*Proof Theoretic Strength* is a measure of the

*Power* of a formal system.

We know, from Gödels incompleteness results and also from later results in the theory of recursive functions, that where the
subject matter of a theory is sufficiently rich the theory will be incomplete.

Since being able to express quite elementary propositions about arithmetic is sufficient to qualify for this limitation

*any* system which is a contender as a foundation for mathematics will be incomplete.

Not only do we know that such systems will be incomplete, we also know that there is no maximal system, which can prove more
than any other.
It therefore is meaningful to rank formal systems according to how much can be proven in the system.

*Proof theoretic strength* is one such measure.

There are several different ways in which the proof theoretic strength can be measured.

The first is also known as ordinal strength.
There are two variants of ordinal strength of which I am aware.
Variant (a) concerns what orderings of the natural numbers can be proven in the system to be well-orderings, the strength
being the smallest which is not provably well-ordered in the system.
Variant (b) concerns what is required to prove the consistency of the system, and is the least well-ordering necessary for
that purpose.