Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.
The theory is strong enough to describe recursively defined integer functions such as exponentiation, factorials or the Fibonacci sequence.
Gentzen showed that the consistency of the first-order Peano axioms is provable over the base theory of primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0.
The additional principle means, informally, that there is a well-ordering on the set of finite rooted trees.
[1] Gentzen's proof highlights one commonly missed aspect of Gödel's second incompleteness theorem.
That said, there are other, finer ways to compare the strength of theories, the most important of which is defined in terms of the notion of interpretability.
A strong form of the second incompleteness theorem, proved by Pavel Pudlák,[2] who was building on earlier work by Solomon Feferman,[3] states that no consistent theory T that contains Robinson arithmetic, Q, can interpret Q plus Con(T), the statement that T is consistent.
So, in the sense of consistency strength, as characterized by interpretability, Gentzen's theory is stronger than Peano arithmetic.
[4] Kleene (2009, p. 479) made the following comment in 1952 on the significance of Gentzen's result, particularly in the context of the formalist program which was initiated by Hilbert.
In contrast, Bernays (1967) commented on whether Hilbert's confinement to finitary methods was too restrictive: Gentzen's first version of his consistency proof was not published during his lifetime because Paul Bernays had objected to a method implicitly used in the proof.
In this language, Gentzen's work establishes that the proof-theoretic ordinal of first-order Peano arithmetic is ε0.
Laurence Kirby and Jeff Paris proved in 1982 that Goodstein's theorem cannot be proven in Peano arithmetic.