Frege’s Principle

Abstract

In his Grundgesetze der Arithmetik,¹ Frege does indeed prove the “simplest laws of Numbers”, the axioms of arithmetic being among these laws. However, as is well known, Frege does not do so “by logical means alone”, since his proofs appeal to an axiom which is not only not a logical truth but a logical falsehood. The axiom in question is Frege’s Axiom V, which governs terms of the form “

$$\mathop \varepsilon \limits^, $$

.Φ(ε)”, terms which purport to refer to what Frege calls ‘value-ranges’. For present purposes, Axiom V may be written:²

$$\mathop \varepsilon \limits^, .F\varepsilon = \mathop \varepsilon \limits^, .G\varepsilon \equiv \forall \left( {Fx \equiv Gx} \right).$$

The formal theory of Grundgesetze, like any (full)³ second-order theory containing this sentence, is thus inconsistent, since Russell’s Paradox is derivable from Axiom V in (full) second-order logic.

In my Grundlagen der Arithmetik, I sought to make it plausible that arithmetic is a branch of logic and need not borrow any ground of proof whatever from either experience or intuition. In the present book this shall now be confirmed, by the derivation of the simplest laws of Numbers by logical means alone (Gg I §0).