Godel's Incompleteness theorum for those of you locked into a certain system of logic

This is a nice little description of it quoted from a book via the website (http://www.miskatonic.org/godel.html):

Pretty entertaining. Just wanted some people to check this out and see what it spawns, if anything.

 

The problem with Gödel's sentence is, it can't be considered a logical sentence, because it is recursively paradoxical. The sentence literally cannot be answered by ANYONE, man OR machine, because it is actually just a play on linguistics, and linguistics are highly flawed. In logic, part of a concept cannot refer to the entire concept itself. That is identical to saying mathematically/conceptually:

a & b = b.

Which is logically flawed, and with every step you evaluate b to, b will change.

In a situation like this, the problem cannot be resolved because, in fact, a & b is never equal to b; as soon as you evaluate a & b, b changes, and then the evaluation of a & b must also change.

Anyway, just wanted to point out that while Gödel thinks he's a genius, he isn't really. The Universal Truth Machine would, instead of saying "true" or "false," would say, "this statement is illogical." I think the UTM might also say afterwards, "And you too are illogical, Gödel!" Haha.

Edit: If ultimate truth comes out to something unresolvable like this ... what does it say about mankind? =P

 

Godel did this on purpose to demonstrate his incompleteness theorum (that the logical system is incomplete because it can be used to create something that does not fit within it's rules - something that is illogical).

Yeah, that is part of what Godel's Incompleteness Theorem(s) are aimed at.

"Godel's First Incompleteness Theorem. Any adequate axiomatizable theory is incomplete. In particular the sentence "This sentence is not provable" is true but not provable in the theory."

"Godel's Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system in not provable in the system."

Here is another quote (please don't hate me- I just like the way this person put it and am to lazy to re-explain it myself)

From Gödel, Escher, Bach by Hofstadter "The other metaphorical analogue to Gödel's Theorem which I find provocative suggests that ultimately, we cannot understand our own mind/brains ... Just as we cannot see our faces with our own eyes, is it not inconceivable to expect that we cannot mirror our complete mental structures in the symbols which carry them out? All the limitative theorems of mathematics and the theory of computation suggest that once the ability to represent your own structure has reached a certain critical point, that is the kiss of death: it guarantees that you can never represent yourself totally. "

I really like the beginning of this web page, it describes the theorem very well: http://www.myrkul.org/recent/godel.htm

I could paraphrase, but it is not necessary.

 

I have to say, when you explained it, it made a hell of a lot more sense than when that webpage tried to explain it. =)

And I agree. With you. But not with the webpage, because the webpage doesn't do an accurate enough job to explain the actual point behind what Gödel was saying.

 

So maybe I was wrong. It was necessary to paraphrase.

Did you like the last webpage I posted? I thought that one was pretty good- had better explanations and the like.

I'm in the middle of reading a book abuot Godel and Einstein, which is why I brought it (Godel's Theorem) up. I really like the analogy to the mind from the book Godel, Escher, Bach.

 

Not sure if i understand the question but if a=0, and b=(anything) a&b=b would be true.

 

a + b = b
a & b = b

There is a big difference between these two statements; that's where I believe you are confused at. a would have to be absolutely nothing for (a & b = b) to be true. However, a does equal something in the cases we are talking about.

 

that's interesting, kharakov.....
don't know if its understanding is worthy of being called a 'conversion experience' though...
!

i like the metaphor it delivers very much. that was a great quote given.

 

Maybe not a conversion experience, but people should know that their system of logic is not complete. When someone brings up something that does not jive with their method of thinking, they should remember that their logical method is not the be all end all-- they have to create new rules to incorporate the new thought into their current method of thinking.

 

I've studied Godel in a graduate-level class on mathematical logic, and I've read most of Godel Escher Bach. I've also studied metaphysics, which ultimately holds each of us responsible for making sense out of Reality, each in our own way. Applying the ideas from Godel to Philosophy (which must ultimately rest on some sort of logical system), I have come to the conclusion that no explanation that is logically self-consistent can ever describe Reality.

