Could someone show me a simple example of something being proved unprovable?

Question

Could someone show me a simple example of something being proved unprovable?

Pretty much what the title says, I want to understand a proof of some statement being proved unprovable.

E: Please read properly the question before marking as duplicates, I'm asking for a proof of a statement being proved unprovable, not examples of unprovable/indecidable statements.

Answer 1

If you are willing to take the completeness theorem for granted, then it can be quite easy.

If $T$ is a first-order theory, then $T\vdash\varphi$ if and only if $T\models\varphi$.

This means that in order to prove that $\varphi$ is unprovable from $T$, then we only need to exhibit a model of $T$ in which $\varphi$ is false.

For example, we can prove that as a field $\Bbb R\models\exists x(x\cdot x=1+1)$, but the Greek also proved that $\sqrt2$ is not a ratio of two integers, therefore $\Bbb Q\models\lnot(\exists x(x\cdot x=1+1))$. And so we proved that the theory of fields does not prove the existence of $\sqrt2$.

Andre suggested in the comments a similar example with group theory and the statement "multiplication is commutative".

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Other even easier examples may include things like a language $\cal L$ with a single constant symbol $c$, and the empty theory. Then $\forall x(x=c)$ is not provable, since in any structure for $\cal L$ with more than one element $\exists x(x\neq c)$ is a true statement.

Or even without constants, just look at $\forall x\forall y(x=y)$.

Answer 2

Consider statement, $P$:
"It is impossible to prove statement $P$."

Suppose it's true. Then, as per $P$, it's impossible to prove statement $P$, even though it's true.

Suppose it's false. Then, as per $P$, it's possible to prove statement $P$, even though it's false.

Asserting either the truth or the falsehood of the statement leads to a contradiction.

The statement appears to be impossible to prove. (Although, perhaps we shouldn't come right out and say it.)