Löb's theorem and provability I learned Löb's theorem. As I understanding, if a statement is formed like "I am provable", the statement should be provable.
I want to ask further about Löb's theorem.
There is two sentences, P and Q.
P: P and Q is provable both.
Q: P and Q is provable both.
.
Now, can we prove that P and Q is provable both? Is this relevant to Löb's theorem?
 A: Löb's Theorem says that,

If certain conditions hold for the theory $T$, then if $T \vdash Prov(\overline{\ulcorner\varphi\urcorner}) \to \varphi$, then $T \vdash \varphi$.

(Here $\overline{\ulcorner\varphi\urcorner}$ is $T$'s formal numeral for $\varphi$'s Gödel number under some given coding scheme: and $Prov$ is a 'provability predicate' suitably expressing the property of numbering a $T$-theorem in that scheme. Details in any textbook!)
Apply this to a Henkin sentence $H$ which is a fixed point for the provability predicate, i.e.

$T \vdash H \equiv Prov(\overline{\ulcorner H\urcorner})$

(so $H$ sort-of-says "I am provable"), and it is immediate that

$T \vdash H$.

Now: suppose (just suppose!) we have a pair of sentences $P, Q$ such that

$T  \vdash P \equiv Prov(\overline{\ulcorner (P \land Q) \urcorner})$
$T  \vdash Q \equiv Prov(\overline{\ulcorner (P \land Q) \urcorner})$.

So $P$ sort-of-says "$P \land Q$ is provable" and likewise for $Q$. Then trivially, we'd also have

$T  \vdash (P \land Q) \equiv Prov(\overline{\ulcorner (P \land Q) \urcorner})$.

Then, by Löb's Theorem again, we'd have

$T \vdash (P \land Q)$, and hence $ T \vdash P$ and $T \vdash Q$.

But there is no novel interest in this as far as I can see.
