Gödel's theorem is a contradiction

This statement is true

If this statement was false then YES, that statement must have been false because not (this statement = true) this statement = false.

If this statement was true then YES, that statement must be true because This statement is true is true

but it can't be both true and false can it?


This statement is false

If this statement was true then no, it can't be because then it would be true, but then again it says its false etc...



Similarly,
"I wrote this statement"

If this is true, then YES its true, because i actually wrote it.

If this is false, then YES its false because i didn't write this statement (look at the assumption).

But then it would be a contradiction because a statement can't be both true and false



Gödel's theorem:
Gödel asks the UTM (Ultimate Truth Machine, which only can say if a statement is true or not) this question
"UTM will not say this statement is true."
and Gödel knows its true, but UTM can't say it thus its not a ultimate truth machine.

_________________________________________________
Is this true? Yes because UTM can't say its true since then he would contradict himself.
_________________________________________________
Is this false? Yes because that would mean "UTM will say this statement is true" and UTM could in practice say its false - thus making the whole statement false.

Thus Gödels theorem is a contradiction, it can't be both true and false.



Funny thing is looking back at the theorem it says:

Gödel asks the UTM (Ultimate Truth Machine, which only can say if a statement is true or not) this question
"UTM will not say this statement is true."
and Gödel knows its true, but UTM can't say it thus its not a ultimate truth machine.

Look at the bold statements, clearly its a contradiction!!! Because if its not a ultimate truth machine then it could in fact say that the statement is true, contradicting itself.. and Gödel would be wrong again, but he can't be both wrong and right can he?


Conclusions:
1. Phrases with Self references are not statements.
2. You can't reason as above by assuming that its either true or false to begin with without first checking that its a valid statement, and selfrerential phrases are not statements.


Ok, I might be wrong here but everyone have a look.

I think maybe you missed the point;). The entire analogy's purpose is to prove that you cannot ever create a universal truth machine, in order to show why it is that the theorum works in other cases as well. In any given set of knowledge, there are still some questions that cannot be answered without outside knowledge.

Quote:

---

Gödel's theorem:
Gödel asks the UTM (Ultimate Truth Machine, which only can say if a statement is true or not) this question
"UTM will not say this statement is true."
and Gödel knows its true, but UTM can't say it thus its not a ultimate truth machine.

_________________________________________________
Is this true? Yes because UTM can't say its true since then he would contradict himself.
_________________________________________________
Is this false? Yes because that would mean "UTM will say this statement is true" and UTM could in practice say its false - thus making the whole statement false.

---

I'm not sure about your logic for this last part here. If UTM says that the statement is false, then it is true, because it cannot say that it is true. If UTM does say that it is true, it becomes false. The point is that there is a known answer (true) but the UTM cannot produce this answer because it does not possess the outside knowledge that we have.

Whether this question is useful or not is a completely different matter; it of course is not. The point it just to show that unanswerable questions will always exist.

I meant

Is this false? Yes because that would mean "UTM will say this statement is true" and UTM could in practice say its true - thus making the whole statement false.

Why could UTM say its true? Because UTM could know that its both true and false - if this was a valid statement.

I think a statement can be answered either yes or no to, not both yes or no.

Btw, kedaman, your original post had a statement, not a question, "asked" by Godel.

UTM is not allowed to give an answer that it knows is incorrect.

The statement is not both true and false. It is true. However, if the UTM were to state that it was true, then the statement becomes false due to the UTM's answer. Thus the UTM cannot say that it is true, even though it is.

If the statement were false, it would not indicate that UTM CAN say that the statement is true, but that it WILL or HAS. Without this certainty, the statement remains true.

Quote:

---

If the statement were false, it would not indicate that UTM CAN say that the statement is true, but that it WILL or HAS. Without this certainty, the statement remains true.

---

Why not will or has as well?

not (UTM will not say it is true) <=> UTM will say it is true

not (UTM has not said it is true) <=> UTM has said it is true

Quote:

---

Btw, kedaman, your original post had a statement, not a question, "asked" by Godel.