Any wonder that I must ultimately reject the adoption of Objectivism (which is really just as subjective a viewpoint as any other, as genuine objectivity is a logical impossibility), just about every Religion (the more strongly it asserts itself to be the One and Only Truth, the more it's rooted in total fantasy), etc., etc.?

And now for the real mind-blower. Consider that one set of paradoxes (observations that contradicted each other) led to Special Relativity, which led in turn to General Relativity. Another set of paradoxes gave us Quantum Physics. Now theoretical physicists have been going nuts trying to reconcile General Relativity with Quantum Physics, and the results promise to open up a new level of understanding. Godel's own work was stimulated by the presence of any number of paradoxes that were cropping up in mathematics at the time; I'm aware of a real corker from Cantor Set Theory, for example. Ultimately, it strikes me that any real increase in understanding (of which knowledge is one subdivision) must ultimately come from the exercise of reconciling paradoxes!

I do try to develop a practical philosophy toward life; after all, Godel's theorems didn't render mathematics useless in any practical sense, and we still have to deal with day-to-day life. But beyond that, I like to deliberately develop contradictory beliefs, input my experiences into the framework of each one, clash them against each other, and see what kinds of new insights develop as a result. I've come up with some really way-out possibilities, and what's more, it often makes surprising sense!

Peace & Love (& Up Yours too)! LOL!

PS The explanation of Godel's Undecidability Theorem, given at the top of this thread, looks like someone got it more or less mixed up with Turing's Halting Problem, another theorem which depends on a paradox for its resolution. As pointed out in Godel Escher Bach, the two problems are somewhat related, but Godel's theorems go a lot deeper.

 

I thought I ought to expand on this a bit.

First of all, before you can even feed a description of the UTF to itself, you have to explicitly describe how P(UTF) can be encoded, and how the UTF might interpret it. This is a central point in Godel's Undecidability (or Incompleteness) Theorem, though Godel doesn't rely on any sort of automaton (see the next point). It is possible to create an internally-consistent logical system, but such a system can never encode or interpret itself. (Philosophically, any logically consistent belief system can't even be aware of its own incompleteness!)

Second, if you are using any sort of automaton, you have to allow for the possibility that it can never come up with an answer at all, simply because the problem space it must analyze is infinite in size, so the automaton will run indefinitely without ever reaching a conclusion. The Halting Problem is based on this idea, and it states that no automaton can analyze every possible program (or statement of "truth", if you will), because it MUST run indefinitely on at least some of those programs.

I hope that this clarifies things a bit. Godel's proofs are actually fairly complex, and require a good deal of background in the theory of mathematical logic in order to be understood at all. I can't claim to understand them completely myself, though I am familiar with their key points. (Hey, it's been over twenty years since I studied this stuff...) Since I am familiar with the results and their implications, I can at least go from there, and I think that this is the real point of this particular thread.

Peace & Love (& Up Yours too)! LOL!

 

That is because reality is literally undescribable. Complete, unadulterated reality, as seen from a "God's-eye" perspective, cannot be described in words or concepts, because we are merely a finite part of reality, and cannot possibly know everything about perfect absolute reality. Beyond that, because all that we know and will ever come to know is based on perceptions, we can only describe our perceptions of absolute reality, but not absolute reality itself.

Yeah, I'm a determinist as well. Reality is not objective, "reality" is a subjective perceptual illusion based on the state of actual reality.

Not necessarily from paradoxes alone specifically, but since paradoxes indicate that we are definately wrong, many people try do try to solve them or make some sense out of them. I agree. =)

Yep. The idea is, you have a concept, called a thesis, and an opposing concept, an antithesis, and you work them out in your head to figure out which concept is correct, or combine both concepts into another concept, called a synthesis.

It's kind of like programming. =P

synthesis = thesis & antithesis;

Where "&," of course, is the binary operator for combination (not necessarily mathematical combination, though).

The idea is, the more syntheses you go through, the more you learn, and the more correct you get.

Damn programmer, UTF is a charset, UTM is the machine, lol! You had me confused for a second.

Also, by P(UTM), you are referring to the viewpoint and perception of the UTM, right?

Right. There will always be recursive paradoxes, in other words. Some things will recurse upon themselves infinitely, and they can never be truly evaluated, but that doesn't mean it makes no sense at all.