In article <email@example.com>, John Baez <firstname.lastname@example.org> wrote: >I assume that by "etcetera" you mean there's one theory like this >per ordinal. [...] >>If one tries to articulate exactly what >>is "implicitly" involved in accepting PA in this sense, then one can >>make a plausibility argument that Gamma_0 is a natural stopping point. > >It would be really great if you could say more about this >plausibility argument.
Let's look more closely at what the notion of "one theory like this per ordinal" means. There's no difficulty figuring out what "Con(PA)" means or how to express that statement in the first-order language of arithmetic. Ditto with "Con(PA+Con(PA))". However, once you start ascending the ordinal hierarchy, a difficulty appears. The language of arithmetic doesn't let you talk about "ordinals" directly---that's a set-theoretical concept. In order to express a statement like "Con(T)" for some theory T, you need at minimum to be able to give some sort of "recursive description" or "recursive axiomatization" of T (where here I use the word "recursive" in the technical sense of recursive function theory) in the first-order language of arithmetic. This observation already yields the intuition that we're not going to be able to ascend beyond the Church-Kleene ordinal, because we won't even be able to figure out how to *say* "T is consistent" for a theory T that requires that many iterations to reach from PA.
There are other problems, though, that potentially get in the way before we reach the Church-Kleene ordinal. Once we realize that what we need is a system of "ordinal notations" to "fake" the relevant set theory, we may (if we are predicavists) worry about issues such as:
1. As we ascend the ordinal hierarchy, isn't it illegitimate to make a jump to an ordinal alpha unless we've already proved, at the level of some ordinal beta that we've already reached, that an ordinal of type alpha exists?
2. And isn't it illegitimate to create sets by quantification over things other than the natural numbers themselves and sets that we've already created?
Condition 1 goes by the name of "autonomy" and condition 2 goes by the name of "ramification." If one formalizes these notions in a certain plausible manner, then one arrives at Gamma_0 as the least upper bound of theories that you can get to, starting with (for example) PA.
One can of course wonder whether 1 and 2 above really capture the concept of "predicativity." Some secondary evidence has accumulated of the following form: Some argument that intuitively seems to be predicative but that is not immediately seen to be provable in the Feferman-Schuette framework is shown, after some work, to indeed be provable below Gamma_0.
It's still possible, of course, for someone---you mentioned Nik Weaver---to come along and argue that our intuitive notion of predicativism, fuzzy though it is, can't possibly be identified with the level Gamma_0. The reason you can't seem to decide immediately whether Weaver's position is nonsensical or not is probably because the critical questions are not mathematical but philosophical, and of course it's usually harder to arrive at definitive answers in philosophy than in mathematics. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences