I wish to ask some question about natural numbers' construction by using logical trees, without using free variables.
First, some background that leads to my question:
True is notated by 1
~True is notated by 0
p and q are two propositions as follows:
p = 0 0 1 1
q = 0 1 0 1
So, we get the 16 logical connectives as seen by the 16 distinct paths of the following binary tree:
Code:
p = 0 0 1 1
q = 0 1 0 1
---------------
/0 Contradiction
/0
/ \1 p AND q
/0
/ \ /0 p not implies q
/ \1
/ \1 p
/0
/ \ /0 q not implies p
/ \ /0
/ \ / \1 q
/ \1
/ \ /0 p XOR q
/ \1
/ \1 p OR q
*
\ /0 p NOR q
\ /0
\ / \1 p NXOR q
\ /0
\ / \ /0 NOT q
\ / \1
\ / \1 q implies p
\1
\ /0 NOT p
\ /0
\ / \1 p implies q
\1
\ /0 p NAND q
\1
\1 Tautology
The complements of a given binary tree with 16 distinct paths are:
Code:
p = 0 0 1 1
q = 0 1 0 1
---------------
/0 Contradiction -----------*
/0 |
/ \1 p AND q ---------------* |
/0 | |
/ \ /0 p not implies q -----* | |
/ \1 | | |
/ \1 p -----------------* | | |
/0 | | | |
/ \ /0 q not implies p -* | | | |
/ \ /0 | | | | |
/ \ / \1 q -------------* | | | | |
/ \1 | | | | | |
/ \ /0 p XOR q -----* | | | | | |
/ \1 | | | | | | |
/ \1 p OR q ----* | | | | | | |
* | | | | | | | |
\ /0 p NOR q ---* | | | | | | |
\ /0 | | | | | | |
\ / \1 p NXOR q ----* | | | | | |
\ /0 | | | | | |
\ / \ /0 NOT q ---------* | | | | |
\ / \1 | | | | |
\ / \1 q implies p -----* | | | |
\1 | | | |
\ /0 NOT p -------------* | | |
\ /0 | | |
\ / \1 p implies q ---------* | |
\1 | |
\ /0 p NAND q --------------* |
\1 |
\1 Tautology ---------------*
---------------------------------------------------
Let's briefly touch 3-valued logic.
True has 3 options which are: True,
mTrue, ~True (
m = middle, ~ = not).
True is notated by 2
mTrue is notated by 1
~True is notated by 0
p, m and q are 3 propositions as follows:
Code:
p = 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
m = 0 0 0 1 1 1 2 2 2 0 0 0 1 1 1 2 2 2 0 0 0 1 1 1 2 2 2
q = 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
A tree of 3-valued logic of these propositions has 3
27 = 7,625,597,484,987 logical connectives.
In this case contradiction is path 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0,
where tautology is path 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2.
Moreover, given any
n-valued logical tree (where
n > 1) it is bounded by contradiction and tautology.
---------------------------------------------
Now let's observe unbounded logical trees by using (without loss of generality) the 2-valued unbounded logical tree (no free variables are used):
Code:
*
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
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 0 1
. . .
This tree is also logically bounded by contradiction and tautology, but it is logically unbounded "below" (each one of its paths is an unbounded "below" distinct logical connective.
It is also observed that the diagonalisation argument can't be used along the 2-valued unbounded logical tree, since given any arbitrary unbounded logical path, its logical complement is already in this tree, which means that there are uncountable unbounded distinct logical paths along that tree.
-----------------------------
Now let's use the 2-valued unbounded logical tree in order to construct the natural numbers along it, by using the notion of radix point, as follows:
Code:
*
|\
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
0---------------1---------------Integers
|\ |\
| \ | \
| \ | \
| \ | \
| \ | \
| \ | \ Fractions
| \ | \
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
...
Code:
*
|\
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
| \
0 1
|\ |\
| \ | \
| \ | \
| \ | \
| \ | \
| \ | \
| \ | \
0-------1-------0-------1---------Integers
|\ |\ |\ |\
| \ | \ | \ | \
| \ | \ | \ | \ Fractions
0 1 0 1 0 1 0 1
|\ |\ |\ |\ |\ |\ |\ |\
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
...
etc.
So it is logically observed that no matter how many times the radix point is "pushed" "downward" along the unbounded logical tree, no logical path "above" the radix point has an unbounded number of bits, which logically means that no amount of bounded logical paths (which are equivalent to collection of natural numbers) is infinite (or unbounded).
By this "direct" logical observation it is realized that there is a straightforward logical linkage between the common property of being logically bounded (as observed among natural numbers, as constructed along the unbounded logical tree) and the logical observation that there is no infinite (or unbounded) collection of bounded paths.
Moreover, if one observes some distinct unbounded path (which is not path 000...) as a measurement value (one logically defines number > 0 without any radix point along it) for the amount of natural numbers (as logically constructed here) one discovers that there are unbounded alternatives to such measurement value (it means that the notion of aleph
0 as the one and only one alternative, is logically insufficient).
Furthermore, being uncountable is based on notions like aleph
0, but since there is no one and only one alternative for the measurement value of the amount of natural numbers (in case that one logically defines number > 0 without any radix point along it), the notion of being uncountable logically does not hold (without aleph
0 as the one and only measurement value of the amount of natural numbers, values like 2
aleph0 have no accurate logical basis).
--------------------------------
If one defines number only in case that there is a radix point along any given unbounded logical path, then one logically observes, for example, The Axiom of Infinity, as follows:
The Axiom of Infinity (as written in Wikipedia):
"There is a set I (the set which is postulated to be infinite), such that the empty set is a member of I AND such that for any x that is a member of I, the set formed by taking the union of x with its singleton {x}, is also a member of I."
By using radix point in order to construct natural numbers (as logically observed here along an unbounded logical tree) one logically realizes that this axiom simply "pushes" the radix point "downward" along the unbounded logical tree, and since no member of that set (which is defined by this mathematical induction) has unbounded bits, this collection has no more than finitely many members (where one of the particular cases of mathematical induction is a set of natural numbers).
---------------------------------
So my question is this: can one please find logical failure(s) in my arguments?
