Goedels Incompleteness Theorem

[Curious6]

I just read an article about Goedel's Incompleteness Theorem, and if I have correctly understood it, it basically means all theorems that we have and that can ever be made are either incomplete or inconsistent. This is also sometimes given as a reason to state that a TOE is impossible because it would be incomplete and/or inconsistent. However, my question is: are all theories really incomplete then, will we never know anything for sure or fully? Will an eventual quantum theory of gravity also be considered incomplete. It is quite a disturbing thought actually, and I would just like to know something more about this.

[HallsofIvy]

Goedel's incompletness theorem is a statement about the fundamentals of mathematics, it is NOT a theorem about physics. it is at least theoretically possible to have a physics theorem that does not correspond to ANY mathematical model. That would be a problem with the mathematics, not the physics.

By the way, most "working" mathematicians believe that the axiomatic theory of natural numbers is consistent, but not complete. "Not consistent" would mean that no theorem means anything, while "Not complete" means that it is not possible to prove EVERYTHING but that the proofs you have are valid.

[Curious6]

OK, thanks for the reply HallsofIvy. If I've understood what you've said, the Incompleteness Theorem does not mean that the proofs we have are invalid, just that it is impossible to achieve a total proof for a theory. However, you mention that it is 'theoretically possible to have a physics theorem that does not correspond to any mathematical model', but as far as I'm aware, most, if not all physics theories correspond or are built on mathematical models. Does this mean they are then inherently inconsistent or incomplete, or maybe both?

[salamander]

Don't know anything about Goedels theorem. However I can't help getting the feeling that you are impling this: if Goedels theorem is right, it would mean the theorem had to be either inaccurate or inconsistent or else it would be the only complete theorem, and thus violating itself... how does that make sense, really?

Cheers... ;)

[pervect]

Don't know anything about Goedels theorem. However I can't help getting the feeling that you are impling this: if Goedels theorem is right, it would mean the theorem had to be either inaccurate or inconsistent or else it would be the only complete theorem, and thus violating itself... how does that make sense, really?
Cheers... ;)

Actually, I don't think this problem occurs. I'm not a mathemetician, so I may have some of the language wrong, but basically Godel's proof used only constructivist methods, so you can't apply the theorem to Godel's proof.

[Integral]

Gödel's incompleteness makes NO statement about the truth or falseness of ANY single theorem. It does make a statement about sufficiently powerful formal SYSTEMS. In general it shows that any useful system will contain statements which cannot be proven, true or false. The common example is self referential:
"This statement is false"
I can formulate this statement but I cannot prove it true or false.

If you want a through lay mans development of Gödel's incompleteness read Gödel, Esher,Bach: An Eternal Golden Braid by Douglas R. Hofstadter. Warning! it is not a trivial read.

EDIT: Oh yeah, this is a math disscussion. Physics relies upon Math for its formal system therefore this is math.

[Hurkyl]

The real trick used by Godel is to create a statement that effectively says "I cannot be proven". If you can do so, this statement must be true and unprovable.

Actually, if you can create a statement that effectively says "I am false", then that will turn out to be an actual logical contradiction.

[TenaliRaman]

if one is inclined towards computer science, then he may like to think of halting problem .... since it has close resemblance with the way godel's incompleteness theorem was proved ... ofcourse the original proof was more mathematically driven than just picking sentences right from the air.

[pervect]

Gödel's incompleteness makes NO statement about the truth or falseness of ANY single theorem. It does make a statement about sufficiently powerful formal SYSTEMS. In general it shows that any useful system will contain statements which cannot be proven, true or false. The common example is self referential:
"This statement is false"
I can formulate this statement but I cannot prove it true or false.

If you want a through lay mans development of Gödel's incompleteness read Gödel, Esher,Bach: An Eternal Golden Braid by Douglas R. Hofstadter. Warning! it is not a trivial read.

EDIT: Oh yeah, this is a math disscussion. Physics relies upon Math for its formal system therefore this is math.

