Re: Peano curve is wrong

qiuzhihong a écrit :
On 4月2日, 下午6时07分, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Tue, 1 Apr 2008 04:10:33 -0700 (PDT), qiuzhihong





<qiuzhih...@xxxxxxxxx> wrote:
In the following opinion, the Peano curve is just a mistake.
1. About Peano curve:
Peano constructed a infinite curve sequence: P = {p1, p2, p3, ...},
and all the points in the unit square will be the limit points of the
curve sequence P. Therefore Peano has a conclusion that the "limit
curve" of the sequence P is filling the unit square.
2. A fault of Peano's deduction:
Assume that all the points on (P1, P2, P3 ..., including the limit
curve) construct a point set.
The limit point of a point set is not certain in the point set, for
example, all the irrational numbers are the limits of rational number
set, but irrational numbers are not in the rational number set.
But the "limit curve" is a member of sequence P. It has no reason to
say that the "limit curve" is equal to the limit point set.
It means, even if the "limit curve" of Peano curves covers the
rational pairs (or something else similar) in the plane only, it still
has the limit point set of P containts the whole plane. Peano's
deduction is not enough to prove the "limit curve" covers the whole
plane.
3. A counterexample of Peano's deduction:
Assume that a sequence of number sequence:
*************************************
S = {s1, s2, s3, ...}.
s1, s2, ... are number sequences:
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
......
*************************************
All the real numbers in [0, 1] are the limit points of S, but it's not
means that there is a "limit number sequence" containts all the real
numbers in [0, 1]. Otherwise the real number system is countable.
This is a good example. But it only proves that your understanding
of the proof that the Peano curve is space-filling is incorrect.

In the construction of the curve we have P_n(t) -> P(t) uniformly.
In your sequences it's not true that sn[j] even has a limit (for
j fixed, as n -> infinity).
David C. Ullrich

?
I don't understand.

This is no news. And your counter-argument "I dont think a curve is a
square" shows that you understand that the Pea,no (and Hilbertn and
countless others, btw) is counterintuitive, but not much more
.

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