Re: Ultimate debunking of Cantor's Theory

On 14 Jul., 00:50, Calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Jul 13, 12:52 pm, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

On 11 Jul., 16:20, Calvin <cri...@xxxxxxxxxxxxxx> wrote:
The original post doesn't restrict set theory to finite
sets. It says that only sets which are finite exist.
Obviously natural numbers exist, and can be used for
example to count the words in this post. Since they
exist, and there are supposedly only a finite number
of them, then there must be a largest one.

That is a naive misunderstanding.

I hope you understood that my words, "there are
supposedly only a finite number of them," did not
mean that I supposed such a thing. I was stating
the proposition that was to be disproved.

To spell it out clearly: There is only a limited number of natural
numbers. Nevertheless there is no largest one among them.

There is a limited number of bits.
From these bits you can form natural numbers.
The magnitude of these numbers is merely
limited by the ingeniousity to introduce suitable
abbreviations.

So, let's say that everyone in the universe finally
runs out of suitable abbreviations, and the last
natural number so abbreviated is recorded according
to those abbreviations. Let's now agree to refer
to that number as George. What is George + 1?

It is not always possible to find n + 1. But it is always possible to
find 2(n-1) say. Here is a simplified model: Consider a tiny universe
in
which only one statement can exist which is limited to seven symbols
(which can be chosen arbitrarily from the key board of a usual type
writer). Of course, contrary to the real universe, the meaning of the
symbols has to be formulated outside. But I think the line of thought
is not disrupted by this disadvantage, unless we admit infinitely
many
ad hoc definitions. For the sake of simplicity let us assume the
decimal system.

In this model we can represent every natural number from 0 to 9999999
by choosing seven symbols from the type writer's set of digits. By
means of abbreviations like 123(9*) which is to represent our number
12333333333 with nine 3's, or 1(999*), our natural number consisting
of 999 1's, we can also represent larger numbers. But we cannot
express every number between them. So we cannot express any number
between 123(9*) and 124(9*), i.e., between 12333333333 and
12444444444
by this simple kind of abbreviation.


Some advanced (nevertheless familiar) abbreviations may enlarge the
set of numbers of the model including what Gauss called "the
measurable infinity" 9^9^9^9 or even larger numbers like 9^9^9! or
9^9^9!! and so on. So there can be very large numbers, but it is
impossible to represent numbers like 9^9^9^9-1 and most others which
are less than this.



Regards, WM

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