>>Logicians [...] know a lot about how much induction >>up to different ordinals buys you. And apparently, induction up to >>Gamma_0 lets us prove the consistency of a system called "predicative >>analysis". I don't understand this, nor do I understand the claim >>I've seen that Gamma_0 is the first ordinal that cannot be defined >>predicatively - i.e., can't be defined without reference to itself. >>Sure, saying Gamma_0 is the first solution of >> >>phi_x(0) = x >> >>is non-predicative. But what about saying that Gamma_0 is the union >>of all ordinals in the Veblen hierarchy? What's non-predicative >>about that?
>The situation is somewhat akin to the situation with the Church-Turing >thesis, in that one is tentatively equating an informal notion >(predicativity or computability) with a precise mathematical notion. >Therefore there is no definitive answer to your question, and Feferman >himself has articulated potential objections to the "standard view" >that Gamma_0 marks the boundary of predicativity.
There's also someone named Nik Weaver who has debated Feferman on this subject:
He seems to claim that Gamma_0 and even larger ordinals have predicative definitions. However, I'm too ignorant to follow this debate. Usually in physics I have a sense for when people are being reasonable even if I don't follow the details. In this debate I can't even do that.
>Having said that, I'll also say that one of the reasons for the standard >view is that Gamma_0 marks the boundary of "autonomous progressions" of >arithmetical theories. The book by Torkel Franzen that you cited is >probably the most accessible introduction to this subject.
This summer I'm in Shanghai without any academic affiliation, so it's hard to get that book. When I return to Riverside in the fall I'll try to read it. But my curiosity is burning right now, so I'll take the liberty of asking some more questions.
>Roughly >speaking, the idea is that if anyone fully accepts first-order Peano >arithmetic PA, then implicitly he accepts its consistency Con(PA), as >well as Con(PA+Con(PA)), etc.
I assume that by "etcetera" you mean there's one theory like this per ordinal. I browsed a paper by Franzen where he was trying to explicate how these theories actually let you prove interesting new stuff.
It's a bit mysterious: I imagine a guy sitting there thinking "Peano arithmetic is true, so I know it's consistent, and I know *that's* consistent too, and I know *that's* consistent...", and so on - and after pondering this way for an transfinite amount of time, all of a sudden he can do new stuff like prove that Goodstein sequences approach zero!
I think Franzen was trying to dispel this naive conception. He said the real action happens at limit ordinals, where the interpretation of everything changes in some sneaky way.
But, my understanding of his comments like an impressionist painting of a surreal painting - Dali's "Sacrament of the Last Supper" as reworked by Monet.
(Hey, I managed to sneak a docahedron into the discussion!)
>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.
>I think you have a better shot at grasping the underlying intuition via >this approach than by staring at Gamma_0 itself and trying to figure out >what is non-predicative about its definition.