Gödel's Incompleteness Theorem

With these scientific achievements, do you think that one-day the humanity will solve all the secrets of the universe and obtain all the knowledge? And with this knowledge, will they able to create a human being (for example, me again ) and totally remove the death from our ways?
I can’t say so because Kurt Gödel has already proven that it is not possible. What do you think?

Gödel's Incompleteness Theorem

“In 1931, the Czech-born mathematician Kurt Gödel 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.

Gödel's Theorem has been used to argue that a computer can never be as smart as a human being because the extent of its knowledge is limited by a fixed set of axioms, whereas people can discover unexpected truths ... It plays a part in modern linguistic theories, which emphasize the power of language to come up with new ways to express ideas. And it has been taken to imply that you'll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself. “ …

…
“The proof of Gödel's Incompleteness Theorem is so simple, and so sneaky, that it is almost embarrassing to relate. His basic procedure is as follows:

1) 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.
2) 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.
3) 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."
4) Now Gödel laughs his high laugh and asks UTM whether G is true or not.
5) 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.
6) 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").
7) "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

Think about it - it grows on you ...
With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics ...
Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it. “

“All consistent axiomatic formulations of number theory include undecidable propositions ...
Gödel showed that provability is a weaker notion than truth; no matter what axiom system is involved ...
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 Gödel's second theorem, which implies that the only versions of formal number theory which assert their own consistency are inconsistent.
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. ”

As a summary, Gödel's Theorem states that there are, in certain formal systems, statements whose truth or falsity cannot be determined from our axioms. If we have axioms for a 'theory of everything' we will then be able to come up with an apparently physical statement whose truth or falsity cannot be determined from those axioms - and so we don't appear to actually have a theory of everything after all.


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

Well, I'm a firm believer in brute facts and, as GIT is consistent with brute facts, I'm not terribly worried about an infinite regress of formal systems.

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I couldn't if I tried.

There are some interesting implications to Godel's work, but I've never been too clear just how well it applies to the universe as a whole. First, we don't deal with much that is certain, and even if we were to express our current understanding as axioms, Godel just showed that there are some statements which are necessary for completeness (eg: "these axioms are complete") which, if added, cause nastiness. It is useful in finite state machines and the mathematical analysis of computer systems, but IMHO not very useful to the world that we live in.

I think it's fairly obvious that there will always be unanswerable questions. And yes, the theorem does apply to the real world; any time you have a system complete enough to do arithmetic, you get incompleteness.

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well it does establish that any formal axiomatic system will never be able to demonstrate it's own consistency. This put s a pretty obvious limit on our understanding, since even if our formal description of the universe is totally correct, we will never even be able to show that our description is self consistent, never mind matches the universe.

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Well, consistency isn't all it's cracked up to be.

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One of the things that interests me about this is the thought that our study of theology might be a formal system. In Godel's terms, such a system must be either inconsistent or incomplete. Or both. Unanswerable questions imply incompleteness, but what about the other side? Could it be true that theology (assuming it is logically coherent) is actually inconsistent too?

God could resolve all questions but he refuses.

The beauty of mathematics is that it does apply to the universe. I have a very good friend who, before he became a docter, was an astronautical engineer. He realized in medical school that the same equations he used to describe the orbit of planets applied to the behavior of certain cellular processes.

It's a mistake to say that a mathematical principle principle like the one Godel described doesn't work in the real world. It does work in the real world, otherwise, science and mathematics would have no value, and this Creation & Evolution forum would be completely pointless. I firmly believe this paradigm that Godel describes tells us what we are capable of understanding, and what we are incapable of understanding, and knowing the limits of what our mind can understand is incredibly liberating, but humbling at the same time. All evolutionists and atheists should VERY CAREFULLY ponder and consider the implications of Godel's proof.

You know, that's the thing, Jet Black. Our human minds are more complex and more powerful than anything we can physically observe in the universe. So while we can't know whether or not the universe is infinite (because infinity is not one of our properties), we can know whether or not our mathematical ideas are consistent with the visible universe. We are capable of understanding concepts such as chance, and from statistics, we know that chance cannot create usable information. So we can know that chance cannot result in life, which is a system of incredibly complex and meaningful information. We can understand biology, but we cannot understand ourselves. We can understand cells, but we cannot understand God.