Disproof of Gödel Theorem

[SelfSim]

@essentialsaltes is right: Gödel's Incompleteness Theorem is about the limits of systems in which one reasons deductively from a set of axioms. It is not about the kind of inductive reasoning one does in science, nor is it about the kind of reasoning one normally sees in theology. Gödel's theorem tells us nothing about the existence of God or the accuracy of the theory of evolution.

I generally agree .. however, both axiomatic systems and theology accept the belief in statements held as being true .. whereas science doesn't .. science tests and verifies then moves forward on those results.
Believed (assumed) truths are not shared with science.

 

[SelfSim]

Please make difference between your word "proven" and my word "confirmed, tested".

The distinction is assumed truth. Verified, 'tested' doesn't require assumed truths.

 

[FrumiousBandersnatch]

That certainly appears to be the inescapable conclusion when going up against Gödel's Incompleteness theorem whilst simultaneously demonstrating total ignorance of the depth of thinking behind it!

It was the ignorance of what the theorem actually said that gave me the facepalm headache.

 

[SelfSim]

It was the ignorance of what the theorem actually said that gave me the facepalm headache.

I'll confess that I'm a bit of a Gödel fan .. the depth of thinking this guy achieved, I think, stands alone. Never mind my admiration either .. he has the admiration of some of the all-time absolute masters of mathematics .. and I don't think any of them have ever come close to challenging the logic in his various works. Simply amazing legacy.

 

[FrumiousBandersnatch]

LOL. If you walk off a cliff, you may test the truth of gravity.

Strictly speaking, confirming a particular instance doesn't make a general truth. Something like gravity may be beyond all reasonable doubt, but - as Hume pointed out - the problem of induction applies to all such observations of the world. Truth and proof apply to statements of logic (which can refer to observations of the world) and mathematics.

 

[FrumiousBandersnatch]

I'll confess that I'm a bit of a Gödel fan .. the depth of thinking this guy achieved, I think, stands alone. Never mind my admiration either .. he has the admiration of some of the all-time absolute masters of mathematics .. and I don't think any of them have ever come close to challenging the logic in his various works. Simply amazing legacy.

Yes - although, as is often the case with masterful insights - his was primed by prior self-referential set-theoretic work, e.g. Zermelo & Russell's paradox, etc.

 

[Wrangler]

Strictly speaking, confirming a particular instance doesn't make a general truth.

Appeal to Strawman. I was responding to the statement that truth cannot be tested. To deny the truth of my example on the grounds that it does not confirm a "general" truth is a Strawman.

Walking off a cliff is a test of the truth of gravity. From a known truth, one can begin to use the powers of logic to arrive at a "general" truth.

 

[VirOptimus]

Appeal to Strawman. I was responding to the statement that truth cannot be tested. To deny the truth of my example on the grounds that it does not confirm a "general" truth is a Strawman.

Walking off a cliff is a test of the truth of gravity. From a known truth, one can begin to use the powers of logic to arrive at a "general" truth.

Are you speaking about logic? Or science? Or the more casual meaning of "truth"?

The theory of gravity can never be proven, as scientific theories arent.

Gravity itself however is a fact. Just like evolution is a fact and the ToE explains how it works.

 

[FrumiousBandersnatch]

Appeal to Strawman. I was responding to the statement that truth cannot be tested. To deny the truth of my example on the grounds that it does not confirm a "general" truth is a Strawman.

Walking off a cliff is a test of the truth of gravity. From a known truth, one can begin to use the powers of logic to arrive at a "general" truth.

OK, my mistake - I thought when you said 'test the truth of gravity' you were referring to testing the theory of gravity as a general truth about the world, i.e. a specific universal attraction between masses. It's trivially true that we have observed things falling towards the earth and have called this phenomenon 'gravity'; that really doesn't need testing, it's observational fact.

But how you go from the known truth of a single observation, e.g. "under these circumstances I observed X", or even multiple observations, "under these circumstances I have always observed X", to a general truth, e.g. "under these circumstances I will always observe X", by the powers of logic, remains unclear. This is the problem of induction.

 

[Wrangler]

Are you speaking about logic? Or science? Or the more casual meaning of "truth"?

The theory of gravity can never be proven, as scientific theories arent.

Gravity itself however is a fact. Just like evolution is a fact and the ToE explains how it works.

My friend. Saying over and over again that scientific theories cannot be proven is not becoming and is wrong. Gravity is not a theory but a scientific law.

The parsing you are doing between logic and science is nonsense. Science IS the application of logic. Logic IS the non-contradictory identification of truth. To parse truth in some way different from casual meaning of "truth" suggests you are playing all manner of word games.

 

[VirOptimus]

My friend. Saying over and over again that scientific theories cannot be proven is not becoming and is wrong. Gravity is not a theory but a scientific law.

The parsing you are doing between logic and science is nonsense. Science IS the application of logic. Logic IS the non-contradictory identification of truth. To parse truth in some way different from casual meaning of "truth" suggests you are playing all manner of word games.

Uh, this is wrong. All of it.

I suggest science 101.

 

[essentialsaltes]

My friend. Saying over and over again that scientific theories cannot be proven is not becoming and is wrong. Gravity is not a theory but a scientific law.

While Newton was justly proud of his Law of Universal Gravitation, one of the great discoveries of the early 20th century was that it is wrong.

We must hope, therefore, that his law was not proven, otherwise we will all disappear in a puff of logic.

For that and other reasons, scientists no longer talk about 'laws' except for those (like Newton's) that have a historical basis.

Gravity is a fact. Theories of gravity are attempts to explain or describe that fact.

 

[Ophiolite]

My friend. Saying over and over again that scientific theories cannot be proven is not becoming and is wrong.

