In article <email@example.com>, John Baez <firstname.lastname@example.org> wrote: >Logicians including Feferman and Schuette have carried out a detailed >analysis of this subject. They 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? > >If anyone could explain this in simple terms, I'd be much obliged.
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.
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. 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. 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. 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. -- 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