[FOM] Gamma_0

[T.Forster@dpmms.cam.ac.uk]

As Nick says, there are indeed other issues, but my question did not 
concern them, and was rather: can someone explain to me in what respect the 
omega-sequence of gammas whose sup is gamma_0 is less predicative than the 
sequence of iterates of lower gammas that one creates in order to obtain 
later lower gammas? Specifically it is alleged that one needs the second 
number class to be a set in order to prove the existence of worders of N of 
length gamma_0. I would be grateful to anyone who can shed light on this 
point.

 
2010, Nik Weaver wrote:

>
>Thomas Forster forwarded the comment
>
>> I can see how to prove the existence of wellorders of the naturals
>> of all smaller lengths without using power set (or Hartogs'..) but
>> it seems to me that i can do the same for Gamma_0 itself.
>
>Is the writer really sure about the first point?  Uncontroversially
>predicative systems typically don't get very far beyond \epsilon_0.
>The usual error in this kind of claim involves failing to distinguish
>between different versions of the concept "well-ordered".
>
>Let's say we have a total ordering < of the natural numbers that we
>know is well-ordered: every nonempty subset has a smallest element
>for <.  Suppose also that the property P is progressive for <, i.e.,
>for every number a we have that [(forall b < a)P(b)] implies P(a).
>Can we conclude that P(a) holds for all a?  Yes, we can prove this
>by considering the set {a: P(a) is false}.  If this set is nonempty
>then it has a smallest element, which contradicts progressivity of <.
>Therefore the set must be empty.
>
>*However*, this argument requires some sort of comprehension axiom in
>order to form the set {a: P(a) is false}.  If the property P involves
>quantification over the power set of the natural numbers, then
>predicativists would reject the relevant comprehension axiom and so
>would not be able to make the stated inference.
>
>This error is so common, I am willing to bet that the writer has made
>an illegitimate inference of this type in his well-ordering proof.
>
>I explain this issue in more detail in Sections 7 and 8 of my paper
>"What is predicativism?", and I discuss it at great length throughout
>the entire first half of the paper "Predicativity beyond Gamma_0".
>Both are available on my website at
>
>http://www.math.wustl.edu/~nweaver/conceptualism.html
>
>Nik Weaver
>Math Dept.
>Washington University
>St. Louis, MO 63130 USA
>nweaver at math.wustl.edu
>