Frege’s Principle

• Richard G. HeckJr.
Chapter
Part of the Synthese Library book series (SYLI, volume 251)

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.

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