As I recall, (it's been awhile), the formal system that's needed to accept Godel's proof does not need to be a system that's powerful enough that one could apply Godel's theorem to it. Basically Godel carefully avoided "controversial" assumptions like the law of the excluded middle when formulating his proof.

[humanino]

Can we say : Godel's theorem justify the need for human-mathematicians ? A (Turing) machine would not be able to prove everything, and the human imagination fills in the missing new postulates the theory needs in order to grow.

This is the way I understand Godel's work, as an answer to Hilbert's old mechanical view of the mathematics.

[marlon]

Is this Godel theorem not the same as falsification ???

A theory is good until we find experiments that have outcomes which are not predicted by the theory. Newtonian mechanics is ok for low speeds but when looked at with light speeds it predicts false results. Just like with QM and Newtobian mechanics ???

This theorem teaches us that we must consider regions in which models of nature are valid, right ???

marlon

[humanino]

Not quite sure Marlon. We are physicists ! Godel deals with pure model, outside of the physical world. He was even worse than a regular mathematician : he was a logician :surprise: :yuck: :biggrin: :tongue2:

[marlon]

Not quite sure Marlon. We are physicists ! Godel deals with pure model, outside of the physical world. He was even worse than a regular mathematician : he was a logician :surprise: :yuck: :biggrin: :tongue2:


personally, i don't see much good in a theorem that states that no theory is ever complete and so on. It is like saying that fysics describes nature. It does not provide a new insight, it just states what everybody with an IQ over 70 allready knows. This is the same Godel that thought he found a solution to the Einsteinequations by using an absolute reference frame right??? I never liked him, he was a very depressed individual, trust me...

marlon

[marlon]

The Only Thing Incomplete To The Theorem Of Godel Is The Theorem Itself !!!

[humanino]

(...) it just states what everybody with an IQ over 70 allready knows. This is the same Godel that thought he found a solution to the Einsteinequations by using an absolute reference frame right??? I never liked him, he was a very depressed individual, trust me...


The Only Thing Incomplete To The Theorem Of Godel Is The Theorem Itself !!!


Marlon, you are hilarious :rofl: :rofl: :rofl:
Fortunately i'm not at work, everybody would have noticed me laughing.
And this is the same crazy Einstein friend.

[marlon]

Marlon, you are hilarious :rofl: :rofl: :rofl:
Fortunately i'm not at work, everybody would have noticed me laughing.
And this is the same crazy Einstein friend.

The only usefull theory on how physics works comes from Einstein : imagination is more important than knowledge.
And Popper learned us the priciple of falsification, which gives a very good insight on how science is to be constructed, i think...


Godel is also useful : he shows us how it should NOT be done...


Keep on laughing, it is good for the bloodpressure.

Good night and "see" you tomorrow evening Humanino

nice talking to you

bonne nuit et à demain

marlon (le plus grand admirateur de Godel, celui qui a voulu battre Einstein mais qui a en vérité seulement battu lui-même...)

[humanino]

Marlon was earlier reffering to Godel's solution of Einstein equation, which exhibits a global rotation. Einstein before beleived it was impossible. This in turn helped understand the Mach principle.

Some Mathematicians claim Godel's theorem is useless because it is a technical proof (which makes it unusually short for such an important theorem/theory, only a few pages. But technical !) and it would kind of lie outside the rest of usefull mathematics. Indeed, Godel's proof is rather a philosophical lesson to us.

"95% of mathematicians don't care about what logician or philosophs might do"

(Dieudonné 1982)
quoting Solovay :
"The general feeling is that Godel's theorem concerns only logicians"

(Kolota 1985)
I found those in Delahaye's 1994 "Information, complexité et hasard"

The "Undecidables" of Godel are of two kinds : the first is basically "I am not demonstrable inside the system" (lier paradox). The second is more interesting, because it makes no call to self-reference. Undecidables of the second kind are equivalent to those of the first kind, but mean "the system is consistent". One can build a Turing machine which, given any consistent system, can build undecidables of the second (or first) form.

