# Goedels Incompleteness Theorem

1. Aug 2, 2004

### 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.

2. Aug 2, 2004

### HallsofIvy

Staff Emeritus
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.

3. Aug 3, 2004

### 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?

4. Aug 4, 2004

### 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... ;)

5. Aug 9, 2004

### pervect

Staff Emeritus
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.

6. Aug 9, 2004

### Integral

Staff Emeritus
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.

Last edited: Aug 9, 2004
7. Aug 9, 2004

### Hurkyl

Staff Emeritus
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.

8. Aug 9, 2004

### 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.

9. Aug 9, 2004

### pervect

Staff Emeritus
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.

10. Aug 9, 2004

### 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.

11. Aug 9, 2004

### 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

12. Aug 9, 2004

### 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: :tongue2:

13. Aug 9, 2004

### marlon

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

14. Aug 9, 2004

### marlon

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

15. Aug 9, 2004

### humanino

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.

16. Aug 9, 2004

### marlon

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...)

17. Aug 9, 2004

### 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.

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

18. Aug 9, 2004

### master_coda

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

19. Aug 10, 2004

### 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.

20. Aug 10, 2004

### marlon

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