Gödel and Determinism

Reference: Is there an absolute Will?

Isene provides the following logic in his article:

  1. For a system to be deterministic, its underlying rules must be consistent.
  2. For a system to be deterministic, its underlying rules must be complete.
  3. No system of rules can be both complete and consistent per Godels Incompleteness Theorems.
  4. Thus, no system can be deterministic.

This is how I see it.

Godel’s incompleteness theorem applies only to axiomatic systems capable of doing arithmetic. I do not know if Godel’s argument can be extended to as complex a system as the universe.

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Definitions:

de•ter•min•ism (noun)
1. the doctrine that all facts and events exemplify natural laws.
2. the doctrine that all events, including human choices and decisions, have sufficient causes.

axiomatic system
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

complete
A set of axioms is complete if, for any statement in the axioms’ language, either that statement or its negation is provable from the axioms.

consistent
A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms.

e·nu·mer·ate verb (used with object)
1. to mention separately as if in counting; name one by one; specify, as in a list: Let me enumerate the many flaws in your hypothesis.
2. to ascertain the number of; count.

effectively generated
A formal theory is said to be effectively generated if there is a computer program that, in principle, could enumerate all the axioms of the theory without listing any statements that are not axioms. This is equivalent to the existence of a program that enumerates all the theorems of the theory without enumerating any statements that are not theorems.

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Gödel’s first incompleteness theorem states that:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory…

Gödel’s theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent.

There are complete and consistent lists of axioms for arithmetic that cannot be enumerated by a computer program. For example, one might take all true statements about the natural numbers to be axioms (and no false statements), which gives the theory known as “true arithmetic”. The difficulty is that there is no mechanical way to decide, given a statement about the natural numbers, whether it is an axiom of this theory, and thus there is no effective way to verify a formal proof in this theory.

This may mean that if this universe (with both its physical and spiritual aspects) can be expressed through a consistent set of principles, then there is a truth about this universe that cannot be demonstrated using those set of principles. That truth may look at this universe (as a whole) exactly for what it is. Such a truth may not be derivable from the set of principles that supposedly describe the universe.

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Gödel’s second incompleteness theorem states that:

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that could be formalized in the theory that is proved consistent. The second incompleteness theorem is similar to the Liar’s paradox, “This sentence is false,” which contains an inherent contradiction about its truth value.

This may mean that this universe cannot contain the ultimate truth about itself. The ultimate truth is unknowable from the reference point of this universe.

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If we go by the definition of determinism that all facts and events exemplify natural laws, we cannot say for certain if that is true or not. In other words, not everything may be predictable ahead of its occurrence.

Manifestations may be related to each other in strict logical sequence meaning that any manifestation may be shown to follow from another manifestation. However, it may be impossible to determine how a manifestation may come to be on its own. This is another version of saying, “Absolutes are unattainable.”

So a system may be deterministic only in a relative sense. It can neither be absolutely deterministic, nor can it be absolutely non-deterministic. 

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Comments

  • Chris Thompson  On July 27, 2012 at 12:33 PM

    Space seems to not have resistance as photons seem to not have mass.

  • vinaire  On July 27, 2012 at 2:34 PM

    (1) Space underlies energy and matter. I believe that energy is ripples in space, and matter is condensation of those ripples.

    (2) The degree to which space has transformed into energy and condensed into matter seems to be perceived as endurance (a property of time).

    (3) Another property of time is perceived as sequence of events. This comes from comparing what is there to an impression of what was there. It is a perception of change.

    (4) There is change in terms of location in space. There is also change in terms of degree of transformation from space to energy to matter.

    (5) The two basic elements are, therefore (a) what is changing [space, energy, matter], and (b) awareness of that change [time].

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  • vinaire  On July 27, 2012 at 2:43 PM

    If one looks at light as ripples in the fabric of space, then one need not worry about the motion of the ‘propagation point’. Speed of these ripples depends upon the nature of space and not on the motion of any ‘propagation point’. One can move in the same direction as these ripples, or in the opposite direction. This would only influence the frequency of light one perceives (Doppler effect) that would appear as change in color. It would not effect the speed of light because light is not something separate from space, and space is not moving even when the ripple is moving.

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    • Chris Thompson  On July 27, 2012 at 9:43 PM

      ah, we are back again to EMR, magnetism, and gravity possibly not so much as particles or waves but possibly as yet another kind of manifestation of space. Something measurable but not understood. Months? ago I had written about the possibility of space flowing like a river.

      Possibily space is “drawn” to and “flows” to mass? Possibly inertia is a quality of space rather than of matter?

    • Chris Thompson  On July 28, 2012 at 1:20 PM

      Vinaire: “Speed of these ripples depends upon the nature of space and not on the motion of any ‘propagation point’.”

