Re: Dial 999 for the real number line


"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
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On Thu, 7 Jun 2007 04:05:54 +0900, toshiaki wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
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On Sun, 3 Jun 2007 02:03:54 +0900, toshiaki wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
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Are you claiming that the axiom of infinity fails to assert the
existence
of an infinite set?

As SixLetters pointed out justlly, its logical
inconsistency is in the argument that the size of a set is
represented by its largest member

What happens if the set doesn't have a largest member?

I don't oppose your opinion one sidedly. That might possibly be
the case of real world. If there is the most distant position
exist, we cannot but imagine space beyond there. But is it a set
completed when the largest number doesn't exist in it?

The question was about the axiom of infinity. This axiom allows us to
conclude that there is a set that has no largest member.

There is no largest member, there are no all members.

Later on you agree that *if* we accept the axiom of infinity, then there
is indeed a set with no largest member.

If there is not the most distant position in the space, we can
not go through all the position of a space. This condition is
the same for N too. We can not refer to all its members.

"We" are not mentioned in the axioms of set theory. A set may exist,
whether we can refer to all its members or not.

Then "We" have nothing to do with set theory. But if someone
utilize the axioms set theory, it is "We".

We use the axioms by drawing inferences from them. Being able to refer
to each and every member of an infinite set doesn't happen to be among
the inferences we can draw.

I think that a subject is important when we think of infinity, or
selfreferential statement. What we can do and see (even though
extended beyond physical restraint ) is finite operation.
Why we cannot refer to its members, while set exists. Because
they exist as long as we refer to.

Why? Do stars exist in distant galaxies? Can we refer to (or even
detect) all of them individually? What makes us so all-important,
anyway?

This interpretation may not be incompatible with the axiom of
infinity.
Even though reals doesn't fill continuum, we can get them from
anywhere we can point out.
This arbitrary position is contraversial. Off course uncomputable
numbers cannot be pointed out. Therefore what we can point out,
are numbers which have explicit numerical value.
I think that the countable reals are sufficient for this purpose. I
shall show how they construct complete ordered field. Chaitin's
number is actually computable. It shows the unrealistic character
of halting problem.
I don't deny Turing's proof, but we can in fact construct Turing's
selfreferencial program and run. We can predict its motion.
The selfreferential statement has alternately opposite results by
adding human manipulation at each turn.

Good luck with proving that the computable reals are complete.

I think that the limit of Cauchy sequence is computable. And the least
upper bound of a set of computable numbers is computable. I think
that If a set of Turing machine is explicitely shown, we can compute
Chaitin's omega, although it solves the holting problem, can't we?
I want to write my opinion about incompletness theorem according to
article in Wikipedia "Incompleteness theorem".

The first incompleteness theorem says that "there is a statement
which is true, but cannot be proved true. The true but unprovable
statement is often referred to as "the Godel sentence". There are
infinitely many statements in the theory that share with the Godel
sentence the property of being true but not provable from the
theory.
I think that G sentense is "G cannot be proven true". Why this
statement are possibly true? If this can be said true, is it not the
same as this has been proven to be true? And why it can be said
from the theorem that there are infinitely many statements in the theory
that are true but not provable from the theory?
I know that there are many statements that are not provable by Peano
arithmetic. Some of them have been proved using other way.
Can the incompleteness theorem be appried except the statement
"G cannot be proven true".
It is by chance that there are the other statements that are true
but not provable. G of "G cannot be proven true" is like nested
mirror image, and not completed statement like antidiagonal number.
I don't deny the statement of incompletness theorem. But I think
that "G cannot be proven true" is meaningless as well as Russel's
self referential set, and should be precluded.
There are many statements that are not provable, but we can not
prove their existence. Mathematics is a empirical science in this
sense. It might shows the inconsistency of set theory that the infinite
set allows various interpritation.

We would have to preclude inappropreate conditions as soon as we
find any flow in a theory, as we did when Russel's paradox was
found. Humans had not understood correctly the axiom of parallel
line for long time, and worked without correct definition of the limit.
This circumstences is the same still now.

The following question has not so much relation to my above
statement. But how many ones we should add to one, to be infinite
for the sum? Can repitition of succesor operation achieve this?

No, it can't. Addition is a finite operation. If we could get something
infinite by adding one unit at a time, then we wouldn't need an axiom of
infinity.

Some people including me say that the axiom of infinity asserts the
existence of a set whose member is generated by never ending
addition. This is more literal interpretation of the axiom.
I think that the reason you choose standard interpretation is only
because the other choice lead to unproductive theory.

