# GÃ¶del's Incompleteness Theorem (logic)

This theorem essentially states that rational thought can never penetrate to the final truth.

A few quotations from a document I read are below. I’d like to know if you guys think this theorem holds any water. Personally, I think it does.

"How can you figure out if you are sane? … Once you begin to question your own sanity, you get trapped in an ever-tighter vortex of self-fulfilling prophecies, though the process is by no means inevitable. Everyone knows that the insane interpret the world via their own peculiarly consistent logic; how can you tell if your own logic is "peculiar’ or not, given that you have only your own logic to judge itself? I don’t see any answer. I am reminded of GÃ¶del’s second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent. "

“Kurt GÃ¶del demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements”

“The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.”

The proof of GÃ¶del’s Incompleteness Theorem is so simple, and so sneaky, that it is almost embarassing to relate. His basic procedure is as follows:

Someone introduces GÃ¶del to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
GÃ¶del 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, GÃ¶del 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 GÃ¶del. Note that G is equivalent to: “UTM will never say G is true.”
Now GÃ¶del 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,” GÃ¶del says. “I know that G is true. UTM is not truly universal.”

http://www.miskatonic.org/godel.html

Paradoxically, you have given a statement that is contingent, not analytic. It goes with this statement :

â€œRational thoughtâ€ is not necessarily in itself final (or ultimate truth). There is a different route we could take, equally acceptable and in line with contemporary thought. And this is the material adequacy. Such that going outside the system does not have to lead to regress. We can stop the regress through intuitionism. The truth could depend on the conditions and time under which the statement is made. Independent, complete axioms might be a fantasy, but we could have material adequacy such that we could have truth (or falsity) just not the kind you imply.

oh wow. Another case of Verbosity.

We could take a different route and then possibly come up with another theorem OK.

Not possibly come up with another theorem. What I think it is, is that your theorem is, perhaps, in itself incomplete and inadequate to address a more coherent account of truth ( or falsity ). This is not to say that your theorem is incoherent. But a little bit of tweaking would make it more, shall I say, intelligible.

Okay, easy with the insults.

You might be right in assuming that all logical systems are incomplete…that is except for one. The only true system would be based on the whole of all things that exist. The entire universe if you will, that is, if it had a limit that anyone knew about. But all that we can even come close to comprehending is within not only a given space but a given time and circumstance. Of course every system we concieve of will be flawed. Does that mean logic is useless? I don’t think so. If logic was useless than I’d of killed myself a long time ago out of depression. What would be the point of anything? I believe we use logic to achieve happiness. I believe that happiness or “good” is the only thing we have to base logic upon. It is our only taste of the truth that we use to barely piece together the smallest of fragments of the entire truth. Your sanity, I believe, should be based not on the proof of that truth to others but the truth that resides in your own feelings of happiness. What you have just discovered in that theorum is our own uselessness and how unimportant we as people are in the whole of things. How incapable we are of being perfect. Ya might wanna get used to that, however, I still wouldn’t go around telling everyone the final truth doesn’t exist. Even though we can’t see the whole of truth because we are confined in time and space we can still confine small truths together and express it as it is in that same confinement that we exist in. Perhaps by perfectly achieving that with true virtues of honesty, faith, hope, and love we might begin to catch bigger glimpes of the final truth.

Apparently, you don’t even understand the magnitude of the topic at hand.
Your statements and questions are incompatible with this thread. Please try to have a small idea of what in the hell you’re talking about before you speak again.

Apparently, you don’t even understand the magnitude of the topic at hand.
Your statements and questions are incompatible with this thread. Please try to have a small idea of what in the hell you’re talking about before you speak again.

Apparently, you don’t even understand the magnitude of the topic at hand.
Your statements and questions are incompatible with this thread. Please try to have a small idea of what in the hell you’re talking about before you speak again.

Did you even skim through the initial post?

“Kurt GÃ¶del demonstrated that within any given branch of mathematics, there would always be some propositions that couldn’t be proven either true or false using the rules and axioms … of that mathematical branch itself. You might be able to prove every conceivable statement about numbers within a system by going outside the system in order to come up with new rules and axioms, but by doing so you’ll only create a larger system with its own unprovable statements”

“The implication is that all logical system of any complexity are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules.”

