# Is a Gödel sentence logically valid?

This might be an elementary question, but I am just beginning to learn logic theory.

From wikipedia article on Gödel’s incompleteness theorems

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250).
The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory. The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory.

My question: Is a Gödel statement logically valid?.

Edit: As Carl answers below, if the Gödel statement is valid, then by completeness theorem, it is provable, which leads to a contradiction. So there exists a model in which the statement is false. Can we construct such a model?

#### Solutions Collecting From Web of "Is a Gödel sentence logically valid?"

No, a Gödel sentence is not logically valid. Because the Gödel sentence for a theory $T$ is unprovable from $T$, it follows from the completeness theorem for first-order logic that there is a model of $T$ in which the Gödel sentence is false.

When the text you quoted says “true” you should read that as “true in the standard model of arithmetic”. Logical validity would correspond to truth in all models. An example of a logically valid sentence is $(\forall x) (x=x)$.