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:
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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")?
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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.
Best Answer
Rom Maimon is describing the program of proof-theoretic ordinal analysis.
First, as you observed in your addendum, it isn't interesting to find some encoding of an ordinal whose well-foundedness implies Con(ZFC), but rather an ordinal such that the well-foundedness of any representation implies Con(ZFC). One hopes that it suffices to consider natural representations of the ordinal, which has been true in practice, but is unproven (and probably unprovable, given the difficulty of making precise what counts as a natural representation).
It's possible to prove that the smallest ordinal which ZFC fails to prove well-founded is computable by noticing that the computable notations for ordinals provably well-founded in ZFC are a $\Sigma_1$ subset of the computable notations for ordinals, so certainly $\Sigma^1_1$, and by a result of Spector, any $\Sigma_1^1$ subset of the computable notations for ordinals is bounded.
As pointed out in the answer Timothy Chow links to above, it's typically true that this notion of proof-theoretic ordinal ends up being the same as ordinals with other nice properties (like implying Con(ZFC)), but there's no proof that that will always happen (and can't be, since there are defective examples that show it's not always true), nor a proof that covers ZFC.
However it's generally believed that for "natural" theories, including ZFC, the different notions of proof-theoretic ordinal will line up.
Finding an explicit description of the ordinal for ZFC is an active problem in proof theory, but progress has been very slow. The best known results are by Rathjen and Arai (separately) at the level of $\Pi^1_2$-comprehension (a subtheory of second order arithmetic, so much, much weaker than ZFC), and after nearly 20 years, those remain unpublished. The results in the area got extremely difficult and technical, and didn't seem to provide insight into the theories the way the smaller ordinals had, so it's not nearly as active an area as it once was. Wolfram Pohlers and his students still seem to working in the area, and some other people seem to be thinking about other approaches rather than directly attacking it (Tim Carlson and Andreas Weiermann and their students come to mind).