

but at times it is misunderstanded as the form that contain some
paradox.
It's meaningful to talk about how to avoid such misunderstanding.

In the definition, a irrational number b is associated with a
fundamental sequence
{bn}, and all the members in the fundamental sequence {bn} is a
rational number. Obviously, according the definition, the irrational
number b
as the limit of the fundamental sequence is not in the fundamental
sequence.

It can be proved by using the induction that the limit not in the
fundamental
sequence has some mathematical reason, rather than subjective
definition.

For example, the fundamental sequence associated with Pi:

{Pn} = {3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, . . .}

Let m be an integer that 0 <= m <= 9, write the numbers in the
sequence
in the form of the sum of the previous number and a decimal fraction,
for
example, 3.14 = 3.1 + 0.04.

(1)P1 = 3.1, is a rational number; and 101 is an integer. m/101 is a
ratio of
two integers, according the definition, it's a rational number.

(2)For any n belong to N, assume that 10n is an integer, then 10(n+1)
is an
integer, and m/10(n+1) is a ratio of two integers, it's a rational
number. Assume
that Pn is a rational number, then P(n+1) = Pn+m/10(n+1), is the sum
of two
rational numbers, is a rational number.

(3)Then for any n belong to N, Pn is a rational number.
What the induction proved is the attribute of a infinite system that
has a
one-to-one mapping with the natural number system. As a irrational
number,
the limit of the fundamental sequence is not in the system.

"Not in the system"? Not in what system. An irrational is not
in the system of rationals. So what? It is in the system of reals.

2 Peano Curve and uncountability of the real number system

In 1877, Cantor proved in theory that there is a one-to-one mapping
between
a line and a plane. In 1890, Peano gave a space-filling curve, named
Peano
curve; and in 1891, Hilbert given an example of such curve, named
Hilbert
curve [1].

This section should talk about Peano curve first. The Peano curve is
generated
by continually split a square into fourths, as Figure.1, after
infinite steps, the
result of this curve is a Peano curve [2].

It's impossible to tell whether this is you speaking or you
quoting someone else. Saying "after infinite steps the result
is the Peano curve" is simply _wrong_. What's true is that
the Peano curve is the _limit_ of the curve constructed at
the n-th step.

Fig. 1. Peano Curve

GREG BREINHOLT and CHRISTOPH SCHIERZ gave a computer recursion
algorithm to generate Peano curve [3]. Though the methods of curve
generation
given by Peano is split every square into smaller squares, it seems a
parallel processing, due to the number of squares produced by every
split is finite,
the nth division produces 4n smaller squares, it can be prescribed a
serial
order to produce those squares one by one. It will change the parallel
processing
generation to a serial one. It means the cure that generated by the
computer
recursion algorithm given by GREG BREINHOLT and CHRISTOPH
SCHIERZ is strictly equal to the curve given by Peano, every point
that can
be covered by the Peano curve also can be covered by the curve
generated by
the computer recursion algorithm.

There is a problem. Draw a coordinatized line across the median line
of the
initial square. Assume that the interval in the square is [0, 1]. Then
generate
the Peano curve according the recursion algorithm step by step.
Therefore we
can number all the points of intersection one by one according the
order of
generation.

And we should note that _many_ points will now be labelled with
the same number. Also that most points will never get labelled,
because they're not in the image of the curve at the n-th step
for any n.

If the Peano curve covers all the points in the square,
the points of
intersection between the Peano curve and the coordinatized line will
cover all
the points in [0, 1]. Due to every point of intersection has a number,
it means
a one-to-one mapping between the real numbers in [0, 1] and natural
number
set. Its contrary to uncountability of the real number system.

But it's not true that every point of that segment has a number.
And it's also not true that each number corresponds to only one point.

Talk about this question in the decimal system, assume that a square
split
into tenths, that is every square has 9 new intersection with the
coordinatized
line. Assume the intersections sequence that produced by the first
split is s1,
the intersections sequence that produced by the second split is
s2, . . .

It is:

s1:

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

s2:

0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09

0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19

. . .

0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99

s3:

0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009

0.011, 0.012, 0.013, 0.014, 0.015, 0.016, 0.017, 0.018, 0.019

. . .

Let S be a sequence that S = {s1, s2, s3, . . .}. If the intersections
between the
Peano curve and the coordinatized line will cover all the points in
[0, 1], that
the sequence S (and it's sub sequence) will include all the numbers in
[0, 1].
Its contrary to uncountability of the real number system.

It seems that the focus is whether the irrational number points on the
coordinatized
line are covered by the Peano curve, or the Peano curve covers the
rational number points only. Before come to this question, it's
meaningful to
talk about the different between the process of constructing the
infinite sequence
and the process of taking the limit of the fundamental sequence. It's
known that there are countabe infinite rational numbers in [0, 1],
these rational
numbers can be produced by an infinite recursion process. And all the
irrational numbers in [0, 1] are the limits of the rational number set
in [0, 1].
It means that the infinite process of constructing the rational
numbers is not
include the process of taking the limit of the rational number set.
Also, it's
known that though the real number set can be produced by taking the
limit
of the rational number set, there is not a one-to-one mapping between
the
rational number set and the real number set. It means the process of
taking
the limit may not keep the one-to-one mapping relation.

