Re: New countable infiniity logic

From: tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 11/21/04 

Date: Sun, 21 Nov 2004 15:06:54 -0800

> From: whit0911@umn.edu
> I am also saying that the set Dec3 = { 0.3, 0.33, ... } is an
> infinite set and therefore *must* have an infinite number of digits
> since it is formed from an infinite sequence (albeit of finite
> sequences of digits).

No, not only doesn't Dec3 have an infinite number of digits in it, it
doesn't have even one digit in it! For example, 0.3 is in Dec3, but 0.3
is not a digit. '3' is a digit, but '3' is nowhere in the set Dec3.
Have you ever written a computer program that deals with arrays of
strings? Consider an array with two strings in it, the string "FOO" and
the string "BAR". Does that array have any characters in it? No, it has
only strings in it. Now look at one of the strings, namely "FOO", which
is itself an array of characters. Does "FOO" have any characters in it?
Yes, it has three characters in it, the character 'F' once and the
character 'O' twice. The array of strings has only strings in it, and
each of the strings has characters in it, but the array of strings
doesn't have any characters in it. Do you finally understand the
difference between arrays of characters and arrays of arrays of
characters? If you program in C, do you know the difference between the
declarations:
  char* p; // Array of characters, or pointer to char
  char** p; // Array of arrays of characters, or pointer to pointer to char
Back to math: Do you know the difference between:
  0.333 // Finite sequence of digits (with decimal point near start)
  (0.3, 0.33, 0.333) // Finite sequence of finite sequences of digits
  0.333... // Infinite sequence of digits (with decimal point near start)
  (0.3, 0.33, 0.333, ...) // Infinite sequence of finite sequences of digits

> the set named Dec3&1/3 may be represented by
> ... Dec3&1/3 = {0.3, 0.33, 0.33, ..., 1/3}

Note that's a different set from what you generated by that formula.
This has an infinite number of finite sequences, and also one infinite
sequence. (Also the notation is sloppy, see where I explained that in
more detail a few minutes ago.)

> cannot also be specified by rearranging elements to give Dec3&1/3 =
> { 1/3, 0.3, 0.33, 0.333, ... }.
> Can this not be placed in a one to one correspondence with the
> naturals? Why is this set of order type omega +1?

When you rearrange the elements, you destroy the order relation.
The original ordered set, and the rearranged ordered set, are different
ordered sets. The original was of type omega+1, the new one is merely
of type 1+omega which is the same as omega. The notation omega is
defined only for ordered infinite sets, not for mere sets without
order. With infinite ordered sets, concatenation is *not* commutative.
O1 + O2 is not necessarily the same as O2 + O1. The original omega+1
set and the new 1+omega (i.e. omega) set can't be put in 1-to-1
correspondence while preserving order.

Transfinite cardinal numbers (comparing sets, allowing rearrangements),
and transfinite ordinal numbers (comparing ordered sets, not allowing
rearrangements that violate the order relation), are not the same!!
For example, as sets without any order, natural numbers, integers,
rationals, decimal rationals, are all the same size, aleph-null.
But as ordered sets, all those four are different types, in fact
transfinite ordinal set theory allows only ordered sets satisfying the
every-subset-has-a-least-element condition, so only the first of those
four sets has an ordinal type defined.

I suggest you set aside transfinite ordinal set theory until after you
get a grasp on transfinite cardinal set theory.

> I believe that the list of naturals are infinite and that each
> accepted individual natural is finite.

That is a basically correct belief.

> I also believe that the number of digits the list of infinite
> naturals generates must also be infinite ...

Nope. Not even one of the natural numbers is itself a digit.
Each natural number contains digits in its representation, but the
number (or representation as sequence of digits) is not the same as a
digit itself. Go look back at that C example for the distinction
between char* and char**, and the distinction between sequence of
digits and sequence of sequences of digits.

> I am concerned that set theory has a contradiction that may require
> naturals that are infinite.

If that's supposed to mean the set of all natural numbers is an
infinite set, that's correct.

If that's supposed to mean that any single natural number is infinite,
that's wrong.

Please clarify what you meant to say there.

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