---

Yes, sorry it wasn't mentioned but it doesn't change a thing

Quote:

---

Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
Now Gödel laughs his high laugh and asks UTM whether G is true or not.
If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
"I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

---

In my reasoning, it is asked whether a statement is true, then also if it is false. In the above reasoning it is only asked whether the statement is true. G is both true and false, but Gödel didn't bother checking if it was false becase he thought that would be a contradiction, but he excluded the possibility that a phrase can both be true and false, constituting a paradox.

Quote:

---

I meant

Is this false? Yes because that would mean "UTM will say this statement is true" and UTM could in practice say its true - thus making the whole statement false.

---

You are saying that the statement is false because UTM could say it is true. UTM cannot say that it is true if doing so would make it false. The statement is only false if UTM does state it to be true, and only true if it does not. Thus the question is one that the UTM can never produce a correct answer to.

Kedaman

You are simply misunderstanding Godel's theorems.

You confine yourself to looking for a flaw in a simplified laymen's version to help ordinary people grasp the concept behind his proof. However, the formal proof it's self is beyond contention. It's as sound as a pound. I've never heard of any mathetician every questioning it.

The Godel sentence is valid because such a sentence can be derived from any formal system (capable of arithmetic). If you reject the godel sentence as invalid then that does not falisfy Godel's proof because he said that any formal system is either incomplete or inconsistant. i.e. If you disallow the Godel sentance, it is incomplete.

As far as I can see, there is no contradiction here. The UTM always gives the correct answer yes? So, if G = 'UTM will not say that this senctence is true' and you ask 'is G true?' if the UTM does not respond, then G is true.

No-one has said that the UTM cannot refuse to answer, if a refusal to answer is the correct answer.

Judd

Quote:

---

As far as I can see, there is no contradiction here. The UTM always gives the correct answer yes? So, if G = 'UTM will not say that this senctence is true' and you ask 'is G true?' if the UTM does not respond, then G is true.

No-one has said that the UTM cannot refuse to answer, if a refusal to answer is the correct answer.

---

The point of Godel's proof is to demonstrate that such a thing as a UTM machine could never exist. There will always be some truth that such a machince could never recognise.

All formal systems are must either be incomplete or inconsistant. Either there is a true statement that a UTM could not utter (incomplete) or, if it did, it would contrdict itself (inconsistant).

Quote:

---

Originally posted by kedaman


This statement is false.

---

That statement is true about it's falseness.

This statement is false. Is true. Since it's true, then it is inherently false. Making the original statement match the conclusion.

Now,

This statement is false. Is false. Meaning the statement would infact be true. But this is a contradiction, so yes, calling it false works.

It can be either.

Because it's truth or lack of truth is rather irrelevent.

It would be similar to asking if "Have a nice day." is true or false.

It's a statement, some statements can be true or false, but not all.

Quote:

---

Originally posted by Alphanos
You are saying that the statement is false because UTM could say it is true.

---

the other way around, I'm saying the statement is false because UTM says it is true.

Quote:

---

UTM cannot say that it is true if doing so would make it false.

---

thats the point, its now proven false.

Quote:

---

The statement is only false if UTM does state it to be true, and only true if it does not. Thus the question is one that the UTM can never produce a correct answer to.

---

Quote:

---

---

"UTM doesn't say this is true" is true
<=> UTM doesn't say this is true

"UTM doesn't say this is true" is false
<=> UTM says this is true

both are independently correct, where's the problem?

DiGiTaIErRoR
Thats my point, and I think Gödel's theorem is one of them

Simon
Where am I misunderstanding it? Even if this is a laymen's version it still has the self reference which must be the big bug in the formal proof as well.

What if the Gödel Sentence is both true and false, doesn't it make itself a contradiction?

Quote:

---

Originally posted by simonm
Kedaman

You are simply misunderstanding Godel's theorems.

You confine yourself to looking for a flaw in a simplified laymen's version to help ordinary people grasp the concept behind his proof. However, the formal proof it's self is beyond contention. It's as sound as a pound. I've never heard of any mathetician every questioning it.