For those who are still interested, I keep on. The \omega number, called Chaitin number, gives the probability that a Turing machine terminates when it is given a random program. \omega\in[0,1]. A formal system S can furnish only a finit number of digits for \omega. The number of digits is bounded from above by an independent constant plus the complexity of the formal system. Any proposition in S giving \omega's digits is an Undecidable of Godel. The knowledge of \omega would allow one to solve most of the famous mathematical conjectures, such as : Is Fermat's great theorem demonstrable in Peano's arithmetic ?

I find it fascinating. But one must be aware, that extracting such proofs from \omega might require a lot of computation time :wink:

[master_coda]

The "Undecidables" of Godel are of two kinds : the first is basically "I am not demonstrable inside the system" (lier paradox). The second is more interesting, because it makes no call to self-reference. Undecidables of the second kind are equivalent to those of the first kind, but mean "the system is consistent". One can build a Turing machine which, given any consistent system, can build undecidables of the second (or first) form.


Of course, this only applies to systems that are powerful enough to formulate such statements.

[matt grime]

I think, Marlon, you need to learn a distinction between a mathematical theory and a physical one.

There are "proper mathematical" statements that are undecidable in ZF(C):
Continumm Hypothesis, which we'll state as there exists a set S and injections N to S to R (N naturals, R reals) and such that S has no bijections to N or R.

Conway also formulated a whole family of Collatz type conjectures which are undecidable.

[marlon]

I think, Marlon, you need to learn a distinction between a mathematical theory and a physical one.

There are "proper mathematical" statements that are undecidable in ZF(C):
Continumm Hypothesis, which we'll state as there exists a set S and injections N to S to R (N naturals, R reals) and such that S has no bijections to N or R.

Conway also formulated a whole family of Collatz type conjectures which are undecidable.

Pfff, to be honest i am not convinced. These mathematical facts you are reciting will undefinately be true. Yet Godel his "work" gives a redundant view on how science in general must work. That is all i am saying. Talking about injections and stuff will not change this view.

regards
marlon

[master_coda]

Pfff, to be honest i am not convinced. These mathematical facts you are reciting will undefinately be true. Yet Godel his "work" gives a redundant view on how science in general must work. That is all i am saying. Talking about injections and stuff will not change this view.


Godel's theorem wasn't talking about how science works. His theorem was about formal systems. And anyone who thinks the theorem is obvious clearly doesn't even know what the theorem actually is.

[Galileo]

And Popper learned us the priciple of falsification, which gives a very good insight on how science is to be constructed, i think...


Yeah, Popper said a theory has to be falsifiable or it isn't scientific...

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

[matt grime]

Pfff, to be honest i am not convinced. These mathematical facts you are reciting will undefinately be true. Yet Godel his "work" gives a redundant view on how science in general must work. That is all i am saying. Talking about injections and stuff will not change this view.

regards
marlon

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...

[Eye_in_the_Sky]

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.

[humanino]

Hey guys, go easy on Marlon !
His opinion is that Godel's theorem is obvious. I feel that all mathematical theorems are obvious once you understand them.

A formal system is either consistent or complete not both. Since one only cares about consistent systems, one if doomed to work with uncomplete systems. That is all. It is a technical road to get there. But once you're up there, it does not look very impressive.

[matt grime]

Why is it obvious that a formal system (containing the natural numbers) must be consistent, or complete (why can't it be neither?) and that it is not possible to be both? What heuristic argument do you have for it? The theorem may be obvious once you understand it, however Marlon clearly doesn't even know the statement of the theorem so he cannot be in a position ot understand it. His views appear to based on the idea that you cannot experimentally prove a theory, only disprove it or validate it. That has nothing to do with Goedel.

[master_coda]

A formal system is either consistent or complete not both. Since one only cares about consistent systems, one if doomed to work with uncomplete systems. That is all. It is a technical road to get there. But once you're up there, it does not look very impressive.

Of course, there are non-trivial systems which are consistent and complete. So perhaps you don't understand the theorem either, since you seem to think it says something that it does not.

[humanino]

Of course, there are non-trivial systems which are consistent and complete. So perhaps you don't understand the theorem either, since you seem to think it says something that it does not.

