mrnicksta
21st-March-2008, 09:49 AM
for those that don't know Gödel is a famous mathematician who came up with this theorem (that has been proved, this is not mere speculation) called, as the title suggests, the incompleteness theorem.
to begin, it might be useful to point out it is reminiscent of the classic Epimenides Paradox: "I am lying.”
The above paradox is neither provably true nor provably untrue, as is the nature of paradox.
Gödel's Incompleteness Theorem takes this notion one step further by stating (and proving) that things fall into 4 mutually exclusive categories:
provably true - quite obvious, we know there are truths we can prove
provably untrue - again quite obvious, there are things which we can prove to be untrue quite easily
unprovably true
unprovably untrue
the last two conjure some deep questions though. if there are things which are unprovably true, then the question begs to be asked is the pursuit of knowledge in some cases completely futile? taking into account Gödel's Theorem, the question of the existence of God may never bear any fruits as it probably falls into the last two categories, making it imposible to prove to true or untrue.
WE CAN NEVER KNOW THE WHOLE TRUTH!
now this sounds quite ridiculous and it takes a lot of time to get your head around, but it is 100% true (ironic that the Incompleteness Theorem falls in the provably true section of the Incompleteness Theorem itself!), for the benfit of those that doubt it's truth i have included an informal proof:
Someone introduces Gödel to a machine that is supposed to be a Universal Truth Machine (UTM), 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.
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 asks UTM whether G is true or not.
(a) If UTM says G is true, then "UTM will never say G is true" is false.
(b) 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!
this is of course an informal proof and Gödel's actual mathematical proof is a lot longer and complex!
for those who have been interested in what i have discussed here, you should check the magnificent book "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter - it is one of the most incredibly intelligent books, that is so broad in it coverage (art, music physics, language, maths, biology, unbelievably clever dialogues), depth and clarity it will have you amazed with every page (at least i have been as i am reading through it). best yet, despite it's broad range of topics it is not a jumble or randomness, it is very focused with it's point and everything is chosen very carefully to fit in with that point.
to begin, it might be useful to point out it is reminiscent of the classic Epimenides Paradox: "I am lying.”
The above paradox is neither provably true nor provably untrue, as is the nature of paradox.
Gödel's Incompleteness Theorem takes this notion one step further by stating (and proving) that things fall into 4 mutually exclusive categories:
provably true - quite obvious, we know there are truths we can prove
provably untrue - again quite obvious, there are things which we can prove to be untrue quite easily
unprovably true
unprovably untrue
the last two conjure some deep questions though. if there are things which are unprovably true, then the question begs to be asked is the pursuit of knowledge in some cases completely futile? taking into account Gödel's Theorem, the question of the existence of God may never bear any fruits as it probably falls into the last two categories, making it imposible to prove to true or untrue.
WE CAN NEVER KNOW THE WHOLE TRUTH!
now this sounds quite ridiculous and it takes a lot of time to get your head around, but it is 100% true (ironic that the Incompleteness Theorem falls in the provably true section of the Incompleteness Theorem itself!), for the benfit of those that doubt it's truth i have included an informal proof:
Someone introduces Gödel to a machine that is supposed to be a Universal Truth Machine (UTM), 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.
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 asks UTM whether G is true or not.
(a) If UTM says G is true, then "UTM will never say G is true" is false.
(b) 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!
this is of course an informal proof and Gödel's actual mathematical proof is a lot longer and complex!
for those who have been interested in what i have discussed here, you should check the magnificent book "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter - it is one of the most incredibly intelligent books, that is so broad in it coverage (art, music physics, language, maths, biology, unbelievably clever dialogues), depth and clarity it will have you amazed with every page (at least i have been as i am reading through it). best yet, despite it's broad range of topics it is not a jumble or randomness, it is very focused with it's point and everything is chosen very carefully to fit in with that point.