The Godel sentence is valid because such a sentence can be derived from any formal system (capable of arithmetic). If you reject the godel sentence as invalid then that does not falisfy Godel's proof because he said that any formal system is either incomplete or inconsistant. i.e. If you disallow the Godel sentance, it is incomplete.

---

Kedaman

You are missing the point because a formal system is not consistant if it can (legitimately) derive a statement that contradicts it's axioms. Godel prooved that they can and you have not demonstrated that it can't.

Furthermore, there will always be true statements that no formal system, no matter how complex, will ever be able to prove.

You are claiming that the Godel statement is invalid despite the fact that, it's formal equivillent, can be legitimately deduced from the axioms of formal systems.

First of all, I'd like to agree with simonm, I'm just continuing the next part because I think your logic is incorrect;).

Quote:

---

the other way around, I'm saying the statement is false because UTM says it is true.

---

Why does the UTM say that the statement is true? You have asserted this several times, but as a universal truth machine, it is unable to utter a falsehood. While the statement is true, if the UTM attempted to state as much it would be uttering a falsehood. Thus UTM cannot ever state that the true statement is true, even though it is.

Judd, that is a truly insightful point you have made! I never would have thought of that, and it is the best solution that anyone has posted so far to this problem. The rest of you are just going in circles (which may not be a bad thing!:D )

can i just point out something:

Quote:

---

"I wrote this statement"

If this is true, then YES its true, because i actually wrote it.

If this is false, then YES its false because i didn't write this statement (look at the assumption).

But then it would be a contradiction because a statement can't be both true and false

---

Isn't this just:
Assume it's true => It's true.
Assume it's false => It's false.
But then it's true AND false.
:confused:

the UTM only answers to questions that can be proven with facts.

don't make the UTM into an UMRM(Ultimate Mind Reading Machine)
UTM only answers to sentences that are questions...
but does not have to answer to all of them... only yes and no
example would be...
if i were to ask the UTM :
"Do I like the color RED?"
how does the UTM will know?? to the UTM, that sentence cannot
be supported with facts unless i said that I like the color Red.

"UTM will not say that this is true" <== is not a question
but if you make G = "UTM will not say that this is true"
and ask if G is true?? then, thats the same thing as asking a
question like if i like the color Red?

UTM will treat those questions the same as if you said
"blah blah blah"

first you are supposed to support the UTM with facts if those
facts are not general(such as personal facts or facts that not everyone knows about)

example:

FACT: I like the color red.
Q 2 UTM: Do I like the color Red?
UTM : YES!

now UTM doesnt have to be a mind reader

FACT: ( UTM will not say that this is true )
or
FACT: ( UTM will say that this is false ) <=== G
Q 2 UTM: is G true?
UTM : YES

hence G is a FACT, and if asked is a FACT true? then UTM will
answer with YES

Hope you dont give UTM a hard time anymore, laters!

i think this stuff is nonsense. there is no such thing as a UTM.

I'd like to say, that UTM cannot answer yes or no to:

1. phrases (ex. "my computer", "ms80SA", "where is my dog?")
2. statements that are contigencies (ex. "P and not Q", "I like the colour red")
3. self referential phrases (ex. "this is false", "UTM will not say this is true"

ask yourself, what is the "this" in "UTM will not say this is true"? Simple substitution: "UTM will not say "UTM will not say this is true" is true", but what is this in here? "UTM will not say "UTM will not say "UTM will not say this is true" is true" is true", and still thats not it, keep on going you wont find out what statement Gödel was refering to!! UTM doesn't classify his phrase as a statement because: "this" cannot be defined.

Thus UTM only answers questions that are either true or false.

kedaman, you excluded questions that you would ask an 8ball...

if you exclude all this, UTM is really an Ultimate Truth Machine.
But let me get this straight about Gödel's Theorem.

Gödel was a wiseass that wanted to ask the UTM a question
that would cause it to malfunction, is that right?

if thats the case, yes, Gödel's theorem is wrong...