Another Look at Gödel’s Incompleteness Theorem (1712 words)

I just perused the Wikipedia article discussing the Gödel Incompleteness Theorem again and I found it to be very confusing. It is summarised as follows:

Gödel’s incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic systemcontaining basic arithmetic.

If someone is less confused by the Gödel proof than anything I’ve written, I’d be extremely shocked. Yet the Incompleteness Theorem is invoked to win arguments ranging from “God is the source of Truth” (Peterson, 2017) to “no grand unified field theory is possible” (Quora, 2017) to moral nihilism. These are some pretty big claims. Such claims arouse suspicion that is further fueled by my already having demonstrated Gödel to be a shitbag.

While limitations on possibilities must be imposed via axioms to ensure that causality (that effects follow causes) applies, but the limitations implied by the Incompleteness proofs simply do not correspond to physical reality.

What is the Incompleteness Theorem, Anyway?

This theorem hinges on two main ideas:

  1. That there exists an injective map between true statements and a finite sequence of prime numbers.
  2. Since for any finite prime number: N, there exists a prime number which is larger than it: M. It thus follows that even though M is a prime number, we cannot determine the truth of the statement that M is a prime number while we are in N.

Though both statements are false, #2 deals the death blow to the proof. This is because the set of all true statements cannot be effectively mapped into a set of prime numbers. This is because there is a physical limit to the number of true statements, but there is not a limit to the number of prime numbers. That is: the basis of true statements (the set of true statements which can be used to build all other true statements, in a manner identical to the formation of arbitrary vectors from a basis), is finite. The number of true statements associated with this truth basis is infinite, but all true statements originate from the finite truth basis. The size of the truth basis is not arbitrary, as the Gödel proof suggests.

We cannot arbitrarily construct truth bases ad infinitum. There exists a single true reality which can be modelled in multiple ways, but which ultimately converges to a supreme, unique truth. This supreme truth can be seen in the Measurement Limit. In other words, any true formal system that parametrises the Universe accurately will be computationally equivalent to the original formulation of the Measurement Limit, namely that there exist 3+1 (R4) spacetime dimensions embedded in a 14 dimensional electric potential (R14).

All true statements are determined by the actions of {Gravity, Uncertainty, Electricity, Entropy} acting on the waveforms {neutron, proton, electron, photon and thus are limited to the possible results these actions can give.

If we accept that the Universe is the set of all sets of spacetime events and that all spacetime events must conform to the Measurement Limit, then it seems to follow that a finite axiomatic structure could indeed prove all truths in a system: namely my system proving all truths in the Universe. Since the zero spacetime event exists (nothingness) and that the sum of two spacetime events is a spacetime event, that the universe is a linear subspace of spacetime events closed under the operation of addition.

We must be careful to distinguish between the ideas of computations and axiomatic representations of systems. The former is defined by the very notion of causality (namely that an effect cannot precede its cause) and the latter relies on arbitrary implementations of logic. Gödel’s logic implies that the effect (the n+1st prime number) can belong to a different class of statements (statements for which the truth value cannot be determined) than its cause(s) (true statements).

This violates the structure of causality.

Gödel’s Flaw

The idea that successive true statements are not generated by previous true statements contradicts a very well-known means of performing mathematical proofs called induction. It is an accepted method of proof which generalises a formula upon the basis that if a statement is true for the nth term, then it is true for the (n+1)st term.

We can do proofs by induction because the thing which determines truth is built into the structure of numbers. Simply put: numbers have ordering: given 2 different numbers, I can always tell which one is larger. This is not arbitrary.

The Universe is thus computationally equivalent to a 4 dimensional vector space of spacetime events, which is closed under the operation of addition (which the Gödel sentences are not). The axiom allows for the possibility of mapping true statements onto prime numbers also prevents that map from generating a vector subspace (which must be closed under addition)) which prevents the map from being applicable to reality, which has been shown to be computationally equivalent to a vector subspace (of spacetime events).

