Gödel’s Theorem: Statement
Earlier we referred to Gödel’s Theorem. We would like to do an explore - so that we can more easily refer to the concepts contained therein. Following is a minor description of it.
Gödel’s Theorem has two parts.
If a System is Consistent, Then it is not Complete.
Also if a System is Complete, then it can’t be Consistent.
This means that no System can be both Complete and Consistent.
This is shown in a more concise form below.
Following is an even more concise version, based loosely in symbolic logic.
Verbally:
If N, then Not M
If M, then Not N
For all S, Not Both N And M.
Both N and M can’t both be true for S, any given System.
Implication: Logic can’t Describe Reality Completely
What are the implications of this theorem?
No one consistent Logical System will be able to encompass all of Reality.
There is no such thing as the building blocks of Existence.
Each feature of Reality will have its own System to describe it.
Some of these Systems might intersect.
Some of these may even be subsystems of other Systems.
But there is no one System that is a meta System to the rest.
There is no one System to which all the rest are subsystems.
Specifically, there is no logical System that will be able to describe Reality in its entirety.
Applying Gödel’s Theorem to itself: A Complete System contains Paradox.
Before looking at some diagrams, let us apply Gödel’s Theorem to itself.
The implications are clear:
Gödel’s Incompleteness Theorem applies to all Logical Systems.
Gödel’s System, which is based in Logic, is necessarily Incomplete.
The Incompleteness Theorem doesn’t encompass all Systems.
There could be another System of description that could be Complete.
This System could not be based entirely in Logic.
However, Logic could somehow be part of this theoretical System.
Although this Complete System could exist - this could be true -
This System could not be completely analyzed with Logic -
Which is inherently incomplete if it is consistent.
Thus this Complete System would not be logically consistent.
It would necessarily and simultaneously incorporate Paradox.
Which is certainly not logical.
Gödel Set Diagrams
Before getting too confused, let’s look at some diagrams containing sets.
First, while no Consistent System can be Complete,
The one Complete System would of necessity include all the Consistent Systems within it.
This is shown in the diagram below.
While containing all the Consistent Systems, our Complete System is not Consistent itself. We’ll see why in the next diagram.
The Inherent Paradox in a Complete System
Let a circled letter equal a theorem that is logically derived from the assumptions of one our Consistent Systems, Con1 & Con3.
In our first Consistent System, Con1, Theorems A, B, C, D, E, & K have all been proved.
In another Consistent System, Con3, Theorems F, G, H, J, K & ~E have all been proved.
There are two features that need to be noted.
First, it is possible that Theorem H could never be derived from Con1.
This is why our first logically consistent system, Con1, is Incomplete.
Second while Con1 derived Theorem E, Con3 derived Theorem ~E.
This is logically impossible.
Something and its opposite cannot both be true in a consistent system.
Therefore, our Complete System contains Paradox.
Examples: The Complete Systems of Science & Math
This conclusion might seem counter-intuitive to our cultural mind-set, where Reason conquers all. However, there are many examples of this type of situation in Reality. This is why we’ve spent so much time developing this topic.
The Complete System of Science contains Light as Particle or Wave Paradox
The most common example is found in Science itself.
We will define Science as the Complete System that deals with Matter.
Because it is Complete, it must contain Paradox.
As an example:
Light behaves as a Wave from one perspective and as a Particle from another perspective.
These are mutually exclusive states.
However, the Complete System of Science contains both Systems:
The logically Consistent System that views Light as a Particle,
And the Consistent System that views Light as a Wave,
Even though the two Systems contain contradictory information.
Complete System of Mathematics contains Euclidean & non-Euclidean Geometry
In the Complete System of Mathematics:
If we assume that parallel lines never intersect (a postulate),
Then we can necessarily draw the conclusion that all triangles have 180 degrees.
This postulate and theorem come from traditional Euclidean geometry.
This conclusion applies to flat surfaces.
If we assume that parallel lines can intersect (a different postulate),
Then we can conclude that all triangles have more than 180 degrees.
This postulate and theorem come from non-Euclidean geometry.
