All I see is the use of the boolean operator AND, which implies there are four possible states.
G_1: sentence is true but not provable <-- this was offered in the original post
G_2: sentence is true and provable
G_3: sentence is false but not provable
G_4: sentence is false and provable

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

If G is false then "this math sentence is false or provable in T".

I edited my OP. Please have a look again if you like.

Have I got it right? — TheMadFool

No.

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

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

G is not provable but it's true. But I'm not really an expert on the fine points so I probably shouldn't have jumped in earlier.

No problem. I'm not sure too. Thanks anyway

Have I got it right? — TheMadFool

Are you equating "true" with "provable"?

Are you equating "true" with "provable"? — Dzung

If you read the wikipedia article then this can't be done.

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"?

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

I don't know.

G = This sentence is not provable

If you can prove G then G is not provable. If you can't prove G then G is not provable

Since, you can either prove or not prove G, it follows that G is not provable.

I think the logic works like that.

You forgot to add that: T is consistent and G is a sentence in the vocabulary of T.

You forgot to add that: T is consistent and G is a sentence in the vocabulary of T. — Nagase

Good to see you Nagase and thanks.

I now see what you meant is similar to Godel's introduction passage in his 1931 paper. It's just an example he stressed the contradiction within a formal system. But it's not all Godel's theorem (1st one) is about. If that answers the question?
1st one says a computerizable set of rules cannot prove all statements of itself. The 2nd is stronger, saying it cannot even prove itself as consistent.
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?

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

Don't give up on math. I think, as Galileo said(?), math is the language of the universe. Consistency may not be be so problematic. It could be that we can't prove every true mathematical statement BUT that may not be required.

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

Yes it broke Hilbert's program, but it didn't prove that nothing is consistent in maths. For example, classical propositional logic is provably consistent (though we obviously want to use stronger logics than a propositional theory).

That said, you can work without consistency. After all, in response to Godel's Incompleteness Theorems, one can choose to go the inconsistent route and adopt a Paraconsistent Logic. This allows one to develop a mathematical theory on inconsistent foundations, yet because Paraconsistent logics lack the Law of explosion, theory is non-trivial.

This can result in an interesting mathematical theory which proves true (and simultaneously false) some of Frege's Logicism, as logicism becomes provable here and one can prove that the Continuum Hypothesis is false (in this formalism).

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Thanks.

I watched a video on it and the reasoning was...

Assume a consistent theory T
G is a statement in T
G = This sentence isn't provable from the axioms of T

Either G is true or false

Assume G is false. That means ''this sentence is provable from the axioms''. But only true statements are provable. That means G is true. But G is false (assumed). A contradiction.

But T is consistent. So it can't be that G is false.

What are we left with? G is true. In other words ther IS a sentence that can't be proved from the axioms.

Have I brushed against the truth or is this the real import of Godel's theorems?

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Thanks, I need to learn more to understand the extent you introduced but the reason I said so was due to my subscription to Godel's own view:
So the following disjunctive conclusion is inevitable: Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified . . . (Gödel 1995: 310).

i.e I am against the scientific trend to treat human as a classical moving machine, especially what inside the brain or thoughts.

Thank you for the explanation. I'm not sure if I understand all that but I feel much closer to the truth than before.

T = consistent mathematical theory
G = The mathematical proposition P is not provable from the axioms of T

G is either provable or not provable

1. G is provable. In other words ''the mathematical proposition P is not provable from the axioms'' is true. G is true

2. G is unprovable. G is true

How do you prove G to be true in T? You can't for then you'd prove G and contradict to its content. So G is true or false in T but we cannot know it (within T). We also have no position beyond T for T does already contain logic & arithmetic. Therefore I think we cannot conclude that G is true, but its truth is unprovable and that's not a proof of G's truth.

I think this shows that mathematics is both intuitive and formal at the same time. We shouldn't forget that mathematics is a creation of more than one brain function and, as such, can appear contradictory in the same way as someone can hold two or more positions in their head simultaneously.

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

For me the problem starts with 'This sentence is not provable'. This is meaningless. It does not state what is not provable. It would make no more sense to say 'This sentence is provable'.'This sentence' is not a statement and is not even a sentence. It is not provable or unprovable.

I wish someone would explain incompleteness in a way that it seems plausible to non-mathematicians. But explanations always it seems to depend on taking the liar paradox seriously, which try as I might I cannot do. , . . .

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Sadly, the whole thing seems beyond me. I've tried different books and sites, but just don't have any idea what he's doing. I get lost at the very first step.

Don't feel badly. I was a professor of mathematics for many years and never encountered a theorem in my area of study that was not provable in PA. Most of us don't. But there are some.

Thanks. But at least you understand what we're talking about. I need to find a professor of mathematics to sit down with me and help me understand it. But they aren't easy to come by.

Thanks. I tried to read the linked article but it goes over my head. It seems it is impossible to explain this issue to non-mathematicians.

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