I. AN EQUATION FOR THE END OF THE WORLD
If one were to view the Incompleteness Theorems through high drama, this would be that adrenaline tale. This epic is charged as an operatic expanse that breaches all. It is overload so exquisite that it veers into propaganda. Let us tell this tale from tomorrow looking back.
On October 7, 1930, none blinked as the planet trembled. The world was oblivious to grand schemes embarking on their venture. The grand began as passing beyond our sighs. Kurt Godel unleashed his Incompleteness Theorems in Kroninburg, Germany. Johnny Von Neumann was one of the few that noticed.
Years later, Einstein goes to Princeton in order to walk Godel home from class. Godel and Von Neumann are members of a group at Princeton. The purpose of this group is to come up with things that are utterly useless. Von Neumann faces banishment from the group for breaking the rules. Von Neumann originated one of the first computers. That is something that the world could use and Von Neumann broke the code.
Von Neumann understood that the Incompleteness Theorems would lay the way for computers. The path of breaking science became the price paid for certainty. Godel obliterated the foundations of mathematics and the fall of science began. There was anarchy in Oz and the shatter of its instruments was deafening.
As Babylonian gods lay dying, the charge of a new uncertainty permeated the midst of things. Fields of logic entered a renaissance and many believed that anything impossible could become probable. Matters were jettisoned into oblivion and indifference surprised our hearts.
II. A THEOREM FOR THE PRONOUNCED
Oblivious is a state of oblivion and that could be a good thing until you get there. DC and Marvel are the two great comic book companies. The DC Universe and the Marvel Universe are where their respective characters unfold their stories. Mephisto is the Marvel version of the Devil, while DC has Darkseid as their resident king of evil. In DC, Darkseid reigns from Apokolips, which is their version of Hell. Darkseid has the Anti-Life Equation. The Anti-Life Equation is like a personal version of Hell that you can take home. It is similar to burned, delivery pizza or an elaborate and extra-dimensional version of a tourism brochure.
The Anti-Life Equation works through a process of reciting the equation, becoming possessed by evil, going mad, embarking on a killing spree, and then you die. One would not expect this to become a fad. In the DC world of things, the Anti-Life Equation will usher forth the end of the world. If one were to contrive a theory for the end of the world, then the Incompleteness Theorems would be a suitable place to begin.
‘Oblivious’ might be a good thing if foundations begin to crumble. If very few know why, then very few will panic before that end takes place. The reason for this state of obliviousness is that many people are not interested in fundamentals, the essence of things. Few people base their interests upon the essence of life, nor do they wish to know such matters until the improbable occurs. Despite the obvious loom of death, many of us live as if their death is improbable. Death does not lie in the impossible and this fact of life has precedence. We just choose to look at it as improbable because such delusions comfort us. Oblivion is never seen coming until it is too late to be saved.
III. THE APOCALYPSE THEOREM
There are various tweaks, translations, and versions of the Incompleteness Theorems. The same principle holds true for a laymen’s synopsis of these theorems:
THE RATIONAL SYNOPSIS
A system can not explain itself.
System A explains system B.
THE SCI-FI SYNOPSIS
If the universe were universal, then it would be accountable.
You need another universe to explain this universe.
THE SIBLING SYNOPSIS
It takes one to know one.
It takes another one to know one.
THE UNIVERSITY SYNOPSIS
If fraternities were universal, then drinking would count.
One drunk needs another drunk to explain himself.
Foundations should start from the beginning. Matters of this nature requires which beginning should first be determined between mathematics, logic, language, and philosophy. When one has satisfied what constitutes the beginning of the beginning, then one may begin with foundations. The questions required are those that determine what constitutes a sentence, proof, truth, and axioms. There is a maddening legitimacy to what seems like pothead philosophy between consenting hippies.
Once things had been settled and organized, Godel turned all of this into symbolic form. When symbolic form had been achieved, Godel then used that form to state that all of life may not be compressed into symbolic form, in fact, it is impossible. He had proven the non-provability of his own proof.
