Proved to be Unprovable

[spanshcastlemagc]

Kurt Godel's Incompletness Theroem mathematically proved that with in a system there are things that cannot be proved or disproved, an example of that would be a system whose axioms included basic arithmetic couldnt be proved. In other words within its own axioms you cant prove that 2+2=4.

I think the implications of this are very interesting. I havent read the book Godel Escher Bach but i think it talks about this.

Anyways, i think this is a good analogy for religious debate, ie within our system (humans- our own minds) it is impossible to prove or disprove things as basic as arithmetic, so this idea of a proof of anything seems pretty ridiculous to me (this often comes up in typical often not overly intelligent religious debates - evolution disproves God, dead sea scrolls prove God)

Correct me on any of my statements that are incorrect regarding Godel, but any thoughts about the implications of the incompleteness theorem, or the nature of proof and if it exists at all for any aspect of religion

[MidGe]

To simplify things heaps, what Godel said was that for a non-trivial formal system there are always statements that cannot be proven. As proof of his theorem he offered his own theorem. /images/graemlins/smile.gif

[chezlaw]

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To simplify things heaps, what Godel said was that for a non-trivial system there are always statements that cannot be proven. As proof of his theorem he offered his own theorem. /images/graemlins/smile.gif

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Here is a list of all the implications for religon:

[MidGe]

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[ QUOTE ]
To simplify things heaps, what Godel said was that for a non-trivial system there are always statements that cannot be proven. As proof of his theorem he offered his own theorem. /images/graemlins/smile.gif

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Here is a list of all the implications for religon:

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True /images/graemlins/smile.gif

[NotReady]

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Here is a list of all the implications for religon:


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If this is true for religion, which it isn't, it's true for all world views and philosophies.

[chezlaw]

[ QUOTE ]
[ QUOTE ]

Here is a list of all the implications for religon:


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If this is true for religion, which it isn't, it's true for all world views and philosophies.

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It is true and you're right its also true for all world views and phhilosophies (assuming you're ignoring logic and maths).

chez

[NotReady]

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It is true and you're right its also true for all world views and phhilosophies (assuming you're ignoring logic and maths).


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It would also be true for logic and math. But it isn't true - there are huge impications for the fact that we all have assumptions we can't prove.

[Silent A]

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Kurt Godel's Incompletness Theroem mathematically proved that with in a system there are things that cannot be proved or disproved, an example of that would be a system whose axioms included basic arithmetic couldnt be proved. In other words within its own axioms you cant prove that 2+2=4.

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You're stating this WAY too strongly. Goedel's theorem doesn't say things like "2+2=4" can't be proven. It says that any sufficently complex system (like arithmatic) includes some statements that are true but cannot be proven true using only the system's axioms. These statements can only be proven true by going outside the system's axioms.

Basically, what Goedel did was to show how you could use arithmatic to construct a complex mathematical statement that says (in effect) "this statement cannot be proven true by arithmatic". It's only when you construct statements that make comments about the system itself that you get these strange "unproveable truths". Simple things like 2+2=4 flow directly from the axioms.

By the way, the reason it's called the "incompleteness theorem" is because his he also showed that any sufficently complex system (for example, arithmatic) that is complete (i.e. all statements can be proven or disproven from the axioms) must be incoherent (i.e. it's possible to construct at least some statements that can be proven true and false from the axioms) and all coherent systems must be incomplete (i.e. if no statement can be proven both true and false then there must also be some unproveable true statements).

With respect to the implications of this on religion, there are none. While the idea of unprovavble truths is interesting, there is NOTHING to suggest that any religious statement can't be proven true yet is true nevertheless.

Finally, Goedel Escher Bach is a great (if rather difficult at times) read.

[chezlaw]

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It is true and you're right its also true for all world views and phhilosophies (assuming you're ignoring logic and maths).


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But it isn't true - there are huge impications for the fact that we all have assumptions we can't prove.

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That's true but nothing to do with godel.

chez

[Silent A]

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there are huge impications for the fact that we all have assumptions we can't prove.

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This isn't what Goedel's theorem is about. It's about unprovable truths (mathematical truths), not unprovable assumptions.

[chrisnice]

There was an article in Scientific American on this in either the current or former issue.

[NotReady]

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That's true but nothing to do with godel.


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I wouldn't know, I've never read him.

[Copernicus]

Ive got an autographed copy of GEB if anybody wants to bid for it /images/graemlins/smile.gif. My sis-in-law was an editor (mentioned in the acknowledgements) and dated Doug Hofstader.

[spanshcastlemagc]

yah i see how i misunderstood, the proof is about true statements that are unprovable, oh well

[MelchyBeau]

a good example of this would be the parrallel postulate. You can't really prove parrallel lines exist universally, what happens is you get 3 types of geometry.

Elliptic, where no parrallel lines exist.

Euclidean, given a point not on a line, there exists exactly 1 line going through that point that is parrallel.

Hyperbolic, there are multiple parallel lines going through one point.

Melch

[Cyrus]

[ QUOTE ]
What Godel said was that for a non-trivial formal system there are always statements that cannot be proven. As proof of his theorem he offered his own theorem.

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"Every rule has its exceptions". As proof of this rule, consider that it has its exceptions as well. Meaning that there are rules without exceptions.

[sweetjazz]

[ QUOTE ]
Kurt Godel's Incompletness Theroem mathematically proved that with in a system there are things that cannot be proved or disproved, an example of that would be a system whose axioms included basic arithmetic couldnt be proved. In other words within its own axioms you cant prove that 2+2=4.

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This is wrong. In any axiomitization of arithmetic, you can prove 2+2=4. What cannot be proven are meta-statements; the first example's (in particular Godel's) used self-reference. Godel's key insight is that it is possible to make self-referential statements in the language of arithmetic.

So, no offense meant as this is hard stuff, I think you are heading off in a wrong direction. I think GEB has a good discussion of Godel's results IIRC, but you have to read it very carefully and avoid misconceptions. A lot of what's going on depends on a subtle understanding of the language used and not confusing concepts that are often closely related but have crucial distinctions.

[Mik1w]

"Informally, Gödel's incompleteness theorem states that all consistent axiomatic formulations of number theory include undecidable propositions."

Also relevant, (his 2nd incompleteness theorum):
"any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent."

From
http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html

[chezlaw]

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This is wrong. In any axiomitization of arithmetic, you can prove 2+2=4. What cannot be proven are meta-statements; the first example's (in particular Godel's) used self-reference. Godel's key insight is that it is possible to make self-referential statements in the language of arithmetic.


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Although its true that you can prove 2+2=4, you can't prove that there isn't also a proof that 2+2=3 (as you can't prove consistency and if inconsistent you can prove anything).

I think this gives the best example of reasonable belief without absolute certainty. We cannot prove that there is no proof that 2+2=3 but it sure seems reasonable to believe there isn't one.

chez

[bearly]

i'm sorry...........is the claim that a logical system can be incomplete? or, that every logical system is incomplete.

[bearly]

please..........any that are struggling so far, read the above post 3 times. i can't imagine it being said better. just to add, "meta" really does have a meaning, and meta-language has a logic of it's own................b