Gödel’s Incompleteness Theorem (GIT)

iSETI Podcast 06/08/2008 (Duration 00:15:14)

 

 

Wikipedia gives Gödel’s Incompleteness Theorem as:

For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

Keeping it simple, ‘theory’ refers to axioms and theorems that are taken as true because they are implied by the axioms; ‘provable in the theory’ means derivable from the axioms, and is ‘consistent’ if it never proves a contradiction; ‘can be constructed’ means that some mechanical procedure exists which can construct the statement; ‘elementary arithmetic’ consists merely of addition and multiplication over the natural numbers.  That is, one can construct true but unprovable statements. Caution: distilling a theorem into a seven-word statement can cause a theorem to lose some of its richness.

Gödel’s Incompleteness Theorem (GIT) is an amazing piece of work. It is amazing in that there are two apparently opposite interpretations. John D Barrow in his book New Theories of Everything expounds the problems of mathematics in understanding the laws of physics of the Universe. As Barrow states,

Superficially, it appears that all human investigation of the Universe must be limited. Science is based on mathematics; mathematics cannot discover all truths. This is an argument that is often heard.

I believe that this pessimistic interpretation was originally given by Hermann Weyl who was noted to have stated that GIT “has been a constant drain on the enthusiasm”.

I see that there are essentially two related problems with the use of mathematics in physics. Don’t get me wrong, mathematics is absolutely essential to physics. However, mathematics is a tool to describe the physical world, while the physical world is reality. It is the reality that makes mathematics invaluable.

The two problems are ‘what are the implications of Gödel’s Incompleteness Theorem’, and ‘what does it mean when a mathematical description of the physical Universe cannot be solved?’

With respect to the first problem, Barrow states that our interpretation of Gödel’s Incompleteness Theorem is a matter of perspective. If one is a pessimist, then GIT takes on Herman Weyl and Stanley Jaki‘s interpretation that (to quote Barrow) GIT ‘prevents us from gaining an understanding of the cosmos as a necessary truth, …, and so is a fundamental barrier to human understanding of the Universe.’

If one is an optimist then GIT takes on Freeman Dyson‘s interpretation that GIT ‘proves that the world of pure mathematics is inexhaustible’. My own view is derived from Roger Penrose in his book The Road to Reality. Dyson and Penrose both have an optimistic view of GIT which I share.

I read Jaki’s paper The Jaki-Godel Theorem. This paper is a ‘must read’ for those who are interested in GIT. Actually, if you are not a Theory of Everything (TOE) guy than Jaki’s interpretation is not pessimistic. It is realistic. If you are a TOE guy than GIT is pessimistic, in that to quote Jaki,

Herein lies the ultimate bearing of Gödel’s theorem on physics. It does not mean at all the end of physics. It means only the death knell on endeavours that aim at a final theory according to which the physical world is what it is and cannot be anything else. Gödel’s theorem does not mean that physicists cannot come up with a theory of everything … such a theory cannot be taken for something which is necessarily true. Apart from Gödel’s theorem, such a theory cannot be a guarantee that in the future nothing essentially new would be discovered in the physical universe which would then demand another final theory and so on. Regress to infinity is no answer to a question that keeps generating itself with each answer.

In my book An Introduction to Gravity Modification I had, without prior knowledge of Jaki’s work, come up with a similar interpretation, and state it as the Preferred Theory Axiom, as that theory, Ti, that explains Nature’s method of operation, MN, for all and any appropriate factors, fk. In reality, however, we are not at the point where we can say for sure that one of these theories, Ti, is the one correct method, MN, that Nature operates by and therefore, we do not know what MN really is.

That is, to formally recognize Gödel’s Incompleteness Theorem in my work I have separated the mathematical description of the theory, Ti, from the physical reality of Nature’s method of operation, MN.

I think the issue is not whether one is a pessimist or an optimist as Barrow suggests, but where one’s focus is. Take a look at the seven words again, ‘one can construct true but unprovable statements’. If one’s focus is on ‘one can construct true statements’ than we get the Dyson-Penrose perspective that what we can construct is inexhaustible. If on the other hand, one’s focus is on ‘unprovable statements’ than we get the Weyl-Jaki perspective that at least some of our constructs will be unprovable.