There are tricks in Godel’s proof as there are tricks in life. The real trick is to make sense of the tricks. The stumbling point of making mathematics fully axiomatic is the notion of paradox. Paradox may legitimately exist in life. It may also rise from flaws within language itself. Since mathematics is a language and If language has holes, then mathematics would also have holes.
IV. GODEL CODING
The Incompleteness Theorem is fundamental and yet not so simple. Godel went to great lengths to illustrate how systems are unable to fully complete themselves. Great lengths were taken to enable those details because Godel had used the whole of mathematics to prove his point. Things were broken down to a workable essence. The essence of any matter is packed, viable, and magnetic in its allure.
Godel numbering is a beauty of code. It was clever to match statements with prime numbers. This was followed by an arithmetical treatment of syntax that enabled “proving a statement” to being replaced with “testing whether a number has a given property”. A self-referential formula is constructed in a way that avoids any infinite regress of definitions. Another way of stating this would be English having a twin nature of mathematics. One may “say something” or “test its logic/grammar”.
A loose simile would also be writing something in English and then testing it through mathematics. One alphabet represents two languages. The language for words shares an alphabet for the language of numbers. Both facets come together as a balance of tongues. One tongue for the epic and sweeping, another tongue for the logically microscopic. Both tongues check and balance one another as an embodiment for metamathematics. The Greeks used their alphabet as a numbering system. That should explain how philosophy would enter into mathematics.
Prime numbers are another method of assigning a unique numbers to statements in order to formulate them into an axiomatic system. These Godel numbers are able to mechanically convert back and forth between formulas and Godel numbers; or English and mathematics. Code is a mere idea
of mapping symbols to meaning. This notion embodies and is also foundational to language and mathematics. The foundational aspect is another example of the obvious being overlooked.
V. AFTER BEDLAM
Godel’s influence is epic and vital to mathematics, logic, and language. It was also necessary in laying foundational aspects to computers and artificial intelligence. Providence seems to have decreed that Johnny Von Neumann should be in the audience listening to Kurt Godel on October 7, 1930.
Since history is notorious for repeating itself, pop culture also has a parallel to this event. There were about thirty-two people in the audience for the debut performance of the Sex Pistols. Members of that audience went on to form bands, record labels, and musical things of that nature. That is the power of influence and the pedigree of an idea and its fans.
Godel’s work addressed the nature of paradox and how language states the context of a given matter. This veer into philosophy which also embodies an aspect that is applicable to computers and artificial intelligence. Newman was one of the first to grasp the aspect of dealing with the unknown when one has limited states of knowledge. This concept lends itself to the nature of thought, creativity, art, strategy, and life. Godel’s influence has yet to peak at fruition. Matters of this nature bear legend. Godel may or may not live up to the hype or the possibilities derived from that influence.
When Godel opened that door, few flinched and even fewer people noticed. Those that concern themselves with such matters are a small cadre of mathematicians, logicians, philosophers, prophets, and those who clearly see tomorrow.
VI. BEFORE APOCALYPSE BEGINS
Let us return to the beginning to better illustrate the essence of the matter. At the turn of the twentieth century, David Hilbert picked up the mantle of laying down a solid foundation of mathematics. His desire was to make all of mathematics axiomatic. This means that mathematics can be broken down to mechanical behavior. If an error or paradox occurs, then give the wheels and cogs a few moments and they will click things into place. A lot of mathematics has this mechanical behavior. Godel had shown the impossibility of Hilbert’s desire to find a complete and consistent set of axioms for all of mathematics. This is illustrated by his two incompleteness theorems. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an “effective procedure”
is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true but not provable within that system. No system can demonstrate its own consistency.
Statements of a formal theory are written in symbolic form and thus it is possible to mechanically verify that a formal proof from a finite set of axioms is valid. This is automatic proof verification and it is related closely to automated theorem proving. The proof verifier checks that a provided formal proof or instructions that can be followed, in order to create a formal proof is that determines if it is correct. This is not hypothetical, there are systems used today to formalize proofs and then they check their validity. Despite the flaws of mechanistic methods, there is still a growth in those methods. When logic veers into mechanical methods, this becomes applicable to computing.
