TITLE: Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?
QUESTION [39 upvotes]: In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically claimed.  More precisely, he argues that the consistency of axioms sufficient for all meaningful mathematics should be provable from "intuitively self-evident" statements about various computable ordinals being well-defined.  Here, of course, he points to Gentzen's 1936 consistency proof, which proves Con(PA) from primitive recursive arithmetic plus induction up to the ordinal ε0; as well as more recent results from the field of ordinal analysis, which show that the consistency of various weaker set theories than ZF (for example, Aczel's constructive ZF and Kripke-Platek set theory) can also be reduced to the well-definedness of various computable ordinals.  Maimon then goes on to say that Con(ZF) should similarly be reducible to the well-definedness of some "combinatorially-specified," computable ordinal, although the details haven't been worked out yet.  (And indeed, the Wikipedia page on ordinal analysis says that it hasn't been done "as of 2008.")  This sounds amazing, especially since I'd never heard anything about this problem before!  So, here are my questions:

Is there a general theorem showing that Con(ZF) must be reducible to the well-definedness of some computable ordinal, i.e. something below the Church-Kleene ordinal (even if we don't know how to specify such an ordinal "explicitly")?
Is finding an "explicit description" of a computable ordinal whose well-definedness implies Con(ZF) a recognized open problem in set theory?  Do people work on this problem?  Or is there some reason why it's generally believed to be impossible, or possible but uninteresting?  Or does it just come down to vagueness in what would count as an "explicit description"?

Addendum: There's a connection here to a previous MO question of mine, about the existence of Π1-statements independent of ZF with lots of iterated consistency axioms.  In particular, using an observation from Turing's 1938 PhD thesis, I now see that it's indeed possible to define a "computable ordinal" whose well-definedness is equivalent to Con(ZF), but only because of a "cheap encoding trick."  Namely, one can define the ordinal ω via a Turing machine which lists the positive integers one by one, but which simultaneously searches for a proof of 0=1 in ZF, and which halts and outputs nonsense if it ever finds one.  What I suppose I'm asking for, then, is a computable ordinal α such that Con(ZF) can be reduced to the statement that there's some Turing machine that correctly defines α.

REPLY [4 votes]: For every c.e. theory T (extending a weak base theory), we already know a polynomial time computable linear ordering $≺$ that captures the $Π^1_1$ strength of T:
Provably in a weak base theory, a $Π^1_1$ statement is provable in T iff for some term s, it is provable in a weak base theory plus "$≺$ is well-founded below s".
Thus, for $Π^1_1$ sound T, the order type of $≺$ is the supremum of order types of provably-in-T recursive well-orderings, and $≺$ is a well-ordering iff T is $Π^1_1$ sound.
Here is an example of $≺$:
$(p_1,q_1,n_1) ≺ (p_2,q_2,n_2)$ if $(p_1,q_1)<(p_2,q_2)$ (lexicographically)
$(p,q,n_1) ≺ (p,q,n_2)$ iff
- $q$ is a T-proof that bounded-time Turing machine $p$ computes a well-ordering, and $p$ accepts $(n_1,n_2)$ (any reasonable choice for 'bounded time' that keeps $≺$ polynomial time works here), or
- $q$ is not a T-proof of the above, and $n_1<n_2$.
The problem is that the above $≺$ is uninformative about T.  A key goal of ordinal analysis is to find a canonical $≺$ that makes the power of T simple and explicit, and thus give us a qualitatively better understanding of T.  Existence of a noncanonical $≺$, combined with existence of canonical $≺$ for weaker theories, suggests that a canonical $≺$ also exists for ZFC, but it is difficult to be certain until we actually find and prove such a $≺$.  Typically, an approach to finding $≺$ can be extended until it becomes too complex, and then a new idea permits $≺$ to become simpler again.
A caution about ordinal representations:
* Every sufficiently large recursive ordinal has different recursive representations that are not arithmetically isomorphic.
* I think that for every recursive ordinal α and a c.e. theory T (extending a base theory), there is a recursive representation $≺$ of α such that T + "$≺$ is well founded" is $Σ^1_1$ conservative over T.  This follows from existence of a recursive pseudowellordering $≺_p$ such that T + "$≺_p$ is well founded" is $Σ^1_1$ conservative over T.