Re: abundance of irrationals!)

In article <1115098696.756233.274310@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

> Virgil wrote:
>
> > >
> > > Not only numbers.
> >
> > Only rational numbers, and not all of them.
>
> The decimal representation contains only multiples of 10^-n.
> Nevertheless you believe that 1/3 can be represented.

Not in your list with only finitely many placed per number, but with a
digit for each n in N it can, in decimal form, as 0.333....
> >
> > > Infinitely many digits.
> >
> > Not in any finite subset of the list, as a finite number of finite
> > numbers adds up to a finite number.
> >
>
> In the whole list. That's enough.

But the whole list is not a number in the list, so that doesn't work.
>
> 1)
> Or could you explain, please, the difference between an unbounded set
> (like the digit-numbers in my list) and an infinite set (like the digit
> numbers of all real numbers).

(1) The first number in your list has only one decimal place, so
contains only finitely many digits.

(2) If any number in your list contains only a finite number of digits,
the next number in your list contains at most one more digit so that
that number also contains only a finite number of digits.

Conclusion: Every number in your list contains only a finite number of
digits.

There are numbers which cannot be represented with only finitely many
digits.

Therefore there are numbers not in your list.

And Cantor is justified.


>
> 2)
> Further, please explain the following difference: If you look over the
> first n lines of Cantor's list,

Cantor does not have a list. You have to privide the list then Cantor
shows that it does not contain listings of every real number.

> you see the first n digits of its
> diagonal number. In order to see the n+1-th digit, which is certainly
> present, you must turn your eyes down to include the next line.
> If you look over the first n lines of my list, you see the numbers with
> n digits. In order to see the numbers with n+1 digits, you must turn
> your eyes down to include the next line.

That is only true for physical lists, but these are all mathematical
lists. They have a next, but not necessarily a down or up.
>
> 3)
> Further, could you please explain why your requirement (that I should
> determine precisely that line of my list which contains 1/3) is more
> powerful than my requirement (that you might name a number which is the
> upper bound of digit numbers in my list).

One of the properties of inductive sets, in fact arbitrary wel-ordered
sets, is that every non-empty subset has has a first or smallest member.

If you claim that the set of listings for 1/3 is not empty, you must
prove that claim.

I, on the other hand, do NOT claim any such upper bound as you request,
so I have nothing to validate.
>
> Unless we have cleared these three points, we cannot come to a
> conclusion - and further discussion is useless, and it is irrelevant,
> how many supporters are there for either of our positions.

I can come to a conclusion easily enough about this dispute in which I
will be joined by all mathematicians, that WM does not know what he is
talking about.

When presented by an axiom system, one can
(1) show that the system is self-contradictory (which WM has not done),
(2) accept the system as presented,
(3) reject the system entirely and create one's own system.

One cannot insist, as WM keeps trying to do, that the axiom system
incorporate properties not inherent in those axioms.
>
> > > But do you know the smallest
> > > infinity according to Cantor? It is that of the natural numbers.
> >
> > And Cantor is right on this point, too!
> >
> Therefore, as my list is not finite and the strings of my list are not
> bounded (not even you would state that, I presume), the number of
> digits is either infinite (which would include 1/3), or 1/3 has digits
> which cannot be enumerated by natural numbers.

False dichotomy. To have a set of string lengths which are unbounded
does not require that any of them individually be unbounded. The
unboundedness is a property of the set, not of its members.

The set N is unbounded, but each of its members is bounded and finite,
as the inductive axiom requires.
>
> There are two alternatives, please choose:
> 1) In my WHOLE list a digit number n is missing. Please secify it.
> 2) 1/ 3 has digits which cannot be enumerated by a natural number n e
> N.

If you mean that the decimal representation of 1/3 has MORE digits than
indicated by any finite natural number, that is true, but it is also
true that every digit in that representation is indexed by some natural
number.

Part of WM's problem is that he cannot be unambiguous.
.

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