There are various methods in developing and testing such mechanistic, symbolic forms of logic. One method occurs in choosing a set of axioms whose goal is to prove as many correct results as possible without proving any incorrect results. A set of axioms is complete if for any statement in an axiom’s language, the statement and its negation are provable from the axioms. Godel illustrates that in some cases it is not possible to obtain and effectively generate complete, consistent theory. All of this leads back to one system being unable to explain another system.
VII. HEAVEN AFTER BEDLAM
One may view this as Godel beating Hilbert and his desire for strict formal systems. Another perception would be that those mechanistic systems that were disproved also enabled both Godel and Hilbert to win. This perception is dependent upon the assumption that truth has always been the goal. Truth has a knack for surprise if it is saddled with expectations. Hilbert expected truth to arrive through a particular paradigm. Godel did not seem to have expectations for how truth would work itself out. Hilbert may not have arrived at his desire, yet he did receive something very interesting and useful. If Godel did not usurp Hilbert’s expectations, then what followed may have never been developed.
The formal birth of computers could have been delayed for a few years. Inventions in retrospect seem crude at inception. Computers were an offshoot of Godel’s success and not the reason for his endeavors. What Godel disproved and discarded had provided excellent parameters for the crude methods of early computers. The axiomatic was not deemed to be useless, but useless for its intended purpose. What was invalid for mathematics became vital for computers, as discarded methods were
linked to a better purpose. Boolean Algebra had fallen out of fashion, but its second act proved suitable in fashioning the logic and language of early computing. New applications for an old idea are not uncommon. If bedlam appears dense and complex, then bedlam may not be what it seems.
VIII. A MADDING CALCULUS
A foundational map for reason and science is necessary in order to retain integrity. These foundations rest upon mathematics. A closer inspection of the foundations of mathematics became necessary. This was to have been a simple affair that offered a complex blessing. Mathematics had its own systematized mess that remained unaccountable. This formed an addition to the already numerous bodies of knowledge that constitute and are related to the foundations of reason. This develops into a maddening strata of foundations having sub-foundations and those also contain additional subsets and their issues.
While the Calculus is not considered to be a part of foundational mathematics, it could easily be included based upon irony alone. Calculus will approach closely but never touch. This exemplifies the plight of foundational mathematics. You are allowed to come near, but the nature of macro and micro infinities will always bear a distance. By definition alone, infinity defies any breach.
Philosophy is usually rendered as the culprit that allowed intangibles to enter into reason. Irony and truth are erratic lovers held by a consistency of their historical continuum. Irony is also saddled to a logician infiltrating mathematics with doubt and intangibles. Mathematics encompasses much of the bedrock that is foundational to science. This hindrance became a contagion possessed by its own set of ideals. The pandemic expanse breached the walls of science and then the world at large. Do not think that truth was shaken. Truth carries an independence that is detached from our perceptions. The viability of proof became the stutter that entered into our nervous conversations. It was never truth, but our perceptions and conceptual views of truth that became the flinch which developed into seizures. The serious usurper began as a nervous sweat. The forged had been rendered for tearing. There was anarchy in Oz. The fall of science had been wrought by a mere logician. The Babylonian gods are still dying.
Godel’s Incompleteness Theorems are favored works of art. A returning glance was required to distinguish between Godel’s statements and his influence. What follows are two translations of The Incompleteness Theorems.
capable of expressing
both consistent and complete.
for any consistent
formal theory that proves
certain basic arithmetic truths,
there is an arithmetical
statement that is true,
but not provable in the theory.
(Kleene, 1967, p. 250).
For any formal
effectively generated theory T
basic arithmetical truths
and also certain truths
about formal provability,
if T includes a statement
of its own consistency
then T is inconsistent.
formal system S
a certain amount of
can be carried out
with regard to statements of elementary arithmetic.
For any consistent
formal system S
a certain amount of
can be carried out,
the consistency of S
cannot be proved in S itself.
THE GODELIAN PSI-FI 3 INCOMPLETENESS VEX THEORIA OO3