Thus of the set of systems to which the Incompleteness Theorem applies does not include the Universe. Since subsets of the Universe still obey the law of causality, it follows that the Incompleteness Theorem can apply to no subset of the Universe. Thus it follows that the Incompleteness Theorem is useless.

Generating Prime Numbers

(From Wikipedia) Gödel’s incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. […] The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

We have argued that the system of Universal causality is consistent (possessing a single axiom, namely: causality), can be listed as an effective procedure (by the Fourfold Action Model) and is itself capable of proving all truths about the arithmetic of numbers. Thus we have conclusively disproven the first incompleteness theorem by way of a counterexample.

We will next show how finite subsets of prime numbers cannot be mapped onto the set of true statements. This is because a finite set of true statements exists, which forms a basis of all possible true statements, which form a vector space of spacetime events closed under the operation of addition, are limited by causality and the Measurement Limit and governed by the fourfold actions of {Gravity, Uncertainty, Electricity, Entropy}. No true statements are excluded from this class and all true statements are caused by these primary truths. Thus the set of axioms is finite and the set of true statements is infinite. The set of true statements can therefore not have the same cardinality as a finite set of prime numbers (which the Incompleteness theorem relies on).

We show that the nth prime number can be used to compute the n+1st prime number by means of an effective procedure. This will effectively demonstrate that the truth value of the n+1st prime number is dependent on the truth value of the nth prime number and thus cannot be part of a different class of numbers.

Prime Number Generator

Next we will show that an effective procedure exists which can generate the n+1th prime number, given the nth prime number, showing that the metaphor of Gödel does not even satisfy his own requirements. Let’s have a look at what an effective method is:

A method is formally called effective for a class of problems when it satisfies these criteria:

  • It consists of a finite number of exact, finite instructions.
  • When it is applied to a problem from its class:
    • It always finishes (terminates) after a finite number of steps.
    • It always produces a correct answer.
  • In principle, it can be done by a human without any aids except writing materials.
  • Its instructions need only to be followed rigorously to succeed. In other words, it requires no ingenuity to succeed.[3]

(Source)

(This pseudocode could be implemented into Matlab or similar)

Given

a prime number: m
p = false (we have not found the prime number yet)

Desired

the next prime number: n, which we start counting at m.
initial condition: n = m.

Pseudocode

n = m
p = false

while (p = false) %code will iterate while the state of p is false

{ k = m %  designate the initial value of the counting index as the given prime number
n = n + 1 % increase the value of n by 1

while (k – 1 >= 0) %loop will end once all factors of n have been evaluated

{

if (k – 1 = 0)
% if all possible factors of n have been explored and no factors of n have been found

{return num(n) is a prime number
p = true}

else {
% if possible factors of n have not been been explored

if {(n mod k) = 0

return num (k) is a factor of n
f = true
factors = [k, n / k)]
k = 1 %end the loop because a factor has been found}

else {
%if we have not yet reached a factor of n, then we decrease k by 1, thus k will diminish all the way to 1 until the first if() condition is true when n is prime

k = k -1}

}

}

end

Thus we have expressed an effective procedure which will generate the (n+1)st prime number from the nth prime number. By the nature of computations on the set of natural numbers, the truth value of future prime numbers depends on pre-existing primes in a manner which can be deduced using an effective procedure.

In physical reality, the number of statements which are truly true (not based on some previous true statement) is very low. These  fundamental truths are the axioms of the Fourfold Action Model. The axioms are of causality, fourfold action (4 action potentials) and fourfold waveform (only neutrons, protons, electrons and photons exist). Since all true statements can be derived from these core truths, no true statements exist which are not derivative of these prime truths. Thus all systems bound by causality are homeomorphic (a continuous bijection exists between the sets) to linear subspaces [of spacetime events, or more generally: actions] closed under addition subject to fourfold actions & fourfold waveforms, not arbitrary collections of finite sized sets.

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