For any natural number n, n+1 is greater than n.

The classifications and are defined inductively for every natural number n using the following rules:

For some natural number n, n is equal to 5 + 5.

The problem: given a natural number n larger than two, determine whether it is prime.

These functions take n arguments for some natural number n and are called n-ary.

For every natural number n, T has only finitely many n-types.

For every natural number n, every n-type is isolated.

Choose a natural number N greater than all types assigned to variables by this stratification.

Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity.

By the compactness theorem, for some natural number n the set would be inconsistent.