Re: Is continuum completely filled up?


<cbrown@xxxxxxxxxxxxxxxxx> wrote in message
news:1171346151.002186.164550@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Let's discuss with two point of view for a while, which have been since
long ago. If you find contradiction in my argument, please point out.
Numerical calcuration of pi doesn't end. I don't know mathematics well,
but if there is a way to mark on coordanates the position of pi
correctly,
please tell me.


Let's take a simpler example.

(1) Let x be the unique real number, greater than 0, such that x^2 =
2.

(2) x is not a rational number; so you would state: "numerical
calculation of x doesn't end".

(3) x is the length of the diagonal of a square with side length 1. So
there is "a way to mark on coordinates the position of sqrt(2)
correctly", even though (2) is true.

Do you have the same problem with sqrt(2) that you apparently have
with pi, or not?

This case is aparently different from pi's as well as 0.999... Sqrt(2) is
distinctly
marked on line with symbol different from decimal expantion. I assume that
there
is pi on line too, though it cannot be distinctly specified. I distinguish
the difinition
of a number from its decimal representation. This treatment is done for
making
useful mathematics. The point is countable. The amount of difinitions are
countable.

Do you mean that there are more reals than decimal representations
in the biginning?

No, he means that the things that we mean by "the real numbers" have
the /properties/ of the real numbers.

Does not this property include the fact that reals have cardinality of power
set
of naturals?

Primary amongst these properties are the usual operations: +, -, *, /.
In any representation, these operations /must/ be understood as part
of the representation; otherwise, it's /not/ a representation of the
reals by itself.

These properties are /independent/ of any particular method of
representing them. That's what lets us talk about numbers base 10, or
base 2; while still talking about "the same thing".

For example, we could represent the real numbers "base pi" if we chose
[*]. In that case, all rational numbers would require an infinite non-
repeating sequence of coefficients in their representation; but we
would represent the number "pi" as "1.000...".

However, if we chose to represent the reals that way, we would have to
change the usual algorithm for multiplying; because clearly pi*pi is
not pi! (i.e., 1.0000...*1.000... should not equal 1.00...!).

So it is not simply a matter of the /representation/ - it is a matter
of the relationships /between/ the elements you are representing.

Yes I understand your explanation and example is interesting. My concern is
the relationships of reals on line or their quantity. Therefor its
expression
contain inevitably infinite parts.

[*] To represent a number x in [0..pi) base pi, use the limit of the
sequence of polynomials {a_0 + a_1/pi + a_2/pi^2 + ... + a_n/pi^n},
where for each n, a_n is the largest integer such that a_1/pi + a_2/
pi^2 + ... + a_n/pi^n <= x.


ZFC is just one of many different ways of talking about "sets". You
are arguing at an even further remove than simply discarding ZFC; you
are completely missing the abstraction that mathematics is based on.

You learned how to calculate addition and subtraction using
(necessarily) finite decimal expansions, and from repeated
applications of addition and subtraction, you learned how to perform
multiplication and division.

From this, you assume that somehow the real numbers are "an extension"
of this idea: that real number expressions such as x + y necessarily
involve /writing out/ the sum, digit by digit, of the decimal
expansions of x and y.

But you are wrong when you imagine that when mathematicians talk about
the real numbers, they are talking about such grossly physical
manipulations as making sums up on a *** of paper.

When we talk about the real numbers, we have logical ristriction even
though
we don't use papers. The limit namet the countable. How do you define more
than countable number of reals? Power set? The meaning of one to one
correspondence is applyed beyond its original one.
It is difficult for us to accept a explanation like that natural numbers
and even numbers have the same cardinality, and this seems to imply the same
quantity. In fact uncountable irrationals build a substance of a real line.

Yes, Cantor's proof is as you explain. My explanation
is as for decimal representation of reals from my view. This view has
existed since before Cantor.


In ancient Greece, it was realized that the sqrt(2) was a
constructible number (i.e., length, via compass and unmarked ruler);
and yet the decimal expansion of sqrt(2) doesn't repeat (i.e., is not
a rational number).

Zeno's argument is based on the endlessness of counting and doesn't accept
infinite set.

By making diagonal number, we can only add new decimal which is not
on a list. If we assume infinite list, it means that this operation
of
making new number doesn't complete.

The word "complete" is not a part of the proof. What are you talking
about? If we know the n-th digit of x for each n, then we know x.

I don't know x, unless I know all digits of x.

So do you think it is senseless to talk about the number x such that
x^2 = 2?

In this sense, my statement is inappropriate. That is about decimal
expantion.

Or do you think that such an x exists, but you can never "know" it?

As for the position of pi on line, I say so.

If the latter is the case, suppose x^2 = 2. Whether you "know" x or
not, wouldn't you still agree that x^2 + x^2 = 4? That is the sort of
question mathematicians are interested in, and not whether the
10,000th digit of of x is a 0 or not!

I am interested in this sort of question, and are designing a theory to
represent
sqrt(2) on line as well as traditional theory. and I am interested in
infinity. It
concerns to all digits, not finite digits.

I cannot garantee for my lack of mathematical skill, but this conclusion
may be correct, as long as we accept ZFC.
My proof is based on the idea that mathematical objects which we can
deal
with, are symbols.
Purpose of my argument is to show outline that a theory only with
countable symbols, has no contradiction.


I think if you investigate what mathematicians actually mean by their
statements, you'll discover that there are already many well-known
theories with even a finite number of symbols which do not "contain
contradictions".

I don't know well constructivism or finitism and another theories, but know
the concept of these two theories. I think it is good that various theories
complement one another. One of problem of constructivism is complexity of
the proof. I accept the law of excluded middle. And don't accept the
assertions
like that sqrt(2) is a rational number. My assertion is that sqrt(2) is
unlimitedly
approximated by the ratinal number.
But when we think of the limt condition, their boundery is removed.
This limit is the condition that infinite decimal is assumed to be
completed, and
0.999... coincide to1, and pi is undistinguishable from rational number.
That is
all of digits of decimal expantion of pi is regarded as unit of repitition
of rational
number, because these are all finite. The important point is that each
numbers
can not be picked up individually in this condition.
In ordinary operation we deal with countable numbers, and don't take into
account this condition. This is used for explanation of the limit
caluculation.
The formula of the lim(n =>oo)sum(1 n)9/10^k is derived by regarding 9/10^n
as 0 (n =>oo). This is only the matter of difinition.

Certainly reals are more than rationals. But it is valid only in
finite
case.

What is the "finite case" of |N| < |R|?

Reals are build from rationals. Rationals are build from naturals.


You are correct; but you don't seem to understand how this "building"
process proceeds.

It is not a matter of the /representation/ (base 10, base 2, whatever)
which is used to move from stage to stage.

It is instead the /properties/ that every valid representation /must/
have that are /defined/ at each stage.

The properties (primarily, the relationships between the elements we
are representing) come first; the representation comes /after/.

I think that origin of these numbers are designed as intermidiate value
between
numbers of former hierarchy.

Regards
Ozaki Toshiaki

.

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