Goedels Incompleteness Theorem

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Gödel's Incompleteness Theorem asserts that within any sufficiently powerful formal system, there are true statements that cannot be proven. This leads to the conclusion that no theory can be both complete and consistent, raising questions about the nature of scientific theories, including a potential theory of everything (TOE). While some argue that all theories are inherently incomplete or inconsistent, others point out that Gödel's theorem applies to mathematical systems rather than physics, suggesting that physics may not be bound by the same limitations. The discussion highlights the philosophical implications of Gödel's work, emphasizing the distinction between mathematical and physical theories. Ultimately, Gödel's theorem serves as a reminder of the limitations of formal systems in capturing all truths.
  • #31
master_coda said:
Euclidean geometry.
Yes, Ok right. Hilbert himself worked on the fondation of it. That is what led him to his program of automatic-mathematic demonstrations somehow.

But apart from that ? This is only one single counter-example. Has anybody since we know Godel's theorem been developping a consistent and complete categorie of formal system ?
 
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  • #33
humanino said:
Yes, Ok right. Hilbert himself worked on the fondation of it. That is what led him to his program of automatic-mathematic demonstrations somehow.

But apart from that ? This is only one single counter-example. Has anybody since we know Godel's theorem been developping a consistent and complete categorie of formal system ?

The theory of real-closed fields was proven to be complete and consistent well after Godel's theorem was published.


Of course, if you actually understood the theorem in any meaningful way then you would already know of examples of complete and consistent systems. However you seem to think that you understand math because you know how to regurgitate explanations that other people have produced.

I'm at a loss to understand why you think the study of physics is any less trivially obvious. After all, physics is nothing more than inventing mathematical theories and testing if they correspond to reality. You've already demonstrated that the math part is obvious, and tests can be understood just as easily - I just have to memorize all the steps of the test. In fact, this applies just as well to all fields of science.

I can certainly see the usefulness of your use of the word "understanding". It reduces every piece of knowledge that ever has been or ever will be discovered to being easy to understand, and obvious. It doesn't allow for any interesting distinctions, but at least it make it easier for me to dismiss areas of study that I have little knowledge of.
 
  • #34
Eye_in_the_Sky said:
1) Do you feel that natural numbers {0,1,2,3, ...} are fairly straightforward objects?

2) Do feel that they are simple enough that, at least "in principle", they are definable by a finite set of axioms?

3) Do you care?

... Please answer "yes" or "no" to each question.

Hello my dear friends. I think you will be very happy to see i am back (?)
My answers :
1)YES
2)NO
3)YES

regards
marlon
 
  • #35
humanino said:
I found the "Presburger arithmetic"
http://en.wikipedia.org/wiki/Presburger_arithmetic
which is consistent and complete ! But "not as powerful as Peano arithmetic"

That's exactly the point. If it were as powerful as Peano arithmetic, then it would be incomplete. How interesting or useful a theory is has nothing to do with how powerful is, so I don't quite see why you think "less powerful" is a problem with a theory.
 
  • #36
matt grime said:
Would you post what you think Goedel's theorem states? Because you obviously are thinking of a different one from the one the rest of us know...

Please, be polite. You are not the only one in here with the ability to read...
The theorem has been stated several times...

regards
marlon

PS thanks Humanino.
 
  • #37
marlon said:
Please, be polite. You are not the only one in here with the ability to read...
The theorem has been stated several times...

regards
marlon

PS thanks Humanino.

It's not rude to ask someone who doesn't understand a theorem to post what they think it is.
 
  • #38
master_coda said:
Of course, if you actually understood the theorem in any meaningful way then you would already know of examples of complete and consistent systems. However you seem to think that you understand math because you know how to regurgitate explanations that other people have produced.
:mad:
I am here, as I told already, only to learn. I feel your posts aggressive. I am certainly not here to fight or justify of my knowledge. I am two years older than you are, you don't look like an old math teacher.
 
  • #39
master_coda said:
That's exactly the point. If it were as powerful as Peano arithmetic, then it would be incomplete. How interesting or useful a theory is has nothing to do with how powerful is, so I don't quite see why you think "less powerful" is a problem with a theory.
see, you have absolutely no sens of humour !
 
  • #40
Galileo said:
Yeah, Popper said a theory has to be falsifiable or it isn't scientific...

His own theory of falsifiability isn't falsifiable. :rolleyes:


This is wishful thinking, I guess. replace scientific by physical.

Falsification itself is not a theory on fysics. It describes how science should work.
 
  • #41
humanino said:
:mad:
I am here, as I told already, only to learn. I feel your posts aggressive. I am certainly not here to fight or justify of my knowledge. I am two years older than you are, you don't look like an old math teacher.

If you were here to learn, then you would have been asking questions and not making assertions. And you wouldn't make assertions like "this field of study is easy because I can memorize the theories you use".

And attaching an "I only want to learn" to your posts does not magically invalidate any criticizms of your remarks.
 
  • #42
Humanino, don't listen to this master-coda-guy...

I think this whole debate is a big mistery to him and he does not get the point. That is why he has absolutely no taste for style. Or he just ate something real bad.


Listen master bla bla bla. Stop interrogating everybody and acting like some professor just because of the first epitheton in your name. Don't call yourself master when in stead you talk like some angry offended boy...

your best friend and mentor

master marlon
 
  • #43
humanino said:
see, you have absolutely no sens of humour !

If you want a link and a quote from that link to be interpreted as humour, you should use a smily face.
 
  • #44
marlon said:
Humanino, don't listen to this master-coda-guy...

