Sentence examples for which is provable from inspiring English sources

I worry that more and more today we lean toward that which is provable, when all the excitement is in the hard-to-prove.

Hence, we have a real statement which is provable in ideal mathematics and not in real mathematics.

saying that for any two sentences there is a third sentence which is provable if and only if either of the first two sentences is provable.

However, Cantor's Theorem (which is provable in type theory and therefore valid in all models) says that D has more subsets than members.

It follows that assuming Δ̰11-determinacy (which is provable in ZFC) one gets that the Σ̰11 sets of reals have the regularity properties (see the exercises in 6.G of Moschovakis (1980)).

Recall that the Reflection Principle (Section 4), which is provable in ZFC, asserts that every true sentence (i.e., every sentence that holds in $V$) is true in some $V_\alpha$.

Detlefsen argues that these conditions are not necessary for a predicate to count as a genuine provability predicate, and indeed there are provability predicates which violate the provability conditions and which give rise to consistency formulas which are provable in their corresponding theories.

Furthermore, might there not be other sentences which are provable and also express the consistency of F? Giving a rigorous proof of the second theorem in a more general form that covers all such sentences, however, has turned out to be very complicated.

Detlefsen's other argument against the common interpretation of Gödel's second theorem focuses on the notion of formalization: That the particular formalization of "T is consistent" by Gödel's formula ConT is not provable does not imply that there couldn't be other formulas, which are provable in T, and which have as much right to be called "formalizations of the consistency of T".

The resulting theory is a conservative extension of ZFC: it proves all the theorems of ZFC about sets, and it does not prove any theorem about sets which is not provable in ZFC.

The First Incompleteness Theorem provides a counterexample to completeness by exhibiting an arithmetic statement which is neither provable nor refutable in Peano arithmetic, though true in the standard model.