Goedel's real accomplishment was to formulate the function G in a system that just contains propositional and predicate logic and the natural numbers with addition. — Pippen
Let S be consistent (assumption). Because of the first incompleteness theorem it follows that if S is consistent, then G is unprovable. From this it follows by mp (and is thus proved) that G in S is unprovable. But this is precisely the content of G (see above, the italic style marked one), so that G in S would be proved, which is impossible according to the first incompleteness theorem (there case 1a), so that the consistency assumption must be false. — Pippen
How would you explain the Second Therorem based on my version with S, G(G is unprovable in S) and my explanation of the Frist Theorem? Maybe that is the best way to show me what you mean, because I will see the difference in my and your version. — Pippen
1) If a system is complete (as in every possible truth can be proved) then it's necessarily inconsistent (contradictions arise)
2) If a system is consistent (as in there are no contradictions) then it is neccesarily incomplete (some truths can't be proved) — TheMadFool
Because of the First Incompleteness Theorem we know that if S is consistent then G is unprovable in S. Since "G is unprovable in S" is our function G (see above) we can shortcut: If S is consistent then G. Now, we just formulate this statement in S and we know it's provable in S. Now, we assume we could also prove in S that S is consistent. Then by mp G would follow (and thus be proven) in S which is impossible due to the First Incompleteness Theorem. Because of this contradiction our assumption must have been false. — Pippen
Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". — Wikipedia
So that is a fancier way to say "This statement is false"? — BlueBanana
Because of the First Incompleteness Theorem we know that if S is consistent then G is unprovable in S. Since "G is unprovable in S" is our function G (see above) we can re-formulate that statement as: If S is consistent then G. Now, we "just" formulate this statement in S and we know it's provable in S. Now, we assume we could also prove in S that S is consistent. Then by mp G would follow (and thus be proven) in S which is impossible due to the First Incompleteness Theorem. Because of this contradiction our assumption must have been false. — Pippen
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