Axiom and the Effect of (Thermodynamic) Information Entropy
Processes such as Randomness etc.
1. The YHWH Axiom
The works of Archimedes of
Syracuse (c. 287 BC--212 BC) were made available to a wide audience
based largely on the commentaries written by Eutocius in the 6th
century. Eudoxus of Cnidus (c. 410 to 408 BC--355 to 347 BC)
wrote a mathematical statement used by Archimedes that has come to be
known as the Archimedes axiom. It can be stated as follows.
Let x be any real number.
Then there exists a natural number n such that n > x.
Since the statement is a consequence
of the completeness of real numbers, it may be called an Archimedean
property instead of an axiom. The axiom concept from Archimedes
pointed out that if there are two different sized lines the shorter
line could be laid end to end a finite number of times to create a distance
that exceeds the longer line. This property has no non-trivial
infinitesimals (i.e. no infinitely large or infinitely small elements.)
In decision making, if a utility
for general understanding is to be assigned to all (earthly) consequences,
the Archimedean axiom that there is no (earthly) consequence that is
infinitely worse than any other could be required. Of course, there
are eternal consequences which can be construed as infinitely worse
or infinitely better especially regarding the decision of eternal salvation.
At times it seems that two
people with the same theological beliefs behave differently. Either
they are self-deceived about their theology or they are dishonest about
their true beliefs. People can claim to be whatever they choose.
Even if a person is honest they are finite and can fail to behave as
they desire due to ignorance or temporarily yielding to temptation.
To avoid fuzzy or relativist analysis of behavior, behavior should be
compared to compliance with or noncompliance with Natural Law of Freewill
(NLF). That can produce a result that can be objectively compared
to other behavior when comparisons are in order, such as for employment
If a Christian church denomination
has a convention and some of its members support a government action
and some members do not support that government action, their opposing
views can be compared to NLF. That does not mean that all Christians
inherently have different views but that there are different levels
of abilities and experience regarding implementing NLF.
The omniscient bracket of game theory leads to Yahweh
or YHWH and the subsequent axiom. A YHWH axiom could be stated
Let x be any behavior.
Then there exists a Natural Law of Freewill, NLF, such that
NLF > x.
A simplified proof could be
thought of as either of the two following statements.
(i) Given any behavioral
choice, there always is a controlling NLF.
(ii) Given any NLF, there
always is a behavioral choice that is a subset of that NLF.
It is often logically concluded
that (i) and (ii) are intuitive.
Intuition is considered to
be direct knowledge that is seemingly without preliminary cognition
or thinking. Most people do not want to be murdered or stolen
from and intuitively accept those two NLF. Counter-intuitiveness
is something real or perceived that is counter to beliefs, derived from
experience, which may be either true or false as determined by proofs.
An example of a truth is the belief that the earth orbits the sun when
it was previously believed that the sun orbited the earth. An
example of a falsehood is the belief that there is no God which is counter-intuitive
to all evidence that proves the necessity of the existence of the God
of the first cause of all effects--YHWH.
Those examples demonstrate
the subjective nature of intuition and limits what is referred to as
counter-intuitiveness because there are different chosen beliefs.
The recent writings of noted Darwinian evolutionists support the murdering
of babies for research and decry the stupidity of the notion
of human dignity contrary to NLF. In June 2008,
the U.S. Supreme Court overruled state laws that allowed for the death
penalty to be justly included for consideration in cases for child rapists.
It is epistemologically rational, for Darwinian evolutionist theology,
to reason that if children can be legally murdered then raping them
should not carry the possibility of being punished by death.
Despite not having any Divine
authority or standard, the former atheist, Antony Flew (now a theist)
chose to value some NLF he called norms while other atheists choose
to value a specific NLF such as peace. (The value of peace
was confirmed to be held by Eric Schechter in an email exchange between
Schechter and Dallas F. Bell Jr. in June, 2008.)
The truth, includes philosophy
and science, will always be generally opposed by those that reject truth
and by necessity embrace untruth expressed in vain philosophy and so-called
science within the arena of information.
Claude Shannon (1916-2001)
is considered to be the father of information theory. Information
theory qualifies information in the academic branches of mathematics
and engineering with applications in statistics, linguistics, cryptography,
neurobiology and computing etc.
Shannon entropy or information
entropy is a key measure of information for information theory.
Information is defined as a measure of the amount of knowledge and not
the knowledge itself. The amount of information does not reveal
anything about itself, its location or its form. This idea could
be juxtaposed with bundle theory where something is described as consisting
of its properties.
Entropy is defined as the information
one person does not have for a situation which may or may not reflect
the information another person does not have for that situation.
Thus, information is subjective even where entropy is small as between
two educated physicists observing the same situation. Information
is measured in bits. With large numbers, Joules per degree Kelvin
may be used. In that instance of entropy, when using the chain
rule, entropy is not increased by additional input. In mutual
information, entropy is concave and mutual information can be both concave
On the other hand, thermodynamic
information entropy defines information and entropy differently than
does information theory. Information is the object that holds
the information and is not the nonphysical information itself.
For example, a book, a computer, a piece of art, a musical composition,
DNA/RNA in biochemistry etc. Entropy is the amount of physical
degrading of the object of information measured in bits or Joules.
