I Godel's ITs & the Physical World: Is a ToE impossible?

[stevendaryl]

Then what is the physical mechanism that implements the self-reference used in GIT?

I think you're confused about Godel's incompleteness theorem. The PROOF involves self-reference, but the conclusion has nothing to do with self-reference.

For example, we don't know how to answer the following question:

Can every even integer greater than 2 be written as the sum of two prime numbers?

That question doesn't involve self-reference. But we don't know how to solve it. (For that particular question, there is no proof one way or the other as to whether it is solvable using standard mathematics, or not.)

A general class of problems is this: Given a polynomial equation with integer coefficients, can we decide whether it has integer solutions? That general problem is known to be unsolvable.

It doesn't have anything to do with self-reference, other than the fact that self-reference is used to prove incompleteness.

 

[martinbn]

Can every even integer greater than 2 be written as the sum of two prime numbers?

That question doesn't involve self-reference. But we don't know how to solve it. (For that particular question, there is no proof one way or the other as to whether it is solvable using standard mathematics, or not.)

Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true. So is your example an example that shows incompleteness or is it irrelevent?

 

[stevendaryl]

Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true. So is your example an example that shows incompleteness or is it irrelevent?

It's an example of the type of problem that is unsolvable: Do all elements of this infinite set have property P?

Every specific yes/no question is potentially solvable someday. But there are sets of yes/no questions such that it is provable that it is impossible to solve every problem in the set.

I've already given one example: Does such and such polynomial equation have an integer solution? For any method you choose to answer such questions, there will be questions that your method gets wrong.

Mathematically, there is no function which given an equation (written in ascii, for definiteness) will return True or False depending on whether the equation has an integral solution.

 

[stevendaryl]

Is it undecideable? Just because we cannot solve the question yet, it doesn't mean it cannot be solved. Take Fermat's theorem, there was no proof for a long time. It turned out to be true.

There is actually a double layer of uncertainty in mathematical proofs: Given a mathematical system (such as ZFC set theory), there are statements that are neither provable nor refutable in ZFC. It would be nice to have natural examples, but often proving that something is unprovable is as hard as proving.

Fermat's last theorem is an example. You might be able to prove it. Or you might be able to disprove it. But you could never prove that it's undecidable, because such a proof could be turned into a proof that it is true:

If it's false, it's provably false. Turning that around, if it's not provably false, then it's true. So if it is undecidable, then it must be true.

So if it's undecidable, you could never prove that it is undecideable.

 

[friend]

It's an example of the type of problem that is unsolvable: Do all elements of this infinite set have property P?

Every specific yes/no question is potentially solvable someday. But there are sets of yes/no questions such that it is provable that it is impossible to solve every problem in the set.

I've already given one example: Does such and such polynomial equation have an integer solution? For any method you choose to answer such questions, there will be questions that your method gets wrong.

Mathematically, there is no function which given an equation (written in ascii, for definiteness) will return True or False depending on whether the equation has an integral solution.

Your examples of unprovability would have been well known before Godel so that if they were definitively shown to be precises unprovable, then Godel would have never had need to construct the GIT. For there would be the examples of unprovability without Godel, and everyone would have know it. So your examples don't mean a thing. They only show that we don't know how... yet.

 

[stevendaryl]

Your examples of unprovability would have been well known before Godel so that if they were definitively shown to be precises unprovable.

Before Godel, it was not known whether there were any undecidable statements (neither provable nor disprovable). So I don't know what you are talking about. Some people (most people?) believed that any meaningful ([edit] mathematically precise) statement could either proved or disproved.

This thread is sounding argumentative for no purpose. Is there actually an issue to discuss, or are we just arguing?

 

[Drakkith]

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