G=This math sentence is true AND not provable in T
Either G is true or false
1. G is true: Then T is incomplete
2. G is false: This math sentence is false AND provable in T. Inconsistent because this math sentence is true. — TheMadFool
No — fishfry
I know the real proof is very complex but it seems to rely on a modified form of the Liar's paradox. Can you explain where I went wrong. Thanks — TheMadFool
Are you equating "true" with "provable"? — Dzung
OK if not, why 1. can be done?
G is provable means it can be proven either true or false, how come "so G is unprovable"? — Dzung
You forgot to add that: T is consistent and G is a sentence in the vocabulary of T. — Nagase
Well that to me broke down any miracles maths had attained. Now if the plain arithmetic cannot be stated to be consistent then what can? nothing on earth. This is exactly a fatal blow to Hilbert as pioneer supporter of maths.
Finally if nothing is consistent then where should you place your trust on? — Dzung
Now if the plain arithmetic cannot be stated to be consistent then what can? nothing on earth. This is exactly a fatal blow to Hilbert as pioneer supporter of maths.
Finally if nothing is consistent then where should you place your trust on? — Dzung
I know the real proof is very complex but it seems to rely on a modified form of the Liar's paradox. Can you explain where I went wrong. Thanks — TheMadFool
Because otherwise, every theorem is provable in T.Assume a consistent theory T — TheMadFool
Ok, but now you have to be careful about exactly what G says.G is a statement in T — TheMadFool
As you have it, G isn't a meaningful expression in T. The problem is with "This sentence." It has to be well defined. Absent that, it's just a variation of the Liar Paradox in English - English isn't math, and Godel's theorem is interesting because it is itself a variation of the Liar Paradox, but expressed in mathematical-logical terms, which the English sentence is not.G = This sentence isn't provable from the axioms of T — TheMadFool
Suppose a mathematical theory/system T.
G=This sentence is not provable in T
Either G is provable or not provable
1. G is provable. So G is unprovable
2. G is not provable
So, there is G in the theory T
Have I got it right? — TheMadFool
And it has nothing to do with Godel's G.For me the problem starts with 'This sentence is not provable'. — FrancisRay
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