First of all, you just repeated what was said above, but your wording was slightly different. Are you really that one track minded? If its not worded in a way that seems appealing to you, you immediately dismiss it?

Secondly, I’m appalled that you actually realise, because I was beggining to lose hope, that no one really understands this topic and what it encompasses. And for that, I give you some credit.

Finally, I’m not in conflict with this particular theorem. I simply wanted to know what peeps thought about it.

Interestingly enough, this problem—in case your awareness is limited within the realm of mathematics—is not limited within the realm of mathematics. There are analytic theses such as â€œAll a priori statements are analyticâ€ that have received an onslaught of criticisms. With just this one little thesis, the problem of infinite regress, demonstrability, and contingency of truth (or falsity) are being discussed.

Indeed, you can find the same line of thought as to “going outside the system” and axioms sufficient enough to confer truth on theorems.

So, no need to be hostile.

• anyway, I suppose I should stay away from this thread *

I concede. Happy?

Whenever someone says proof I think of Logic, and when I think of Logic I think of formal systems and final truths.

Tell me, o’ wise man, what system would you use to compile these proofs and attempt to prove anything?

How does one judge without making the system for judgement suspect? And if the system is suspect, all of them… well aren’t you really just saying that all knowledge is incomplete? You dirty little Skeptic you!

Wait, you will prove by falsification. Not what is, but what isn’t. Just say nothing is and drink the kool aid.

And Hi Marie… De’trop will call me names, but Hubba Hubba!

Hi GCT, forget about de’trop and that commotion he caused in the other thread. Thanks.

You are most welcome, and I don’t worry about De’trop that much, as I know what his problem is. He hates the fact that he was born from woman, instead of being birthed by a jackal, as befitting the Anti Christ.

I was anticipating and therefore hopefully deflecting any and all complaints of your hubbaness

okay my apologies for the interruptions, I return you now to this thread, in progress.

What I am saying McGrady, is that when you pose this incompleteness theorum of all mathematical branches what you are really saying in a sense is that nothing is true because the only way you can prove something is true is by explaining it with outside means which then only broadens the scope of what you are saying and allows for falsities to occur in that newly broadened system.

For example, if I say there is a tree and it exists, I can say whatever I want about the tree; what its made of, what it looks like, and how it survives. All those statements will be true so as long as the tree exists. But the only way I can prove it actually exists as a tree is if I say there exists something else that is not a tree beside it to actually note the difference between it and the other. As soon as I do that I have stepped outside the box and now have to account for the proof of that other thing I compared it to to prove its existence. As soon as you make a statement it becomes false because you cannot prove it. This is as you said an ever-tighter vortex of self-fulfilling prophecies.

It is as well the same incompleteness theorum without numbers. Just because I applied it outside of math doesn’t mean that I have no clue what I’m talking about. The point I was trying to make is that it would be ineffectual to disregard our sanity and sense of truth simply because we cannot perfectly prove the completeness of the systems we use in math and logic. It would be better to assume some things in order to achieve desirable results or in other words, to be happy.

I completely see your point, Nientilin. I don’t necessarily agree that its “the only way you can prove something to be true”, but yes, its a sensible point you’ve made.

netlinin: your points a right on… this is where the latter wittgenstein comes in I think…

anyway I thought godels incompletness was supposed to apply to all natural languages… like that’s not radical or anything for you to point that out…

anyway, it allows for the realisation of the importance of ineffable forms of life as a grounding of communication and language… its an important thing to keep in mind… especialy when people start assuring you that we can explain what we mean completely with language and more language to explain that and more to explain that etc etc etc… which is why people from radicaly different communities of thought and action can’t settle their disputes with “reason” “objectively”… because they allways hit up against this incompleteness as the language hits its descriptive limits against ineffable forms of life…

he provides a way of looking at language as inevitably interweaved with the practical conduct of everyday life… the coexistence of language (via godels ANY language including maths) with “what cannot be said”…

in his latter work the prosaic and mundane… the practices that comprose forms of life are what give linguistic terms meanings… not the internal rules of the language…

which I thinnk, if you accept Godels, becomes an important aspect of language and communication…

Actually that is an old hypothesis. I think most of the researchers have abandoned that one.