Thoth Hermes comments on Why do the Sequences say that “Löb’s Theorem shows that a mathematical system cannot assert its own soundness without becoming inconsistent.”?

[Thoth Hermes]

But that requires that for all T, T is provable, in order to get that T and not T are provable at the same time.

That additional assumption requires a system (not PA) in which T and not T (everything or anything at all) are provable already. But we aren’t assuming that kind of system, only a system that trusts itself.