To get infinite set, infinite repitition should be done. What is the
latter infinite? How much succesor is needed?

Read the previous statement again.

To obtain N, there have to be infinit numbers from the bigining.
Therefore we call it completed infinite set.

I call it an infinite set. "Completed" is redundant. And "from the
beginning" means "from the axioms", including the axiom of infinity.

There is no other choice for me than to write redundantly, if
"infinite" has been registered already.

I don't this view onesidedly, but my
attitude for infinity is the suspention of judgement.

In this reason I think it is inappropriate to treat as if
completed
infinite set exist and that Tarski's opinion about AC is
reasonable.

I don't see the connection. It's possible to accept the axiom of
infinity without accepting the axiom of choice.

ZF is more preferable for me. But this choice leads to another trouble.
One candidate which cut through this trouble is a theory of countable
reals, that is computable reals. It is computable, so that it has no
problems related to AC.

Does the power set of the computable reals exist?

It is the same as the case of a finite set. Infinite case is a extention
of a finite case. Only the interplretation of "infinite" and cardinarity is
defferent from your's. There is not cardinarity in my theory. Nonzero
measure is not derived from uncountable reals. A set is measured by
specifiable members.

As I told SixLetters, it's ok if you want to claim that infinite
sets
do
not exist. I have no problem with that. The point is, that *if*
you
accept that a set of natural numbers exists, *then* it follows that
the
set is infinite.

I agree this, and Six Letters too seems to do. A problem is the view
about infinity. We dont agree an interpretation which derives the
view
that the infinite decimals correspond to transcendental numbers.

Not all of them. Some infinite decimals correspond to algebraic
numbers.

This is a reply that I have been afraid of. In addition some correspond
to rationals. Sorry I saved trouble.

According to whether we accept such an idea or not, we get
remarkably different results. I wrote the reason why I cannot accept
completed infinite set.

What is the difference between an "infinite set" and a "completed
infinite set"?

An infinite decimal 3.14... =/= pi, a completed infinite decimal 3.14...
=
pi.
An Infinite set doesn't have a size, a completed infinite set has a
size.

All sets in ZF are "completed".

What is the inevitability of it? "Infinite" seems to have a meanig
"endless" too.

An Infinite set has a measure zero, a completed infinite set has a
volume.

Some infinite sets have positive measure. Some may be unmeasurable, if
you accept AC.

We can prove the uncountability of the reals from the Baire Category
Theorem, or by using Lebesgue measure theory. Neither of those methods
has anything remotely to do with decimal digit strings.

I don't know well the Baire Category theorem. Though I am going to
study measure theory since long before, I cannot take time to do so.
I'm afraid that my theory might be considerably different from
standard one.
I think that if we avoid completed infinite set like all irrational
numbers,
we may buld theory similar to Lebesgue's. The reals can be dealt with
as continuum, though they are countable. It is a measure theory for
computable number.

One of the consequences of measure theory is that every countable set of
reals has measure 0. Since the unit interval [0,1] has measure 1, it
follows that there are uncountably many reals in that interval.

My explamation was insufficient.
In my theory a point is a mark or a rule to measure a space. When we
can put points on a space densely it has a measure corresponding to its
area. In other words dense set has nonzero measure.
Therefore a measure of [0,1] covered with reals is equal to [0,1] covered
with rationals.
The reason is the same as that a measure of [0,1] is equal to that of
(0,1).

The results is depend on the difference between whether
to accept that infifinite decimals represent all reals , for example
pi,
or not. I wonder that this might be only the matter of deffinition or
stand point of view.

All that is needed is the fact that the reals form a complete ordered
field. Absolutely no reference to decimals is needed.

The reals form a complete ordered field, it is countable. I would show
this in my old thread. This is based on the idea that the more two
numbers approach, the less they are not distinguishable.

There goes Lebesgue integration theory. If there are only countably many
reals, then every integral evaluates to 0, hence the whole theory is
useless.

In my theory the integration of function on dense set is the same as
that on connected space in Lebesgue integration theory.
No uncomputable reals are needed.
My countable reals can be picked up individually in a distance,
but looked as continuum at very near position.


The gap that I mean, is not visible one. Because it is the limit of our
conprehention and counting is a probe. I said before that a line is
made of interval. But when the number of points increase, these
intervals seems as if not to exist.
"countable" is a subjective word.


Regards

Ozaki Toshiaki


.

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