FOM: Infinity
Richard G. Heck, Jr.
heck at fas.harvard.edu
Fri Sep 28 12:57:13 EDT 2001
Just as a point of information (for me). As I learned GB, it's basically
ZF, with the background logic extended to predicative second-order logic;
so we have comprehension just for first-order formulae (with second-order
parameters). ACA_0, conceived one way, stands to PA in this relation; hence
Ferreira's remark. (Hi, Fernando!) So I take it that the sensible finite
axiomatization of GB to which Shipman alludes below isn't the one I know,
since that isn't any more finite than the first-blush axiomatization of
ACA_0. What is it, then? And how is it that it gets to be so much nicer?
Richard
At 09:40 AM 9/28/2001, you wrote:
>Ferreira:
><<ACA_0 is a well-known conservative extension of PA, and it is finitely
>axiomatizable. It has two sorts of variables, but it can be reformulated
>with only one sort of variables (at the cost of introducing new predicates).>>
>Shipman:
>The problem is that this finite axiomatization is extremely complicated,
>at least as it is done in Simpson's book "Subsystems of Second-Order
>Arithmetic". [snip]
>
>The system of Godel-Bernays conservatively extends ZFC and the axioms are
>easy to write down and intuitively understandable, though not at all
>elegant. [snip]
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