The crux of Hilbert's problem set forth in 1900, lay in a logical contradiction first attributed to Bertrand Russell as the Liar's paradox. It is the paradox inherent in any set containing itself as an element.
The fine point at issue, is the difference between truth and provability. Godel opted for provability.
As he put it on page 123: "the existence of undecidable propositions is not just a matter of incompleteness but a threat to the [very] integrity of the whole works." If a statement F were true but not provable, then the statement not-F would not cause any contradictions because there would be nothing in the system to disprove it. However, in such an unsettling circumstance, 'one obtains a consistent system in which a false proposition is provable.'" And "it can easily be shown that any formula whatsoever would be provable even an outright absurdity like 0=1." Such a situation would be an intolerable logical no, no.
So how did Godel go about constructing a proof to find Hilbert's pest?
First he devised a method for transforming "statements about process of proof," into "statements about properties of numbers."
After doing this, he could then write metamathematical assertions about the construction of proofs, as simple arithmetic formulas.
This turned meta-mathematical propositions into simple mathematical propositions. From here, any mathematical formula could then be expressed as a unique number.
Thus, a self-referential coding system was born that allowed Godel to isolate and exhibit any propositions that were contained in the system but not provable. QED.
Anyone seeking more details, cannot do better than the blow-by-blow description given in the appendix, or that in Douglas Hofstatder's Godel, Escher and Bach, or Rebecca Goldsten's "Incompleteness: The Proof and Paradox of Kurt Godel." I have reviewed them both on Amazon.com. Easily a five star effort.
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