both the rule and the axiom are well known from the basic modal logic
K.
Similarly, we can show that if there is a proof of the sentence A in T, then T itself can check this proof, so T proves Prov(⌈A⌉)—we shall call this principle Prov-completeness. Again, though in a less straightforward way than in the case of modus ponens, we can formalize the principle itself and see that T actually proves:
Prov(⌈A⌉) → Prov(⌈Prov(⌈A⌉)&#x 2309;).
When we rephrase the principle of Prov-completeness and its formalization in modal logical terms, we get the modal rule that is usually called necessitation:
(3) � 0A0;,
and the modal axiom
(4) � 0A0;□A → □□A,
which is the transitivity axiom 4 well known from modal systems such as K4 and S4.
Finally, one might wonder whether T proves the intuitively valid principle that "all provable sentences are true," that is, whether T proves Prov(⌈A⌉) → A. Unexpectedly, this turns out not to be the case at all. Löb proved in 1953, using Gödel's technique of diagonalization, that T proves Prov(⌈A⌉) → A only in the trivial case that T already proves A itself!
Löb's theorem has a formalization that can also be proved in T. Writing both the theorem and its formalization in modal terms, we get the modal rule
(5) � 0A0;,
and the modal axiom
(6) � 0A0;□(□A → A) → □A,
usually called W (for well-founded) by modal logicians.
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