De hát az inkriminált állítás az nem az, hogy "minden krétai hazudik", hanem az, hogy "egy krétai azt mondja, hogy minden krétai hazudik". Lényeges különbség, és ezen bukik a "ebben a megfogalmazásban nincs olyan csapda, amiből lehetetlen kimenekülni" állításod.
Az egésznek az a szépsége, hogy Gödel előtt azt gondolták, hogy a matematika zárt, vagyis bárki állít valamit, az megoldható, vagyis illeszkedik egy formalizmushoz. Ehhez képest Gödel bebizonyította, hogy létezik olyan, a formalizmus nyelvén (annak jeleivel) felírható állítás, ami nem illeszkedik a formalizmushoz, vagyis nem bizonyítható. És itt ne keverjük az állítások igaz, hamis és nem bizonyítható voltát...
Valóban lehetne egyszerűbben is, íme egy angol nyelvű szöveg:
The proof of Gödel's Incompleteness Theorem is so simple, and so
sneaky, that it is almost embarassing to relate. His basic procedure
is as follows:
- 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.
- 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.
- 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."
- Now Gödel 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," 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.
Rucker, Infinity and the Mind (http://www.miskatonic.org/godel.html)
És van egy jóval részletesebb oldal is: http://www.ddc.net/ygg/etext/godel/
