FOM: Infinity

Richard G. Heck, Jr. heck at
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?


At 09:40 AM 9/28/2001, you wrote:
><<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).>>
>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]

Richard Heck
Professor of Philosophy
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Harvard University
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