      Chris: Yes, understood – but this has not been addressed has it? This constant speed of light for all frames of reference is a great counterintuitive mystery of physics, is it not? So much so that it has been allowed to sit in the corner collecting dust, though the law has been used to move the whole of physics forward. Am I mistaken?

    • Chris Thompson  On July 28, 2012 at 1:26 PM

      We have to be careful with our analogies. This conversation gets into similes such as “water through a hose” when comparing “electricity through a wire.” — Has very limited usefulness and is useful mostly for conversational purposes.

      I get the sense of what you are describing and you may be right but we should be careful when comparing radiating EMR to ripples on a pond. One problem that I am looking at is the source of the disturbance. These similes may seem similar like magnetism and gravity seem similar, but these two phenomena are wholly different from one another are they not?

  • vinaire  On July 27, 2012 at 2:49 PM

    A photon does not experience resistance rom space because it is not moving against space, just like a ripple in water is not moving against water.

    Photon does not seem to have mass because it is representation of a ripple and not any kind of medium.

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  • vinaire  On July 27, 2012 at 2:52 PM

    ‘Havingness’ would have to do with manifestation. When there is manifestation then there is ‘have’. When there is no manifestation then there is no ‘have.’

    This universe is quite a manifestation. Space itself is a manifestation.

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  • vinaire  On July 28, 2012 at 6:25 AM

    OK. Let me try it once again:

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    Hubbard: “Granting of Beingness (life) to something. The preclear is as well as he can grant life to things, an action which involves the creation of energy.”

    In my opinion, Granting of Beingness would be letting something (or somebody) be for what it is, and making no efforts to alter it by adding or subtracting to it. You are not giving it life. It is what it is. You are not creating energy. The energy is already there. You are simply becoming aware of it. By granting beingness you are not changing anything out there. You are simply becoming more non-judgmental.

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    Hubbard: “The basic granting of beingness is the thetan duplicating himself as another thinking being.”

    This does not make sense. First of all, ‘Thetan’ is merely a label for a combination of locations, movements, particles and considerations. Such label may be applied to other combinations of locations, movements, particles and considerations. But this is being judgmental.

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    Hubbard: “In the mechanics of the granting of beingness we have ‘orientation point’ and the ‘symbol’. An orientation point is that point in relation to which others have location.”

    There is simply the process of looking and becoming aware of what is there. ‘Orientation point’ and ‘symbols’ are consideration put forth by Hubbard. As far as locations go, they are all relative to one another. There is no absolute location. Points orient themselves to one another in a mutual fashion and become aware of one another.

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    Hubbard: “It is also that point from which the space containing the locations is being created.”

    Locations are the properties of space. Any ‘orientation point’ is also a location and, thus, it is part of the space. There is no point beside space that creates space. If awareness is looked upon as space then it is simply there. It may be made to concentrate at any point within that awareness.

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    Hubbard: “In the orientation point we have our basic definition of space: ‘Space is a viewpoint of dimension’. “

    (1) Space is simply there. It is also awareness. So awareness is simply there.

    (2) Awareness may focus, or concentrate, itself at any point within itself, and one may call that an ‘orientation point’.

    (3) Awareness may also focus, or concentrate, itself on another point within itself, and one may call that a ‘symbol’.

    (4) Awareness may then differentiate between these two points, and one may call that a measure or ‘dimension’.

    (5) Awareness, or space, shall then consist of points (locations) and measurable properties (dimensions). This may be represented mathematically by scales.

    (6) But the awareness, or space, is already there. It is not created. It simply focuses and concentrates itself in various ways.

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    The above is what appears consistent to me. 🙂

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  • Chris Thompson  On July 28, 2012 at 1:09 PM

    You are firing on all cylinders! 😀

    • vinaire  On July 28, 2012 at 4:58 PM

      Well, the above observations are definitely exciting for me… enough for me to put them together in a new essay here.

      Granting of Beingness & Space

      I would not call LRH wrong. He took knowledge to a certain level and, thus, paved the way for better observations to be made..

      LRH is proving to be a great springboard toward greater consistency.

  • Chris Thompson  On July 28, 2012 at 1:11 PM

    The idea that space could not move does not seem consistent to me. This would give space a static property unlike anything else that we know anything about in the universe.

    • vinaire  On July 28, 2012 at 4:49 PM

      I think that any movement, or lack of it, attributed to space can simply be attributed to consideration, because a ‘location’ by itself (when nothing is there) is just a consideration.

      • Chris Thompson  On July 28, 2012 at 5:15 PM

        Physically then, what do you suppose gravity is? What do you suppose is happening?

        • vinaire  On July 28, 2012 at 6:41 PM

          Gravity is associated with mass. Mass seems to be standing waves formed by ripples in the fabric of space. I don’t know how mass affects the surrounding space. Mass seems to produce attractive force in its surrounding, which is the gravitational field.

