Re: Characterization of Gamma_0


Rupert wrote:
Rupert wrote:
Daryl McCullough wrote:
Rupert says...

Daryl McCullough wrote:

Instead of working with PA, let's switch to a weak second-order theory.
In this theory, we have two sorts of objects, naturals and sets of naturals.
The sets of naturals are closed under comprehension.

What sort of comprehension? First-order comprehension?

I think that's sufficient for my purposes.


Yes, well I always thought something like your statement was right, but
I'm afraid I don't know for sure. I hope someone qualified can clear
this up.


I think we might have to use 1-consistency instead of just consitency.
I think in T_0 the well-foundedness of epsilon-null is equivalent to
the 1-consistency of PA, not just the consistency.


I've actually been having my doubts. Take T_0+1Con(T_0). Presumably you
want the ordinal of that to be delta_0, the first fixed point of the
epsilon function. But I suspect it might be epsilon_1. How would you
prove epsilon_1 is well-founded in T_0+1Con(T_0)?

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Characterization of Gamma_0
    ... Rupert wrote: ... In this theory, we have two sorts of objects, naturals and sets of naturals. ... The sets of naturals are closed under comprehension. ...
    (sci.logic)
  • Re: Characterization of Gamma_0
    ... Daryl McCullough wrote: ... In this theory, we have two sorts of objects, naturals and sets of naturals. ... The sets of naturals are closed under comprehension. ...
    (sci.logic)
  • Re: Characterization of Gamma_0
    ... Rupert says... ... In this theory, we have two sorts of objects, naturals and sets of naturals. ... The sets of naturals are closed under comprehension. ...
    (sci.logic)
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