Therefore it has two simple but useful conclusions:

1. The infinite process is not certain include the process of taking
the limit;

2. The process of taking the limit may not keep the one-to-one mapping
relation.

Though the process of constructing the Peano curve is infinite steps,
it's not
certain means the process include taking the limit of Peano curve. And
though
the limit of Peano curve is a square, it's not enough to prove there
is a one-toone
mapping between the curve and the square. Base on the sequence S given
above, it can has a conclusion that before taking the limit, the
intersections
between the Peano curve and the coordinatized line cover the rational
number
set in [0, 1], the graphics is a curve but not a square; after taking
the limit, the
intersections will cover all the number in [0, 1], and at that time
the graphics
is a square but not a curve.

Fig. 2. Hilbert curve

3 Hilbert curve

The generation of Hilbert Curve is similar with the generation of
Peano curve,
it is continually divide a square into fourths and connected the four
smaller
squares centers as Figure.2, the result of this curve is a Hilbert
curve [4]. As
the discussion in section 1, Hilbert Curve also can not covers the
whole square.
This section should talk about another thing, the one-to-one mapping
between
a line segment [0,1] and a square [0,1]*[0,1] base on Hilbert Curve.

Coding the squares in binary system. The four squares that generated
by first
split are coded clockwise as (0.00, 0.01, 0.10, 0.11). After a split,
the smaller
squares' codes are generated by adding two bits after the former
square's
code. For example, in the second split, the four squares that
generated from
the square coded 0.00, will be coded coded as
(0.00000.00010.00100.0011).

This coding is hoped to give every point in the square a single code
in [0, 1],
to finish the mapping the square [0, 1]
[0, 1] to the line segment
[0,1].

This coding is just similar to the real number's definition given by
Cantor,
which using the limit of a sequence to define a point. For example,
the point
that the center of the square is defined by the sequence {H ?
1(1/2),H ? 2(1/2), . . . ,H ? N(1/2), . . .}, and the center of the
square will be coded as
1/2. But according the definition, every point in sequence {H ?
1(1/2),H ? 2(1/2), . . . ,H ? N(1/2), . . .} is a constant point,
every point is the limit of
itself, that is every point in the sequence will be coded as 1/2.
Therefore the
number 1/2 will map to a infinite square, but not only the center of
the square.

By using the reduction to absurdity, it can be proved that there is
not a oneto-
one mapping between a line segment [0, 1] and a square [0,1]*[0,1]
base on
Hilbert Curve. Assume that there is a one-to-one mapping, and a point
p in
the median line that parallel to the y-axis, point p is map to x(0 <=
x <= 1).
That is, the limit of the sequence {H-1(x), H-2(x), . . . , H-
N(x), . . . } is point p.

Due to the Hilbert Curve is symmetry about the median line that
parallel to
the y-axis, the limit of the sequence {H-1(1-x), H-2(1-x), . . . , H-
N(1-x), . . . } is point p, too. Because of it's a one-to-one mapping,
it has x = 1 ? x = 1/2.
That is, all the points in the median line are map to 1/2, it's
inconsistent with
the one-to-one mapping postulate.

WHAT postulate is that? People have explained to you that these
space-filling curves are _not_ one-to-one. Nobody ever said they
were.

This reduction to absurdity not only works in the initial square, but
also works
in any smaller square that generated by Hilbert Curve. Also, the
absurdity
is not caused by self-intersecting, or it can not be eliminated by
permit to
self-intersecting.


4 Cantor's dimension reduction

In 1877, Cantor proved in theory that there is a one-to-one mapping
between
a line and a plane. His proof is like this: Assume y is a real number,
and
y = 0.a1b1a2b2 . . . anbn . . .

Let

x1 = 0.a1a2 . . . an . . .

x2 = 0.b1b2 . . . bn . . .

It finished a mapping from y to (x1, x2).

The proof has used the induction unconspicuous. As the discussion in
section
1, what the induction proved is the members' attribute in the
fundamental
sequence, not include the limit of the sequence.

5 Conclusion

In Cantors definition of real number, using a fundamental sequence to
definite
a number in the real number field. This paper points out that
according the
definition, in the real number field, a rational number a is equal to
a sequence
{a}. It will construct a set that containts itself: number a is equal
to a set,
the set has an element that a itself. The set will lead to the
Russell's paradox.

What?

Though Cantor's definition of irrational number didn't contain such
paradox,

What???? It leads to this paradox although it does not lead to this
paradox?

This paper also discusses that Peano curve and some other plane
filling curves
contain such paradox.

References

[1] Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy
of the
Infinite, Princeton University Press, Princeton, USA, 1990.

[2] G. Peano, Sur une courbe, qui remplit toute une aire plane,
Mathematische
Annalen, 36 (1890) 157-160.

[3] G. Breinholt, C Schierz, Algorithm 781: generating Hilbert's space-
filling curve
by recursion, ACM Transactions on Mathematical Software (TOMS), 24
(1998)
184-189.

[4] D. Hilbert, Ueber die stetige Abbildung einer Line auf ein Fl?
chenstck,
Mathematische Annalen, 38 (1891) 459-460.

David C. Ullrich
.

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