So I correct myself : any system able to define natural numbers bla bla bla
Please master_coda, what kind of consistent and complete system has been seriously investigated ? (this is a real question, I'd like to learn.)

[master_coda]

So I correct myself : any system able to define natural numbers bla bla bla
Please master_coda, what kind of consistent and complete system has been seriously investigated ? (this is a real question, I'd like to learn.)

Euclidean geometry.

[humanino]

Why is it obvious that a formal system (containing the natural numbers) must be consistent, or complete (why can't it be neither?) ...

If a formal system is not consistent, then it is complete.
[A & no(A)] implies B (whatever B)


... and that it is not possible to be both?

This is precisely Godel's theorem. I say "any demonstration is obvious once you know how to demonstrate it". I claim that mathematics are obvious. The reason for that, the reason why mathematics are so easy compared to physics, is that in mathematics, one knows what one is talking about. There are definitions. In physics, we can only applies clever ideas to model reality.

I actually went through Godel's demonstration. Several times. :wink: It's technical, but one can understand every single step.

Now one can understand Godel's argument without understanding the technical demonstration. Actually, I would probably never have undestood the technical part of it, without the first section of the article, where he explains the "lier paradox" thing. Even though, of course, I had already heard it somewhere else, would it just be in the beginning of my book where Godel's article is reprinted.

In my previous post where I quote Delahaye's 1994 book, I think I clearly showed that many very respectable mathematicians dare saying they don't care about Godel's theorem. So I personnaly forgive Marlon for making fun of it. I thought he was joking, that's all. Besides, he and I are physicists. I talk for myself, but I am just here to learn, so I want people to correct me whenever I post something wrong.

[humanino]

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 ?

[humanino]

I found the "Presburger arithmetic"
http://en.wikipedia.org/wiki/Presburger_arithmetic
which is consistent and complete ! But "not as powerful as Peano arithmetic"

[master_coda]

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 explainations 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.

[marlon]

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

[master_coda]

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.

[marlon]

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.

[master_coda]

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.

[humanino]

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 explainations that other people have produced.

:mad:
I am here, as I told already, only to learn. I feel your posts agressive. 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.

[humanino]

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 !

[marlon]

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.

[master_coda]

:mad:
I am here, as I told already, only to learn. I feel your posts agressive. 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.

[marlon]

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

[master_coda]

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.

[master_coda]

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.

[marlon]

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

[humanino]

Marlon, I love you.

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

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.

[master_coda]

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.

[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.


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

[master_coda]

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:

[marlon]

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...

[humanino]

may the farce be with us...

I feel mathematics like a supreme art. The most constrained of all, which makes it the most interesting.

[marlon]

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...

[master_coda]

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.

[humanino]

it has never been the victim of renormalization...
:rofl: :rofl: :biggrin: :approve: :tongue2:

[humanino]

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:

[marlon]

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...

[master_coda]

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.

[humanino]

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... :uhh:

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:

[marlon]

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

[humanino]

Trust me, we don't like tensors
Neither did He ! :rolleyes:

GET RID OF THE CONCEPT OF A DISTANCE.

:surprise: :eek: :cry:
I'm outta here ! :tongue2:

[humanino]

We certainly have to understand non-perturbative features of QFT. I know Marlon will one day solve the confinement problem. This should indeed come from topological properties. At least everybody hopes so by studying toy models for QCD.

But this is far from Godel ... :rolleyes:

[master_coda]

we cannot tell nature how to work or what to do

Of course you can. I do it all the time. Nature doesn't seem to listen too well, though. :tongue2:

[humanino]

Let's get back to Godel !
Could someone explain how diophantine equations raised interest in Godel's theory ?

[humanino]

Of course you can. I do it all the time. Nature doesn't seem to listen too well, though. :tongue2:
Master ! I trusted you ! you cannot fail ! :surprise: :cry:

[master_coda]

Let's get back to Godel !
Could someone explain how diophantine equations raised interest in Godel's theory ?

One of Hilbert's problems was to produce a general method for determining if a given diophantine equation was solvable or not. It was not actually proven that no such method exists until the 70s.