You have been corrected on this already by two other members. However, your misunderstanding here is so egregious that a third correction should not go amiss.

Science does not prove things. It constructs plausible explanations for observed phenomena, based upon observation and experiment. When the observations and experiments are sufficiently "dense" and the explanation sufficiently detailed, we have a theory. A theory is as good as it gets in science and that is very good indeed.

Scientific laws are a historical anachronism.

The above is offered not as an attack, but an educational experience.

 

[sjastro]

My friend. Saying over and over again that scientific theories cannot be proven is not becoming and is wrong. Gravity is not a theory but a scientific law.

The parsing you are doing between logic and science is nonsense. Science IS the application of logic. Logic IS the non-contradictory identification of truth. To parse truth in some way different from casual meaning of "truth" suggests you are playing all manner of word games.

This is not correct.
The field equations for General Relativity is a theory of gravity.

Rₐₑ - (1/2)Rgₐₑ = (8πG/c⁴)Tₐₑ

The terms on the left hand side of equations are tensors.
The Tₐₑ term on the right hand side of the equation can be pseudotensor.
From a pure mathematical perspective the equations are illogical as a tensor cannot be a pseudotensor.
Einstein described the equations as the left hand side being made of marble (maths and logic) and the right hand side made from straw (physical insight and intuition).

The equations are examples of a mathematical model which by definition is not as rigorous as pure mathematics.
The reason is straightforward; mathematical models are approximations and are not designed to prove anything.
Theories that are built on mathematical models are not subject to “proof” or “truth”, but on observations and experiments which either supports the theory or contradicts it.

When the field equations are subjected to the conditions of weak gravity fields and velocities well below the speed of light they breakdown to the following equation.

∇²V = -4πGρ.

This is Newton’s theory of gravity expressed as a gravitational potential V.
Newton’s theory is therefore an approximation of Einstein’s theory.
Each successive theory of gravity is more accurate than the theory it replaces; it is not based on truth.

 

[SelfSim]

... Walking off a cliff is a test of the truth of gravity. From a known truth, one can begin to use the powers of logic to arrive at a "general" truth.

Walking off a cliff is a test that gives meaning to the term 'gravity'.
There is no objective test that we can apply which can lead to the conclusion that gravity exists independently of that meaning. (If you don't agree, please cite such a test). Put more simply, 'gravity' is a word which describes a concept (or model) we came up with to describe what scientifically thinking minds observe, when someone walks off a cliff, (etc).

The idea that gravity exists independently from the meaning the scientific process has developed is a belief where: 'A belief is a notion held as being true for any reason'.

The scientific process is what defines science. It is published and taught widely around the world (feel free to check up on this). A highly simplied version is: speculation, formation of a testable hypothesis, testing (of that hypothesis), recording of the results and forming a conclusion (feel free to check up on this from a reputable source). There is nothing more to it than that and .. it requires no preassumptions such as the existence of truth or some kind of mind independent reality. The latter is just added baggage that comes with beliefs.

Every testable object in science has an operational definition .. Science only uses operational definitions in testing .. there are no exceptions. In general, these definitions are already tested and are never held as 'being' fixed assumptions held independently from these tested meanings (as in 'truths').

 

[SelfSim]

... The equations are examples of a mathematical model which by definition is not as rigorous as pure mathematics.
The reason is straightforward; mathematical models are approximations and are not designed to prove anything.
Theories that are built on mathematical models are not subject to “proof” or “truth”, but on observations and experiments which either supports the theory or contradicts it.
...
Each successive theory of gravity is more accurate than the theory it replaces; it is not based on truth.

Just wanted to repost the above words. They capture the delicate interplay between the 'hardness' of math logic (linguistically .. 'models') used in science, and the never ending human quest to improve on their accuracy by way of comparison with observational measurements.

(I hestitated a little over the term 'approximations' ... I interpret that as meaning approximation with observations, however others (Realists) sometimes interpret this as being approximations to some kind of independent Reality, the latter of which I claim as being a belief .. albeit by way of a testable definition of belief). Either way, these differing interpretations don't impact getting on with doing the science anyway .. which is kind of 'unifying'.

 

[DavidFirth]

First, Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. They are not all inclusive of every axiomatic system you can think of.

Second, you cant disprove a theorem. A theorem is, by definition, a proven conclusion based on the premises presented in the theorem. It is, by the strictness of basic arithmetic, a proven mathematical fact. There is no formal axiomatic system capable of modelling basic arithmetic, period, end of story. There is no logically acceptable debate about it whatsoever.

 

[SelfSim]

Yes - although, as is often the case with masterful insights - his was primed by prior self-referential set-theoretic work, e.g. Zermelo & Russell's paradox, etc.

Yes .. 'twas a very important phase of history in that they were probing the limits of math's applicability to 'giving access to' what they accepted as being 'reality' (aka Realism).

I'm not a real fan of Russell's philosophy. Wasn't he the one who distilled his so-called Law of Thought?:

i) The law of identity: 'Whatever is, is.'
ii) The law of non-contradiction (alternately the 'law of contradiction'): 'Nothing can both be and not be.'
iii) The law of excluded middle: 'Everything must either be or not be.'

What a pile of truism word-salad! All depends on what is meant by 'is' and 'be'.
Circular reasoning!

 

[joinfree]

First, Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. They are not all inclusive of every axiomatic system you can think of.

Any physical system can be described by the case of Gödel, e.g., a model of a car is formal axiomatic system.

 

[PloverWing]

Any physical system can be described by the case of Gödel, e.g., a model of a car is formal axiomatic system.

Hmm. Okay, then. Can you give me the axioms you would use as a starting point, and some examples of theorems that you would prove about cars?