The doubled interval (0, ω)

[WM]

Cantor created the sequence of the ordinal numbers by means of his first
and second generation principle

0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ..., 2ω, 2ω+1, 2ω+2, 2ω+3,
.., 3ω, ... . (9.1)

In 1884 he exchanged the positions of multiplier and the number to be
multiplied with the result

0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ..., ω*2, ω*2+1, ω*2+2, ω*2+3,
ω*3, ... . (9.2)

This is mentioned only in order to avoid confusion. We will stick to his
second notation (9.2).

This sequence, except its very first terms, has no relevance for classical
mathematics. But it is important for set theory that in actual infinity
there does nothing fit between ℕ and ω. Likewise before ω2 and
ω3 there is no empty space. What is the alternative? Only ordinal
numbers. According to Hilbert we can simply count beyond the infinite by a
quite natural and uniquely determined, consistent continuation of the
ordinary counting in the finite. But we would proceed even faster, when
instead of counting, we doubled numbers by a factor of 2. This leads to
the central issue of this chapter: Consider the setℕ  {ω} = {1, 2,
3, ..., ω} and multiply every element by 2 with the result

{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2}.

What elements fall between ω and ω2? What size has the interval
between 2ℕ and ω*2? The number of doubled numbers is precisely |ℕ|.
But half of the natural numbers are not in it. If all natural numbers
including all even numbers are doubled and if doubling increases the
value, then not all doubled even numbers fit below ω. Numbers greater
than all even numbers are not possible.

The natural answer is (0, ω)2 = (0, ω2) with ω or ω+1 amidst.

Every other result would violate symmetry and beauty of mathematics, for
instance the claim that the result would be ℕ U {ω, ω*2}. All numbers
between ω and ω*2 which are precisely as many as in ℕ between 0 and
ω, would not be in the result? Every structure must be doubled! Like the
interval [1, 5] of lengths 4 by doubling gets [1, 5]*2 = [2, 10] of length
8, the interval (0, ω)2 gets (0, ω*2) with ω*2 = ω + ω =/= ω
where the whole interval between 0 and ω2 is evenly filled with even
numbers like the whole interval between 0 and ω is filled with natural
numbers before. On the ordinal axis the numbers 0, ω, ω*2, ω*3, ...
have same distances because same number of ordinals lie between them.

This means that contrary to the collection of visible natural numbers
ℕ_def which only are relevant in classical mathematics the whole set ℕ
is not closed under multiplication. Some natural numbers can become
transfinite by multiplication. This resembles the displacement of the
interval (0, 1] by one point to the left-hand side such that the interval
[0, 1) is covered. Of course these natural numbers are dark like every
result of ω/k with k in , for instance ω/2 or ω/10^10^100.

Regards, WM

[Ross Finlayson]

Post by WM
Cantor created the sequence of the ordinal numbers by means of his first
and second generation principle
0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ..., 2ω, 2ω+1, 2ω+2, 2ω+3, ..,
3ω, ... . (9.1)
In 1884 he exchanged the positions of multiplier and the number to be
multiplied with the result
0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ..., ω*2, ω*2+1, ω*2+2, ω*2+3,
ω*3, ... . (9.2)
This is mentioned only in order to avoid confusion. We will stick to his
second notation (9.2).
This sequence, except its very first terms, has no relevance for
classical mathematics. But it is important for set theory that in actual
infinity there does nothing fit between ℕ and ω. Likewise before ω2 and
ω3 there is no empty space. What is the alternative? Only ordinal
numbers. According to Hilbert we can simply count beyond the infinite by
a quite natural and uniquely determined, consistent continuation of the
ordinary counting in the finite. But we would proceed even faster, when
instead of counting, we doubled numbers by a factor of 2. This leads to
the central issue of this chapter: Consider the setℕ  {ω} = {1, 2, 3,
..., ω} and multiply every element by 2 with the result
{1, 2, 3, ..., ω}*2 = {2, 4, 6, ..., ω*2}.
What elements fall between ω and ω2? What size has the interval between
2ℕ and ω*2? The number of doubled numbers is precisely |ℕ|. But half of
the natural numbers are not in it. If all natural numbers including all
even numbers are doubled and if doubling increases the value, then not
all doubled even numbers fit below ω. Numbers greater than all even
numbers are not possible.
The natural answer is (0, ω)2 = (0, ω2) with ω or ω+1 amidst.
Every other result would violate symmetry and beauty of mathematics, for
instance the claim that the result would be ℕ U {ω, ω*2}. All numbers
between ω and ω*2 which are precisely as many as in ℕ between 0 and ω,
would not be in the result? Every structure must be doubled! Like the
interval [1, 5] of lengths 4 by doubling gets [1, 5]*2 = [2, 10] of
length 8, the interval (0, ω)2 gets (0, ω*2) with ω*2 = ω + ω =/= ω
where the whole interval between 0 and ω2 is evenly filled with even
numbers like the whole interval between 0 and ω is filled with natural
numbers before. On the ordinal axis the numbers 0, ω, ω*2, ω*3, ... have
same distances because same number of ordinals lie between them.
This means that contrary to the collection of visible natural numbers
ℕ_def which only are relevant in classical mathematics the whole set ℕ
is not closed under multiplication. Some natural numbers can become
transfinite by multiplication. This resembles the displacement of the
interval (0, 1] by one point to the left-hand side such that the
interval [0, 1) is covered. Of course these natural numbers are dark
like every result of ω/k with k in , for instance ω/2 or ω/10^10^100.
Regards, WM