Clearly if someone wants to prove that a particular problem in arithmetic is unsolvable, it's probably worthwhile to study the results of people like Godel. The incompleteness theorem was directed at arithmetic, after all.

[Eye_in_the_Sky]

1)YES
2)NO
3)YES ... :smile:

[matt grime]

This is precisely Godel's theorem. I say "any demonstration is obvious once you know how to demonstrate it". I claim that mathematics are obvious. The reason for that, the reason why mathematics are so easy compared to physics, is that in mathematics, one knows what one is talking about. There are definitions. In physics, we can only applies clever ideas to model reality.

I actually went through Godel's demonstration. Several times. :wink: It's technical, but one can understand every single step.

Now one can understand Godel's argument without understanding the technical demonstration. Actually, I would probably never have undestood the technical part of it, without the first section of the article, where he explains the "lier paradox" thing. Even though, of course, I had already heard it somewhere else, would it just be in the beginning of my book where Godel's article is reprinted

I see no argument there supporting your assertion it is obvious. If mathematics is easy can you post a proof of the Riemann Hypothesis then? Or do you mean the mathematics of the proof of Goedel's theorem is obvious? Or that each step is simple? "obvious" is a subjective statement, and tends to indicate when used in mathematical proofs that the result can be deduced with less than an hour of thinking about it.

[arildno]

This reminds me of how a certain Dr. John Watson thought once the steps of a deduction had been explained to him ("Gosh, was it that easy?" (or elementary..))

[humanino]

Well, I am quoting Michel Berger. Besides, I remember my math exams a few years ago. One can search for an answer really hard for several days without finding it. And then, when it comes to working the answers with the teacher, and once you get the answer, it is pretty obvious.

I don't mean to be provocative here.

[arildno]

No, you're not provocative.
You just sound like Dr. Watson..:wink:

[matt grime]

There is a difference between thinking, "gosh, why didn't I think of that" and the problem/theorem/solution actually being obvious in a sense that a professional mathematician would use, since there's not guarantee that you would *ever* have actually thought of it without hindsight. It can only be obvious if you've thought of the step without being told. So, I will accept that you consider Goedel's theorem to be obvious if you can with hand on heart say that you would, without prompting, have firstly thought it up as a conjecture, and then proved it. Thinking that the steps in someone else's proof are obvious is not a particularly note worthy thing since, if it is a basic explanation (ie omits no steps), written by a good mathematician, then it *will* seem obvious, but that is a function of the writer not the reader. If you're using it in that sense then you don't mean it's "easy" in the sense you could have proved it, you mean you've read a good explanation if it.

[humanino]

If mathematics is easy can you post a proof of the Riemann Hypothesis then?
Stay tuned ! :wink:

Seriously, I feel sad everybody is taking what I said the wrong way :frown:


Or that each step is simple?
Yes, it would be closer to that. But once you know each single step, you get a general overview of a proof. I guess I just have difficulties to express myself in a foreign language. Obvious might ot be the word I need.

[humanino]

So, I will accept that you consider Goedel's theorem to be obvious if you can with hand on heart say that you would, without prompting, have firstly thought it up as a conjecture, and then proved it.

So I must admit I was wrong I guess.

I was basically trying to single out a specific feature of mathematics as compared to physics for instance. But then, mathematicians most often tend to think physics is just applied math.

[humanino]

When confronted to a new result in math, one can work out the proof and understand it because everything is clearly defined.

When confronted to a new result in physics, one can undestand the mathematics without getting the feel for the physical process involved.

Well, maybe I should just forget it...

[matt grime]

"obvious" is perhaps not the right word for *mathematical* reasons (but as I hope I said, this is very subjective, and it is only MY belief that that is what obvious *should* be reserved for) and is I suggest not reflective of the fact that just because there appears post facto no other way of thinking that that was how you'd have thought about it. Here's an example of someone missing the point, it's hearsay, allegedly said by a fellow at Trinity Hall at high table one evening. After watching Panorama program about Wiles's proof of FLT, the fellow, not a mathematician, said to a mathematician, "yes, I think I understand the proof".