Entropy rates vary due to the potential of the object and its environment.
Entropy or loss to the object will occur unless there is an infusion
from outside the object's system.
An object of information, such
as a computer program, cannot become more complex than its potential
and in time will degrade beyond the point of providing useful information
unless it is infused from an outside source. To say otherwise
is to not just promote the notion of self-perpetuation but to imply
a form of self-creation. Those two ideas are provably false and
are counter-intuitive to most people.
Unfortunately, there are highly
capable atheists like Stephen Wolfram who reject their intuition.
Wolfram's recent writings and book of 1,000 plus pages explains how
he used a computer to run his Cellular Automata (CA) program in an attempt
to disprove the existence of God by serving to disprove laws of thermodynamics.
He had to first ignore that he himself existed from DNA/RNA biochemistry
which is subject to entropy to a future certain date of physical death.
Second, Wolfram had to reject the reality that the computer was made
by other humans and will eventually deteriorate. Third, he had
to reject that his finite CA program only achieved the potential of
what he himself had dictated by his input.
An observer could deduce that
a higher energy or intelligent designer created Wolfram's DNA/RNA,
other humans intelligently created his computer, and Wolfram intelligently
created his program. From Wolfram to his program, entropy is shown
by energy being lost. An intelligent design of overall creation complexity could
be condensed into the following three logical steps.
- The universe has
both order and complexity or ordered complexity.
- Ordered complexity
could not be without a designer.
- A designer created
All sane people agree with
(1), and (3) cannot be disproved. Wolfram inadvertently proved
(2) to be true and the reality of freewill contrary to section 12.7
of his book.
(In an email exchange between
Catherine Boucher, at Wolfram research, and Dallas F. Bell Jr.
during June, 2008, that section was recommended concerning the subject
of freewill.) Wolfram wrote that he believed that freewill
is an illusion because all the controlling details are not known.
This means that his CA program behaved only as he programmed it and
could not have witnessed complexity coming from non complexity without
his input. If he had incorrectly argued for freewill, his CA program
would then have had to have self-analysis or consciousness as humans
with freewill have within the bounds of NLF. Either choice proved
that there was a designer of his CA program's ordered complexity as
perceived by him.
Thereby, efforts in information
entropy, such as Wolfram's, end up demonstrating the reality of truth
and NLF. They also provide a vehicle to address related effects
that are often described with names like randomness.
4. Randomness etc.
Probability is the study of
coincidence, arbitrariness and chance or randomness. Coincidence
is the alignment of two or more seemingly unrelated events that are
perceived as having a non causal origin. The perceiver(s) of that
occurrence usually places more importance on that event than those not
directly affected by it. Arbitrariness (Latin from arbitrarius
meaning someone that must judge a matter) is considered to be choices
made without underlying logic. Chance or randomness is used to
describe a seeming lack of order from cause to effect. (That
is a view of randomness held by Louis Anthony Cox Jr. expressed in an
email exchange with Dallas F. Bell Jr. in November, 2007.)
In a Newtonian sense, there
are no probabilities if all the variables are known. If all the
numerous conditions are known for a person shooting an arrow with a
bow, then it would be perfectly obvious where the arrow would strike.
The notion of randomness proposes that there is a state where no law
exists and where everything is possible, such as an apple becoming an
Randomness describes lawless
variance in change. A randomness proof would need to consider
both the physical realm (e.g. electrons etc.) and the nonphysical realm
(e.g. numbers etc.) Likewise, disproving randomness would need
to address those same issues. Disproving randomness could be as
*Observe the variables of a
-Something in state A
changed to state B.
*Observe the known laws:
-Something cannot not be something.
It is, if not the sum of its parts.
-Cause is necessary for state
A to change to state B.
-The cause is not without causal
-The change is entropic unless
inputted by energy from outside the process/system.
*Then, something is caused
to change from state A to state B within causal parameters.
So whether all variables are known or not, true randomness cannot occur
on any known level. That proof of no randomness would be applied
to quantum phenomena and other unknowns.
The Axiom of Choice (AC) was
formally created around 1904 by Ernst Zermelo. It described the
mechanism of arbitrary choice. A simple AC form may be the choice
from a group of nonempty sets. A choice can be made from a member
of each set in the group. A choice function exists for each set
even if there are no rules. If a choice is to pick an apple from
each set of a group of baskets full of the same type of red apples,
picking the apple on the top of each basket could require using AC.
However, if the baskets contain only one green apple and only one red
apple each, an algorithm of always choosing the green apple first could
The AC could be applied to
infinite untruth such as randomness where untruth is defined as the
group of sets of all possibilities that are not truth. It would
not be applied for truth such as eternal salvation where formal logic
could be applied.
The YHWH axiom demonstrates
the intuitive link between NLF and behavior. Information entropy
in thermodynamics is shown to be as useful a definition as is found
in information theory. The disproval of randomness shows how its
counter-intuitiveness is indeed false. The choice is then axiomatic
of whether to accept, by freewill, the truth of the God of the first
cause of all effects or by necessity reject Him and suffer the present
and eternal consequences.
RIGHTS RESERVED © 2008 DALLAS F. BELL, JR.---------------