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. It was first proven, informally, by Gottlob Frege in his Die Grundlagen der Arithmetik, published in 1884, and proven more formally in his Grundgesetze der Arithmetik, published in two volumes, in 1893 and 1903. The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics known as neo-logicism.

Overview

In The Foundations of Arithmetic, and later, in Basic Laws of Arithmetic, Frege attempted to derive all of the laws of arithmetic from axioms he asserted as logical. Most of these axioms were carried over from his Begriffsschrift; the one truly new principle was one he called the Basic Law V : the "value-range" of the function f is the same as the "value-range" of the function g if and only if ∀x. 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. 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 of the fundamental propositions of arithmetic from a single consistent principle." This achievement has become known as Frege's theorem.

Frege's theorem in propositional logic

In propositional logic, Frege's theorems refers to this tautology:
The theorem already holds in one of the weakest logics imaginable, the constructive implicational calculus. The proof under the Brouwer–Heyting–Kolmogorov interpretation reads.
In words:
"Let f denote a reason that P implies that Q implies R. And let g denote a reason that P implies Q. Then given a f, then given a g, then given a reason p for P, we know that both Q holds by g and that Q implies R holds by f. So R holds."
The truth table to the right gives a semantic proof. For all possible assignments of false or true to P, Q, and R, each subformula is evaluated according to the rules for material conditional, the result being shown below its main operator. Column 6 shows that the whole formula evaluates to true in every case, i.e. that it is a tautology. In fact, its antecedent and its consequent are even equivalent.