Why Brouwer Was Justified in his Objection to Hilbert's Unqualified Interpretation of Quantification

We define a finitary model of first-order Peano Arithmetic in which quantification is interpreted constructively in terms of Turing-computability, and show that it is inconsistent with the standard interpretation of PA. 1 Hilbert’s interpretation of quantification Hilbert interpreted quantification in terms of his ε-function as follows [Hi27]: IV. The logical ε-axiom 13. A(a)→ A(ε(A)) Here ε(A) stands for an object of which the proposition A(a) certainly holds if it holds of any object at all; let us call ε the logical ε-function. 1. By means of ε, “all” and “there exists” can be defined, namely, as follows: (i) (∀a)A(a)↔ A(ε(¬A)) (ii) (∃a)A(a)↔ A(ε(A)) . . . On the basis of this definition the ε-axiom IV(13) yields the logical relations that hold for the universal and the existential quantifier, such as: (∀a)A(a)→ A(b) . . . (Aristotle’s dictum), 1