Leon Henkin put forward the question whether the sentence expressing its own provability ("I am provable") is true or false, and provable or not (Henkin 1952).
For example, though Wittgenstein vacillates between "provable in PM" and "proved in PM" at (RFM App.
As the incompleteness results in particular teach us, there are sets which are only weakly but not strongly representable (the key example being the set of statements provable in the system).
In linear logic already we have t → T provable without its converse, for instance.
It is this faculty of intuition that allows us to see, for example, that the formula saying "I am not provable" must be true, even though it defies proof within the system where it lives.
On the other hand, if we view a synchronous formula as an agent whose evolution is determined by the corresponding right-introduction rule, then it is possible for a provable sequent to evolve to a non-provable sequent (for example, by applying the ⊕ right-introduction rule).
For example, note that the lemma only provides a (provable) material equivalence between D and A ⌈D⌉) (which states that both sides must have the same truth-value) and does not claim any sort of sameness of meaning.
(φ might be 'x=x', expressing the property being self-identical, for example — in this case, of course, '□∀x x=x' is provable in SQML. Or it might simply be the formula 'Px', where 'P' is assumed to express some arbitrary property that necessarily holds of everything).
Heyting also proposes a provability operator +, where +p means "p is provable".
Another key point about numeralwise expressibility is that although we informally interpret, for example, Prov Sb ru1…unZ x1)…Z xn))), by: 'the formula with Gödel number r is provable if the Gödel number for the xi th numeral is substituted in place of the i th variable,' neither the formal statement within the theory P nor anything we prove about it appeals to such meanings.