I think this whole debate is a big mistery to him and he does not get the point. That is why he has absolutely no taste for style. Or he just ate something real bad.


Listen master bla bla bla. Stop interrogating everybody and acting like some professor just because of the first epitheton in your name. Don't call yourself master when in stead you talk like some angry offended boy...

your best friend and mentor

master marlon

I'm reminded of the cranks in theory development who try and cover up the fact that their ideas make no sense by saying that other people just haven't achieved their level of understanding.
 
  • #45
master_coda said:
If you want a link and a quote from that link to be interpreted as humour, you should use a smily face.


oooooh, thanks for this nice incomplete (i.e. USELESS) statement
 
  • #46
Marlon, I love you.

I want to quote someone, again, since that is the only thing I can do :biggrin:
Michel Berger said:
Contrary to a commonplace idea, mathematics are the easiest science. Why that ? Because they live in their own reality, the deepest and most concrete of all. Indeed, a mathematician (...) works without ad hoc hypothesis. (...) a mathematical result is never challenged : it is forever.

Head of the Institut des Hautes Etudes Scientifiques
Former president of the Société Mathématique de France

I am sorry master coda, I only have french references.
 
  • #47
marlon said:
oooooh, thanks for this nice incomplete (i.e. USELESS) statement

Ha ha!

humanino, if you want to be funny you should use irony like marlon.
 
  • #48
master_coda said:
I'm reminded of the cranks in theory development who try and cover up the fact that their ideas make no sense by saying that other people just haven't achieved their level of understanding.


hahaha, yeah... you the man coda...

The fact that YOU of all people make such a statement.

you are hilarious and i don't need smilies to express that.

VIVA GODEL
 
  • #49
marlon said:
hahaha, yeah... you the man coda...

The fact that YOU of all people make such a statement.

you are hilarious and i don't need smilies to express that.

VIVA GODEL

It's one of the great advantages of having expertise in a field. Something that I say can be true while the same remark is cranky coming from the ignorant. :smile:
 
  • #50
okelidokeli

on a more serious note though. I think that math is NOT a science in itself. It is the language used to describe the world. Although it is one of the most essential parts of the human race and it is our universal language, we should keep in mind that math on it's own does not explain anything.

To be more positive (otherwise the master will be mad again) math is the best approximation of perfection , ever know to man kind. Even master-coda will agree on this last statement, right yoda (sorry, coda).

marlon, and may the force be with me and my friend from Paris...
 
  • #51
may the farce be with us...

I feel mathematics like a supreme art. The most constrained of all, which makes it the most interesting.
 
  • #52
humanino said:
Marlon, I love you.

I want to quote someone, again, since that is the only thing I can do :biggrin:


I am sorry master coda, I only have french references.


humanino, you don't have to apologize to the self-proclaimed Yedi-master. His forgiveness has no boundary, it has never been the victim of renormalization...
 
  • #53
marlon said:
on a more serious note though. I think that math is NOT a science in itself. It is the language used to describe the world. Although it is one of the most essential parts of the human race and it is our universal language, we should keep in mind that math on it's own does not explain anything.

I would agree with this. Math is more of a tool of the other sciences, rather then being a science by itself. In particular, math doesn't really worry about falsifiabilty. Primarily because math by itself doesn't really attempt to describe the physical world.
 
  • #54
marlon said:
it has never been the victim of renormalization...
:smile: :smile: :biggrin: :approve: :-p
 
  • #55
I fear math is considered as a tool in the US, and that would have terrible consequences to american physicists. They shall understand by themselves the great superiority of math truth.

Math is art :wink:
 
  • #56
humanino said:
I fear math is considered as a tool in the US, and that would have terrible consequences to american physicists. They shall understand by themselves the great superiority of math truth.

Math is art :wink:


right on, brother

math is art, RAP is a tool...
 
  • #57
marlon said:
humanino, you don't have to apologize to the self-proclaimed Yedi-master. His forgiveness has no boundary, it has never been the victim of renormalization...

Hey, I would have been happy with a world based on Euclidean geometry. It's you physicists who keep insisting on picking more and more complicated theories.

I keep telling physicists that if they shoot themselves in the foot it'll hurt, and they keep insisting on trying it.
 
  • #58
master_coda said:
It's you physicists who keep insisting on picking more and more complicated theories.
We shall find something simple in the end. Just a little detour... :rolleyes:

master_coda said:
I keep telling physicists that if they shoot themselves in the foot it'll hurt, and they keep insisting on trying it.
Yeah, we love it. Again ! :biggrin:
 
  • #59
master_coda said:
Hey, I would have been happy with a world based on Euclidean geometry. It's you physicists who keep insisting on picking more and more complicated theories.

I keep telling physicists that if they shoot themselves in the foot it'll hurt, and they keep insisting on trying it.



no no no this is wrong, man...

We cannot be taken responsible for the way nature looks and reacts. We only describe it, we cannot tell nature how to work or what to do. Trust me, we don't like tensors but we need them because they are the best covariant entities known to us until know. They were introduced just because of this demand of the human mind to spacetime is always lorentzian in our vicinity...

If you want to simplify math and thus fysics try this : GET RID OF THE CONCEPT OF A DISTANCE. It is a crazy and almost stupid idea, yet some mathematicians as well as fysicist are playing with that thought in order to eliminate them damned tensors of GTR
 
  • #60
marlon said:
Trust me, we don't like tensors
Neither did He ! :rolleyes:

marlon said:
GET RID OF THE CONCEPT OF A DISTANCE.

:surprise: :eek: :cry:
I'm outta here ! :-p
 

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