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        • Chris Thompson  On July 28, 2012 at 8:33 PM

          yes, Vinaire, but since we are speculating . . .

          Why is it that the rate of acceleration of a falling object at 32ft/sec/sec is the same for a feather or a bowling ball? My speculation is that these items are being drawn along floating on a river of space.

        • Chris Thompson  On July 28, 2012 at 8:34 PM

          . . . and it is that space which is moving — pressuring these objects to gravitate toward the mass object.

        • vinaire  On July 29, 2012 at 7:01 AM

          The gravitational acceleration of 32 ft/sec/sec is the proportionality constant between weight and mass. Weight is proportional to mass. That is the mathematical significance.

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        • Chris Thompson  On July 29, 2012 at 11:10 AM

          Yeah, well it doesn’t seem we are exchanging concepts on this. Maybe I could ask this differently.

          If we begin with the idea that mass is attracted to mass, then what are we saying? What is occurring?

          In an atmosphere, when I change the local pressure, I create a vacuum and it is the air pressure which is pushing objects like balloons, etc., toward the lower pressure area. Is there a reason to think that space cannot behave somewhat like this? Can it be that it is space which is attracted to matter as well as matter?

          We have this paradigm that fixates on space being nothing and yet, why would we continue to think this.

        • vinaire  On July 29, 2012 at 11:28 AM

          Gravitational force between two stationary objects is proportional to the product of the two masses. It is also inversely proportional to the square of the distance between the two centers of mass.

          The inverse square law tell me that the gravitational attraction spreads out from the center of mass in a spherical configuration, similar to light from a point source.

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        • Chris Thompson  On July 29, 2012 at 6:58 PM

          We haven’t quite defined “gravitational attraction” have we?

  • vinaire  On July 29, 2012 at 5:16 AM

    I think I can now state my understanding of THETAN with much more clarity as follows:

    A thetan is not a thing. It is a label applied to a point where all spiritual energies and forces are treated as being concentrated, much like we use the label “center of mass” to a point where all mass of an object is treated to be concentrated for the mathematical application of forces. Please see CENTER OF MASS.

    So, underlying the abstract concept of thetan there is an actual structure of energies and forces. This may be what Buddha was describing here: THE STRUCTURE OF “I”.

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  • Chris Thompson  On July 29, 2012 at 11:50 PM

    Magnetic charges attract and repel particles.

    Gravity is different. And not to be confused wtih the magnetic pole phenomena of Earth and some other planets. Magnetic alignment of a planet is different than gravity.

    The intuitive statement of inertia is that, “matter at rest tends to stay at rest and matter in motion tends to stay in motion.”

    Then consider matter accelerating toward matter because of the “attraction” of gravity. The intuitive feeling toward this is that particles are being pulled toward one another because matter draws itself toward other matter. But this is not because of an electrical charge, is it? No. Then what? Can it be possible that matter has a character that draws space-time to itself?

    And if it could be possible, then could we postulate that space-time flowing into matter is the “attractive force” of gravity?

  • vinaire  On July 30, 2012 at 8:26 AM

    Electromagnetic waves are self-propagating. Does that mean they are self-determined?

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    • Chris Thompson  On July 30, 2012 at 3:34 PM

      ah! good one. Maybe.

      Why don’t they diminish as they ripple through space?

    • Chris Thompson  On July 31, 2012 at 10:35 PM

      Does space contain a self-catalyzing quality which is stimulated and activated by an EMR disturbance?

    • vinaire  On August 1, 2012 at 5:32 AM

      Somewhere here there may be a structure, which would explain self-determinism.

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  • Chris Thompson  On January 11, 2013 at 8:11 AM

    Vin: “I don’t think everything is knowable.”

    Chris: This is a good example of “the liar’s paradox.” Everything, in its greatest sense is synonymous with universe. Nothing in its greatest sense is synonymous with “not of the universe.” In the context of those two examples, the following two statements are consistent:

    Everything is knowable.
    Nothing is unknowable.

    Because my two statements cover everything, I shall claim that they are complete. If this were to be true, does it follow that I have proved your original post when you wrote, “This is how I see it. Godel’s incompleteness theorem applies only to axiomatic systems capable of doing arithmetic. I do not know if Godel’s argument can be extended to as complex a system as the universe.”

    Or have I shown something else, such as my ass?

    • vinaire  On January 11, 2013 at 10:17 AM

      Language seems to condition our mind. How can one get out from this limitation of language that even imprisons our thinking.

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      • Chris Thompson  On January 11, 2013 at 10:25 AM

        Ah, the language is solid; manifested; more solid layer than thought. To express, we must create a structure to hold a thought, which is already some type of structure. So the language, once spoken or written does indeed imprison the expressed concept and makes it “hard.”

        • vinaire  On January 11, 2013 at 1:02 PM

          Each symbol is a compressed thought fixed in space.

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