But this raises the second problem, put more succinctly it is, ‘can the Universe be describe in purely provable mathematical terms?’ Barrow’s tentative conclusion is that GIT will limit the sorts of questions we can answer. This is also in part a challenge to GIT as an area of physics, in that if there is a part of the Universe that cannot be described in arithmetic terms, then GIT no longer holds.

Earlier I had presented the second problem as ‘what does it mean when a mathematical description of the physical Universe cannot be solved’. This question places a wedge that separates mathematics from physics.

The Russian mathematician Vladimir Arnold had posed a challenge whether it was possible to have a general mathematical criterion to decide whether any equilibrium was stable. Francisco Doria and Newton da Costa showed that this was not possible, and also showed that a similar question, ‘will the orbit of a particle become chaotic?’ is also undecidable. By undecidable one means if neither that formula nor its negation can be proved within the theory, or keeping it very simple, an outcome can be found to be part of a theory in a finite number of evaluations.

If we look at the Universe, all its processes are ‘solved’ and ‘known’ outcomes. Each ‘iteration’ of the Universe’s physical processes leads to a known or unknown ‘solution’. One can infer that the Universe consists of solved, decidable processes. ‘Decidable’ in that the outcome is known in the next iteration or maybe prior to the next iteration. There is no question that a physical reality cannot be ‘solved’, as in the manner a mathematical equation cannot be solved. The Universe does not stop in mid-process because it cannot ‘solve’ the completion of its processes. The Universe, as we know it, does not have a region or kenos, where the laws of physics are indeterminate. The Indeterminate Kenos (see An Introduction to Gravity Modification for definition of kenos).

Note, I use the term ‘indeterminate’ and not ‘unknown’.  ‘Indeterminate’ means that the Universe cannot know or figure out or solve for a solution. ‘Unknown’ means that the Universe knows ‘what is next’ but we as humans do not know.

One could argue that in quantum theory outcomes are not known because they are probabilistic, that the outcomes are indeterminate. This is indeterminate as in ‘not fixed’. It is not indeterminate as in ‘undefined’. Arnold, Doria and da Costa’s work showed that mathematical descriptions of the physical Universe have outcomes that could be undefined because they are undecidable. However, the Universe has all the outcomes clearly defined and also knows which outcome to select, and when. In quantum theory we don’t know which outcome will be selected and therefore term this type of behavior ‘probabilistic’.

So what does this mean? It means that all descriptions and laws of physical reality aka Universe, have both known and unknown solutions. This is where mathematics diverges from the laws of physics. In the physical universe indeterminate or undecidable outcomes are not allowed, but are allowed in mathematics.

That implies a mathematical description of the physical Universe that produces indeterminate, undecidable or unsolvable outcomes is either representationally insufficient because it cannot forecast how a physical process will unfold or incorrect because it cannot facilitate further decidable enhancements.

This also implies that for substantial theory driven technological progress we require physical theories that meet four criteria consistent, solvable, decidable, and verifiable. My work in, An Introduction to Gravity Modification meet all four criteria.

With this in mind let us take at look at Michio Kaku‘s classes of impossibilities. Michio Kaku in his book Physics of the Impossible describes three classes of impossibilities. Class I is impossible today but do not violate known laws of physics. Examples are teleportation, and antimatter engines. Class II impossibilities are at the very edge of our understanding of the physical world. Examples are time machines, hyperspace travel, and wormhole travel. And Class III impossibilities violate the known laws of physics. We are in Class 0 impossibility.

It is clearer if we work backwards. Within the limits of our knowledge base we can therefore infer that Class III impossibilities are, inconsistent, unsolvable, undecideable and unverifiable. That is Class III impossibilities do not meet all 4 requirements for a physical theory.

Class II impossibilities are consistent, unsolvable, undecidable and unverifiable. That is they meet only the first of the 4 requirements.

Class I impossibilities are consistent, unsolvable, undecidable but verifiable, that is they meet only 2 of the 4 requirements for a physical theory.

And therefore, we can say that Class 0 impossibilities are consistent, solvable, decidable and verifiable. Note, however, that if a hypothesis is consistent, solvable and/or decidable but unverifiable, it is only of a theoretical interest with no practical or physical value.

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About Benjamin Solomon
Ben Solomon is a Committee Member of the Nuclear and Future Flight Propulsion Technical Committee, American Institute of Aeronautics & Astronautics (AIAA), and author of An Introduction to Gravity Modification and Super Physics for Super Technologies: Replacing Bohr, Heisenberg, Schrödinger & Einstein (with Kindle Version)

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