TITLE: Consistency of Analysis (second order arithmetic)
QUESTION [12 upvotes]: Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic? 
Update:
Which (different) methods can be used to prove the consistency of Analysis? and where can I find such proofs?

REPLY [14 votes]: As Noah says, the direct successor of Gentzen's method, cut-elimination, has been generalized up to $\Pi^1_2$-comprehension.  This was shown separately by Rathjen and Arai; the full results have never been published, but fragments have appeared in various papers.  Rathjen published "An ordinal analysis of parameter free $\Pi^1_2$-comprehension", which covers a fairly strong subtheory of $\Pi^1_2$-comprehension.  The strongest results of Arai's I'm finding actually published only go up to $\Pi_3$ reflection, "Proof theory for theories of ordinals II: $\Pi_3$-reflection".  My paper, "Ordinal analysis by transformations", is rather vague about the comparison between the system it's analyzing and usual hierarchy, but I now think the system it analyzes is around the strength of parameter free $\Pi^1_2$-comprehension.
However the consistency of analysis has been shown by other means.  First, as Carl mentions, Spector's proved the consistency of analysis in "Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics".  Spector's proof is considered non-constructive (it uses bar recursion).
Girard gave a constructive consistency proof of analysis: he proves strong normalization proof for System F, which is an equivalent system.  This proof can be found written up quite nicely in his book, Proofs and Types.

REPLY [9 votes]: Although the proof-theoretic ordinal of second-order arithmetic is very hard to determine, there is another standard method for the proving consistency of arithmetic: Gödel's Dialectica interpretation.  This was originally used by Gödel to give a different relative consistency proof of Peano arithmetic by reducing its consistency to the consistency of a quantifier-free theory of functionals of finite type known as system $T$.
This work was later extended by Spector and Howard to give a relative consistency proof for second-order arithmetic. The weaker system used is the same system $T$ augmented with bar recursion. The details are spelled out in section 6 of Gödel's Functional ("Dialectica") Interpretation by Jeremy Avigad and Solomon Feferman from the Handbook of Proof Theory.
Although this is not a Gentzen-style analysis, it does have a certain analogy. Gentzen showed that the consistency of Peano Arithmetic reduces to that of a weak theory augmented with transfinite induction. The Dialactica-style relative consistency proof for second-order arithmetic reduces its consistency to that of a (different) weaker theory $T$ augmented with bar recursion, which can be seen as a scheme for constructing objects by transfinite recursions. The induction scheme dual to bar recursion, bar induction, is a kind of transfinite induction scheme. The proof also gives a characterization of the provably total computable functions of second-order arithmetic, much like the consistency proof for Peano arithmetic does.