# B Unbounded Logical Trees

1. Aug 3, 2016

### Look

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 (Text):

p = 0    0  1 1

q = 0    1  0 1
---------------

/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 (Text):

p = 0    0  1 1

q = 0    1  0 1
---------------

/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 (Text):

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 327 = 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 (Text):

*
/ \
/   \
/     \
/       \
/         \
/           \
/             \
/               \
/                 \
/                   \
/                     \
/                       \
/                         \
/                           \
/                             \
/                               \
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 (Text):

*
|\
| \
|  \
|   \
|    \
|     \
|      \
|       \
|        \
|         \
|          \
|           \
|            \
|             \
|              \
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 (Text):

*
|\
| \
|  \
|   \
|    \
|     \
|      \
|       \
|        \
|         \
|          \
|           \
|            \
|             \
|              \
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 aleph0 as the one and only one alternative, is logically insufficient).

Furthermore, being uncountable is based on notions like aleph0, 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 aleph0 as the one and only measurement value of the amount of natural numbers, values like 2aleph0 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?

Last edited: Aug 3, 2016
2. Aug 3, 2016

### Staff: Mentor

Isn't your construction just the same as the one we are used to, only written binary?
Where it has to be mentioned, that the usual construction is binary, too, only that numbers are written $∅, \{∅\}, \{∅, \{∅\}\}$ etc.
And isn't $p → q$ simply the Peano axiom?

3. Aug 3, 2016

### Look

My question is about the possible conclusions that are derived from using an unbounded logical tree, in order to construct natural numbers.

These possible conclusions are logically related to the cardinality of natural numbers, and more generally to the very notions of transfinite cardinals, as observed by using unbounded logical trees (were the unbounded binary tree is some particular case of unbounded logical trees).

4. Aug 3, 2016

### Staff: Mentor

In this case I miss a rigorous set of axioms including deviation rules and a rigor definition of a logical tree and the proof of your assertions according to these axioms.

However, this is only my personal view, because I want to avoid a debate based upon a long text full of interpretable trapdoors instead. E.g. I read fraction in your print outs, and failed to find a definition for them.
The fact you highlighted the term conclusions makes me ask for allowed conclusions, for otherwise it's simply a "nice that we talked about". But as I said, my personal opinion.

5. Aug 3, 2016

### Look

A logical tree can't be but rigorous, otherwise it can't be considered as logical, in the first place.

Please observe how one can use an unbounded logical tree, in order to research an axiom (and in this case ZF Axiom Of Infinity)

At this stage the discussion is only about what "above" the radix point, or the absence of radix point along an unbounded logical tree.

At this stage the "allowed" conclusions is about cardinalty as determined by using or not using the notion of radix point along an unbounded logical tree.

6. Aug 3, 2016

### micromass

Staff Emeritus

7. Aug 3, 2016

### Look

I observed it by myself.

As much as I know, using unbounded logical trees and their logical implications on cardinality, are not found in any professional material of the issue at hand.

8. Aug 3, 2016

### Staff: Mentor

Then it may not be discussed here at the PF. Once you publish your work in an accepted peer-reviewed journal, it can be discussed here. Thread is closed.