Godel's incompletness theorem - proving a statement is false

Question

I have two question regarding Godel's incompletness theorem. The theorem says that every axiomatic system is either incomplete or inconsistent. If it's consistent, then there are true statements that are unprovable.

If so, then it must also be the case that there are false statements that cannot be proven to be false, becuase if we could prove that every false statement is false, then we would be able to prove every true statement - by contradiction.

If every false statement was provably false, then I could just take the unprovable true statement $A$, prove the falsehood of its negation and I would have a proof of $A$.

Then there must exist both true and false statements that cannot be proven true and false respectively.

Question: where is the logical error I'm making here?

Answer 1

How do you know that a statement is true if one cannot prove it? If a statement is not decideable from the given axioms you can decide whether it is true or false by adding the statement or its negation to the axioms.

Answer 2

According to G's Th, it is correct to say that :

"there must exist false sentences that cannot be proven false".

If a "suitable" theory $T$ is consistent, then - by G's Th - there exists a true sentence $G_T$ such that:

$T \nvdash G_T$ and $T \nvdash \lnot G_T$;

this is the incompleteness of $T$.

But if $G_T$ is true, then $\lnot G_T$ is false.

To say that a sentence $A$ is "provably false (in the theory $T$)" must be transalted as :

if $A$ is false, then $T \vdash \lnot A$.

Thus, $\lnot G_T$ is a false sentence of $T$ and $T \nvdash G_T$, i.e. :

$T \nvdash \lnot (\lnot G_T)$

and so we have that $T$ has a false sentence (i.e. $\lnot G_T$) that cannot be proved false.

Answer 3

It is not true that EVERY axiomatic system is incomplete. Godel himself proved the completeness of first order logic. It is true that a system sufficiently rich enough to do mathematics with (i.e. one that can formulate the Peano axioms) is incomplete.

In this case what you say is true. There are false statements which are not provably false; I mean, we can't decide on their truth value.