[FOM] Finite axiomatisation

[Harvey Friedman]

THe two occurrences quoted there of ACA_0 are typos. They should be  
ACA, which is ACA_0 together with the full induction scheme. ACA is  
not finitely axiomatizable.

Harvey Friedman

On Aug 24, 2008, at 7:03 AM, pax0 at seznam.cz wrote:

>
> I react to the posting from Stephen G Simpson <simpson at math.psu.edu>
>
>>  Yes, this is very much in the literature.  See for instance my book
>>  "Subsystems of Second Order Arithmetic," where the significance of
>>  ACA_0 in reverse mathematics and foundations of mathematics  
>> generally
>>  is discussed.  There it is pointed out that ACA_0 is a finitely
>>  axiomatizable conservative extension of PA, analogously to how  
>> NBGC is
>>  a finitely axiomatizable conservative extension of ZFC.
>
> where he points out that ACA_0 is finitely axiomatizable.
> But I found a paper by Harvey Friedman, where he claims (on the 1.  
> page) the opposite:
> "RCA_0 cannot prove TST <--> ACA_0 since ACA_0 is not finitely  
> axiomatizable."
> The pdf can be found here:
> http://www.math.ohio-state.edu/~friedman/pdf/BabyBRT100301.pdf
> LECTURE NOTES ON BABY BOOLEAN RELATION THEORY
>
> Where is the problem? Jan P.
>