It happened, as surely as every other tangible, provable event in my life.
The difference between the two notions of provability: "provable in a given formal system S" and provability by any correct means — manifests itself as a consequence of Gödel's Second Incompleteness Theorem, as follows.
As a consequence, this also settles Henkin's original problem: assuming that the arithmetized provability predicate is again "normal" (i.e., satisfies Löb's derivability conditions), all sentences "asserting their own provability" are provable.
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 in Boole's case the set-theoretic consequences that he relies on are all easily provable by formal proofs in first-order logic, not even using any set-theoretic axioms; and by the completeness theorem (see the entry on classical logic) the same is true for first-order logic.
He defined the real numbers to be the cuts (L, U) just described that is, as partitions of the rationals with each member of L less than every member of U. Cuts included representatives of all rational and irrational quantities previously considered, but now the existence of greatest lower bounds became provable and hence also the intermediate value theorem and all its consequences.
Finally, in 1974, Martin proved the landmark result that Δ̰11-determinacy is provable in ZFC.
Hence, if the arithmetic sentence were provable, p would also be provable contradicting the previous result.
Heyting also proposes a provability operator +, where +p means "p is provable".
Valid inferences among propositions are reflected by the provable formulas, because (for any A and B) A ⊃ B is provable if and only if B is always a logical consequence of A. The propositional calculus is consistent in that there exists no formula in it such that both A and ∼A are provable.
