Let's summarize it as follows:
It is known that there is a bijection from
A={0,1,2,3,…} to
B=(2^0,2^1,2^2,2^3,…} .
B is a set that may be arranged as 2-valuead infinite 0;1 logical tree from contradiction (notated by branch 000…) to tautology (notated by branch 111…), where the root is not bit 0 or bit 1.
Moreover, although the number of levels of that tree is countably infinite, the number of its branches is uncountable, simply because given any branch of that tree, its complement is also in this tree.
By Cantor's diagonal argument along a set that is arranged as a "list" of branches, it is shown that there is always a 0;1 branch that is not in the "list", which enables to conclude that the tree arrangement has cardinality that is strictly greater than the cardinality of the "list" arrangement.
Can we define a "list" of infinitely many branches such that any given branch is actually included within it?
In order to construct such "list" we observe that given any branch of the form ..01
… (where .. is any finite number of bits and
… is an infinite repetition of the written right bit) along the 2-valuead infinite 0;1 logical tree, ..10
… is its immediate branch, as follows (the ellipsis … (unlike
…) is used here by its traditional meaning) :
This is the beginning of the tree form:
Code:
*
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
0 1
/ \ / \
/ \ / \
/ \ / \
/ \ / \
0 1 0 1
/ \ / \ / \ / \
/ \ / \ / \ / \
0 1 0 1 0 1 0 1
/ \ / \ / \ / \ / \ / \ / \ / \
. . .
and this is the beginning to the "list" form of the immediate pairs:
01
...
10
...
001
...
010
...
101
...
110
...
...
that also can be written as
011111
…
1
00000
…
00
1111
…
010
000
…
1011
11
…
11000
0…
...
where a branch that is not in the "list" (according to Cantor's notion) starts, in this case, by the following bits:
110101…
A careful observation of the immediate branches of the forms ..01
… and ..10
… discovers that branch
110101… is already included in the set of branches (whether it is arranged as a tree or as a "list") as follows:
In case of the tree arrangement, it is trivially observed that
110101… is defined along the infinity many ..01
… and ..10
… immediate branches:
Code:
*
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
0 [B]1[/B]
/ \ / \
/ \ / \
/ \ / \
/ \ / \
0 1 0 [B]1[/B]
/ \ / \ / \ / \
/ \ / \ / \ / \
0 1 0 1 0 1 [B]0[/B] 1
/ \ / \ / \ / \ / \ / \ / \ / \
. . .
In case of the "list" arrangement
110101… is defined along the infinity many ..01
… and ..10
… immediate branches, by the following bijection:
Code:
[B]1[/B]0[U]…[/U]
↑
1[B]1[/B]0[U][U]…[/U][/U]
↑
11[B]0[/B]1[U]…[/U]
↑
110[B]1[/B]0[U]…[/U]
↑
1101[B]0[/B]1[U]…[/U]
↑
11010[B]1[/B]0[U]…[/U]
↑
110101[B]0[/B]1[U]…[/U]
…
Since any given infinite branch is already included in the set of infinity many ..01
… and ..10
… immediate branches, whether it is arranged as a tree or as a "list", the following bijection holds, such that no branch is missing:
1 → 011111
…
2 → 100000
…
3 → 001111
…
4 → 010000
…
5 → 101111
…
6 → 110000
…
…
By this observation there is no strict distinction between countably infinite 0;1 branches and uncountable 0;1 branches.
By further observation of the issue at hand, one enables to conclude that, for example, numbers of the form 1… > 01… > 001… > … are bigger than any natural number and each one of them can be used as an infinite cardinal number.
In that case one is no longer observes the size of infinite sets by strict distinction between countably infinite sets and uncountable sets, and instead of this strict distinction, a "spectrum" of infinitely many infinite cardinals is used to define the size of a given infinite set.
The immediate branches argument that is used here by 2-valued infinite logical tree, is without loss of generality (as seen, for example in
http://www.internationalskeptics.com/forums/showpost.php?p=11497546&postcount=2150).
------------------------
The form of an infinite set, whether it is arranged as a tree or as a "list", has no influence on its cardinality, as follows:
Let's take, for example, these two infinite logical trees:
Code:
* 0
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
0 1 0 1
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
0 1 0 1 0 1 0 1
/ \ / \ / \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \ / \ / \
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
/ \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
. . . . . .
The left tree is a set of distinct logical connectives (branches) from contradiction (000
...) to tautology (111
...), and there is no problem to define a bijection from an infinite set of natural numbers to the left tree, as observed by the immediate branches argument.
The right tree is a set of distinct logical connectives (branches) from contradiction (000
...) to 0111
..., and there is no problem to define a bijection from an infinite set of natural numbers to the right tree (as observed by the immediate branches argument) even if there are infinitely many branches that are not included in the right tree, simply because missing branches is not a guarantee that there is no bijection from an infinite set of natural numbers to the right tree (the missing branches are not included in the right tree,
in the first place).
Moreover, the fact that there are P(S) members that are not included in the mapping from S to P(S), is equivalent to the fact that there are infinitely many branches in the right tree that are not included in the mapping from an infinite set of natural numbers to the right tree, where this fact does not prevent the bijection from S to P(S) (the missing P(S) members are not included in the mapping,
in the first place, similarly to the observation between the left and right trees).
