Note that whether or not a property can be expressed by a formula of arithmetic has to do exclusively with the power of the language, whereas the question of whether or not it can be captured has to do, in addition, with the power of the selected axiom system. Thus, for example, the formula

there-exists-a-number  z  such-that   x * z = y.

is a formula with two free variables which expresses the relation “y is divisible by x”; but if we choose a very weak axiom system – for example, one consisting only of the axiom “0 + 1 = 1” – the property “y is divisible by x” will not be captured by this (or any other) formula.

— Gödel’s Incompleteness Theorems – A Brief Introduction

— Malgosia Askanas

2014.02.01 Saturday ACHK