Frege's theorem

In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle.

[2] The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work.

However, not only did Basic Law V fail to be a logical proposition, but the resulting system proved to be inconsistent, because it was subject to Russell's paradox.

[4] The inconsistency in Frege's Grundgesetze overshadowed Frege's achievement: according to Edward Zalta, the Grundgesetze "contains all the essential steps of a valid proof (in second-order logic) of the fundamental propositions of arithmetic from a single consistent principle.

For all possible assignments of false (✗) or true (✓) to P, Q, and R (columns 1, 3, 5), each subformula is evaluated according to the rules for material conditional, the result being shown below its main operator.