An obvious proposition is, say, if n is even, then n^2 is even.
Goedel is not obvious in the sense I understand the word. If it were obvious how come there were no well examples of statements that were unprovable before the conjecture was proven, that it caused some mild outrage (mainly for philosphical reasons, apparently), and that for a while no reasonable mathematical statements were known to unprovable? Ok, so you've read a good explanation of the proof, that doesn't make it an obvious result which was how it sounded to me.

[matt grime]

When confronted to a new result in math, one can work out the proof and understand it because everything is clearly defined.

When confronted to a new result in physics, one can undestand the mathematics without getting the feel for the physical process involved.

Well, maybe I should just forget it...

It is only a result AFTER it has been proven. Mathematics behaves like physics in this regard. You need to look at the evidence and formulate a reason as to why it's true, then prove that the conjectured reason is true. If you think you can quickly work out the proof, then I suggest you've not done any difficult mathematics.

Let me give you an example of what part of my PhD is motivated by.
Several people in the 80s (and before) noticed that there were many similarities between representation theoretic facts of a group and normalizers of some p-subgroups. The so-called local representation theory. However there were no known links between the actual theories that worked in all situations. For instance a "Morita equivalence" would explain it, but it was true for things that were known to be not morita equivalent. Eventually M. Broue conjectured that a "derived equivalence" was the reason for it. He made this in the early 90s. Now, more than ten years on no one has proved the conjecture in full generality but they have for almost every reasonable group (reasonable in terms of size or important characteristics, for instance it has been shown for all permutation groups). If you still think maths is simple then heaven help anyone who does a difficult subject like physics (hint, tongue in cheek parting shot). But perhaps it might convince you research mathematics is more like a science than you realize.

[mathwonk]

While it is fun philosophically to play with the scary implications of Goedel's theorem, i.e. that there may exist lots of true but unprovable theorems in a given system,

in real life, it seems harder to find interesting problems or conjectures that some smart mathematician cannot eventually (or rather quickly) prove or disprove.

I.e. it often seems the problem we have is the exact opposite of the one we are worried about here. It is hard to think up good problems that actually last awhile in the face of the onslaught of research progress.

Of course there are a few famous problems that have lasted quite a while, like Goldbach's conjecture, and the Hodge conjecture, but it may be that Goldbach (although not Hodge) has lasted so long because not too many people are
interested in it.

FLT (Fermat) of course lasted a while, but finally bit the dust, so these time frames may actually be measured in hundreds of years, making it hard to know.

But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel.

I think Hilbert showed his greatness with his famous list of problems in 1900. They really inspired people. The centennial celebration of that event in 2000 at UCLA failed pretty badly in my opinion to replicate his feat with a new list of interesting problems, although there were some nice talks.

What do you guys think?

[matt grime]

"But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel."

However, they aren't the only ones. Continuum hypothesis, Collatz type conjecture, etc. The type of set theory you use can be important and not just to set theorists (Shelah showed that Ext_{\mathbb{Z}}^1(A,\mathbb{Z})=0 implies A is free, depends on the set theory you adopt.)

[marlon]

"But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel."

However, they aren't the only ones. Continuum hypothesis, Collatz type conjecture, etc. The type of set theory you use can be important and not just to set theorists (Shelah showed that Ext_{\mathbb{Z}}^1(A,\mathbb{Z})=0 implies A is free, depends on the set theory you adopt.)

Matt Grime...snizzlefizzle, that the answer!

regards,
MaRLoN

[humanino]

"But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel."

However, they aren't the only ones. Continuum hypothesis, Collatz type conjecture, etc. The type of set theory you use can be important and not just to set theorists (Shelah showed that Ext_{\mathbb{Z}}^1(A,\mathbb{Z})=0 implies A is free, depends on the set theory you adopt.)

Yes yes yes. Those are the important ones.

At some points, mathematicians get into a position like "we have that very important assumption : let us look closer if it is independent of our already classified axioms". And : either the new assumption is demonstrable, or it is not in which case we are glad to say "We found a new axiom to classify"