Understanding Universe from laws extracted from the Universe?

[ExNihilo]

Hi,

Based on Kurt Godel's incompleteness theorem, may be we could never understand the Universe completely because all our knowledge originated from within the Universe itself. Is it a valid application of the incompleteness theorem?

 

[DennisN]

Based on Kurt Godel's incompleteness theorem, may be we could never understand the Universe completely because all our knowledge originated from within the Universe itself. Is it a valid application of the incompleteness theorem?

No, not according to me. The theorem is about mathematics/axioms in mathematics. Physics is about (more or less) accurate models, verifiable explanations and predictions. Two different animals. But this does not mean I say we will come to understand the Universe completely, that question is simply beyond my map.

 

[ExNihilo]

Hum ... I read somewhere longtime ago, that Godel's theorem is generic. Something like you cannot use the knowledge inside a set to claim to understand the whole truth of the set itself. For example, using the precepts of a religion, you cannot claim knowing the whole truth about that religion.

So basically, could the incompleteness theorem could be that generic?

 

[DennisN]

"I read somewhere longtime ago, that Godel's theorem is generic."

Generic in mathematics or outside mathematics? Some might propose this, others will not (e.g. here and here). I wouldn't propose it. This is the basic problem in my opinion:

• A mathematical theorem can be proven or disproven (mathematical proof).
• A scientific theory cannot be proven in the same way, it still needs to be falsifiable.

So if you prove a mathematical theorem and strictly apply it to e.g. physics, physics will not be science anymore. This is one of the reasons why I personally avoid doing such "hard" comparisons between mathematics and physics. And if I venture any further into this, I think the thread eventually will be moved to Philosophy, so I stop here.

 

[Jolb]

I actually started a thread about Godel's Incompleteness theorem and its applicability to physics a few days ago, you may want to read it to see if it answers your question:
https://www.physicsforums.com/showthread.php?t=583692

Anyway, Godel's theorem certainly does not apply in general. In fact, Godel also proved Godel's Completeness Theorem, which basically shows that in a first order logic, all true statements can be proved from the axioms, and thus the true statements in a first-order logic are in fact enumerable (I hesitate to say countable since I don't want to sneak in any mathematical logic). In other words, you can't "Godelize" a formal system of logic without introducing the Peano axioms of number theory, or some other axioms.

Godel's theorem is not a very general statement at all: Godel's proof is merely a construction of a non-provable but also true statement--he just disproves the general idea that "all truths are provable" by demonstrating a counterexample. Another nice example of a proof like this is Cantor's diagonal slash: He provides a proof that real numbers are uncountable by taking an arbitrary countable collection and constructing an additional real number not in the collection, so he proved that there are at least two different infinities, countable and uncountable. To this day, nobody knows whether there are any infinities between these two--this is known as the Continuum Hypothesis.

That kind of proof contrasts with other types of proof. For example, take Archimedes' proofs by exhaustion of the areas of circles/spheres/cones/etc. They go along the lines of "suppose a circle had an area greater than ∏r², let's say ∏r²+ε". He then shows that you could circumscribe a polygon with enough sides such that its area is greater than ∏r² but less than ∏r²+ε. Then likewise he inscribes a polygon and shows its area can be made greater than ∏r²-ε. The proof gives an extremely definitive statement of what IS TRUE, as opposed to Godel or Cantor showing showing that a certain statement is NOT TRUE.

 

[ImaLooser]

I actually started a thread about Godel's Incompleteness theorem and its applicability to physics a few days ago, you may want to read it to see if it answers your question:
https://www.physicsforums.com/showthread.php?t=583692

Anyway, Godel's theorem certainly does not apply in general. In fact, Godel also proved Godel's Completeness Theorem, which basically shows that in a first order logic, all true statements can be proved from the axioms, and thus the true statements in a first-order logic are in fact enumerable (I hesitate to say countable since I don't want to sneak in any mathematical logic). In other words, you can't "Godelize" a formal system of logic without introducing the Peano axioms of number theory, or some other axioms.

Godel's theorem is not a very general statement at all: Godel's proof is merely a construction of a non-provable but also true statement--he just disproves the general idea that "all truths are provable" by demonstrating a counterexample. Another nice example of a proof like this is Cantor's diagonal slash: He provides a proof that real numbers are uncountable by taking an arbitrary countable collection and constructing an additional real number not in the collection, so he proved that there are at least two different infinities, countable and uncountable. To this day, nobody knows whether there are any infinities between these two--this is known as the Continuum Hypothesis.

Godel constructed a theorem of mathematics that was not provable using only mathematics. It was however provable using some extra axioms that allow transfinite induction (Gentzen).

The Continuum Hypothesis is the statement that there are no infinities with cardinalities both greater than the integers and smaller than the reals. It was shown you can assume it is true and get a consistent mathematics. You may also assume that it is false and get a consistent mathematics.

I think that Godel has little or no relevance to physics. They aren't especially concerned about axiomatic proof, what they want are effective theories.