co-author: Alexis Clancy
An important coding project of the New Computer Science is to program a Universal Incompleteness Generator in Java.
The program will generate undecidable arithmetical statements from axioms.
A major first step towards achieving this has been taken by Stephen Lee of Tachyos.org.
Lee writes: “Given a set of axioms for arithmetic, Gödel tells us that there is an arithmetical statement which can neither be proved or disproved from those axioms. The purpose of this project is to write a computer program which will take the axioms and use them to generate a statement which is undecidable in that axiom system.”
(the following 5 lines are adapted from the Wikipedia article on Kurt Gödel)
Kurt Gödel was an Austrian-American logician, mathematician, and philosopher.
One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A.N. Whitehead, and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.
Gödel is best known for his two Incompleteness Theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna.
The more famous Incompleteness Theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms.
To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
(end of adaptation from the Wikipedia article on Kurt Gödel)
(beginning of aphorisms by Alexis Clancy)
No matter how sophisticated a logic machine is, there exists some proposition somewhere in the universe that will choke such machine.
Every logic machine has its limit of applicability.
Without Incompleteness, a logic machine is not a true model.
Incompleteness must play a role in an accurate model.
No matter how sophisticated a universe is, if it’s built on rigid axioms, there will always exist somewhere in that universe an unproveable statement.
The mechanics which are permitted by Incompleteness we call Third Space mechanics, that which is beyond the visible and the invisible.
Beyond the seen and the unseen is the Epsilon Cleft.
It is the lip of a Möbius when the Möbius is Twisted.
The Epsilon Cleft is a novelty space.
The Epsilon Cleft is the source of new dimensions which cannot be accessed from a binary system.
When symmetry breaks and more dimensions are required.
An Epsilon quantity must be taken on faith, or else the computation would take an eternity.
We are interested in PII – the Positive Inferences of Incompleteness.
Incompleteness is an opportunity.
Concepts of creative evolution.
We are interested in the infinite characteristics of the Gödel space.
Gödel also made important contributions to proof theory by clarifying the connections between classical logic, intuitionist logic, and modal logic.
Gödel showed that the Continuum Hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent.
But I (Alexis) detect a possible violation of the Continuum Hypothesis in elaborating a varied typology of Incompleteness.
Georg Cantor has established the Aleph notations for countable and uncountable infinities.
Aleph subscript zero is countable infinity.
Aleph subscript one is uncountable infinity.
Aleph subscript Epsilon is a new Aleph subscript, developed by Alexis to describe the inner infinities of Gödel (Incomplete) type spaces.
In our model, there is a Gap and a Jump.
The Gap is the first type of Incompleteness.
The Jump is the second type of Incompleteness.
Exclusion Incompleteness is a third type of Incompleteness.
The information regarding symmetry breaking is instantaneous everywhere.
The Hebrew or Fractal Aleph, which looks like an N, or like the Möbius Twist.
The form of this Glyph fits with the central importance of reversibility, or with the polarity of vectors.
Reversibility in a four-way sense, not just a two-way sense.
The Aleph is the quantum unit present in all Glyphs.
Making it the implicit operator.
The End of Time.
We will be Outside of Time.
The Möbius Twist in the infinity lemniscate [a lemniscate is the mathematical representation of a figure-eight] is a child analogy of raw Incompleteness; the first child, an Adam…
(end of aphorisms by Alexis Clancy)
Incompleteness is fractal or holographic.
We don’t want to build a universal computing device.
We want to build something holographically representative of nature.
A radically ecological computing device.
Based on faith, fiction, cultural history, recursivity, cybernetics.
One valley of Incompleteness opening up to another Incompleteness, another representation of Incompleteness, in a fractal or holographic sense.
Incompleteness is a super-analogy.
We adopt the prefix “super” because it is absolute and time-independent in its superiority; all derivative analogies are subordinate always.
Analogies of a different order.
A subject represented as a lower ordinal analogy than the object.
But all analogies inherit from the first ordinal analogy, which is the Möbius or the Aleph.
The New Computer Science is founded on a paradoxical elementary implicit symbol, the Aleph.
To refound computing on paradox instead of on certainty.
The Aleph exists both symbolically and substantially: a paradoxical foundation.
The first Glyph sets the tone for all subsequent encryptions of narrative.
Aleph is the implicit, silent aspiration; the breath upon which all other utterances are carried.
It visually suggests two hems of timespace, hopping over each other.
The information about symmetry hopping.