This conclusion applies to curved surfaces such as globes.
For decades, scientists, such as Einstein, hoped to resolve the Light/Wave Paradox with a higher level of understanding. For two millenium, mathematicians tried to prove that parallel lines never meet, so that this wouldn't have to be an assumption and logically contradictory conclusion could not be drawn. Everyone was hoping for one truth, not contradictory truths.
Yet there is no need to be dismayed by the apparent contradictions of the logically Consistent systems within the Complete Systems of Science and Math. Because Science and Math aspire to be Complete Systems, they must of necessity contain Paradox. Differing underlying assumptions can lead to mutually exclusive conclusion, as we saw in our mathematical paradox. While the ins and outs of these logical problems can go on forever, they are used here as signposts to another way of approaching Reality - based upon the relativity of Truth.
More Examples: Some Complete Systems of Life
Corporate vs. Individual Systems
Other examples closer to home: Let us contrast the theorems derived from the underlying assumptions of the Corporation with the theorems derived from the underlying assumptions of the Individual. While there are many similarities, there are some theorems that are entirely different and others that are contradictory. For instance the Corporation treats the Worker like a Machine, which produces services. Its assumption is the Work must be done as economically as possible. This is in direct contradiction to the Person’s underlying assumption that being paid the most for the least work is best. The resolution of these two polar tendencies leads to negotiations between the Corporatation and the Individual. The Corporation and the Individual are both logically consistent Systems that belong to the Complete, yet Inconsistent, System of Humans.
Systems of Death and Life
The Theorems of Death are different from the Theorems of Life in the same way - similar, yet Different and Contradictory. For instance: an assumption based upon the inevitability of Death might lead to detachment from all earthly pursuits, while assumptions based upon Life might lead to a heavy involvement in the world of things. Most take an intermediate course, balancing opposites: embracing involvement in Life’s many treasures, then retreating when things become too stressful. Life and Death yield different underlying assumptions, which lay the foundation for Consistent Systems contained in the Complete System of Existence.
Theorems of Person and Being
The Theorems of the Person coincide, contradict, and are different from the Theorems of Being. The Inherent underlying Assumptions attached to Person and Being lead to Consistent Systems that are contained in the Complete yet contradictory System of Conscious/Awareness, for lack of a better word. As an example, the Theorems based around the assumption of Person would lead to the importance of everything that surrounded the Person, including his external circumstances. Then everything becomes very important. This leads to stress - as we attempt to fulfill our Person’s goals, which we assume are our goals. The Theorems associated with the assumption of Being lead to the glory of existence independent of circumstance. While interested in our Person’s hopes and dreams, Being doesn’t invest any importance to them.
The Systems of Longevity and Vitality
The systems of Longevity and Vitality lead to similar and opposed Theorems. For instance, those who are invested in Longevity assume that a long life is most important. Accordingly, they take great pains to protect their Health, perhaps at the expense of Living - protecting Life but afraid to Live. Vitality, on the other hand, assumes that Living is most important and would never consider sacrificing Living for Health. Both merge when Person realizes that Vitality is connected with Longevity - although not necessarily correlated with it.
Self Reflection: to step out of Logical Boxes and embrace holistic Reality
The beauty of Self Reflection is that it enables us to step out of our Logical Boxes –
The Logically Consistent Systems that we set up to deal with Reality.
We aren't required to stick with just one Consistent System.
We easily jump from System to System - depending upon Circumstances.
We choose the System that is most applicable to our Situation.
Our Freedom to Choose allows us to step out of our Logic Boxes - if we want -
To view our Boxes from a more Complete perspective.
This Complete perspective contains Paradox,
As there are contradictions between the logically Consistent Systems
That are contained in the Complete System.
Stepping out of the Box usually has to do with the grander perspective.
Thus we add a layer of Complexity
To encompass these Consistent systems in a more Complete system.
We humans have the ability to embrace the Complete Systems
That are based upon a more holistic view of Reality
Which encompasses and yet transcends logic.
This is one of the Emergent features of Self Reflective Awareness.
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