# Are provable statements true

logicphilosophyprovability

I recently started reading an introductory logic textbook, and I haven't got yet to the chapter that talks about completeness theorem, but I just couldn't wait to read about it using shortcuts. I just read Completeness Theorem and had the below question.

So in the link above, there is a proof for "every true statement is provable" by using Henkin's Theorem: if $$T$$ is syntactically consistent, then $$T$$ has a model.

But I am wondering what would be the proof for the converse: every provable statement is true.

Intuitively, this sounds obvious because if $$T$$ is provable, then in any model, we should be able to use the proof that $$T$$ is provable. This implies $$T$$ is provable in any model, (which implies there exists a proof of $$T$$ in any model???)$$\leftarrow$$ not sure.

Also, let's say we have a proof that every provable statement is true. But then that just tells me "every provable statement is true" is provable, not necessarily true???

The completeness theorem is a result about the traditional systems of proof that states $$(T\vDash \phi) \Rightarrow (T \vdash \phi).$$ That is, if $$\phi$$ holds in every model where $$T$$ holds, then there exists a proof of $$\phi$$ from $$T$$. Conversely, the soundness theorem states $$(T\vdash \phi) \Rightarrow (T\vDash \phi),$$ so that if you can prove $$\phi$$ from $$T$$ then in every model where $$T$$ holds, $$\phi$$ must also hold. The proof of the soundness theorem is considered elementary and goes like this: