[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

Quoting Richard Heck <rgheck at brown.edu> who wrote, referring to my earlier
posting:

> Surely this isn't quite right: You're going to need some sort of axioms
> governing the class quantifiers to get anything this strong. 

Richard is, of course, absolutely right. 
You need at least the Induction Axiom (IA), though not the Induction Schema
(IS). Working from memory, I was focusing solely on the weak (arithmetical)
comprehension schema, which can - if I am not wrong (?) - derived from the
rule of existential instantiation for class variables, and forgetting the
need of IA.  

I am sorry for having been so sloppy. Thanks to Richard for pointing this out. 

So, if I am not confused again, you can - by adding the elementary defining
axioms of S, + and x, and IA to the natural formalized SO logic* - get
something amounting to ACA_0  (right?); which gives (more or less) all
ordinary mathematics. 

(* this logic is of course not the real full SO logic, but a partial
two-sorted FO formalization of it; something that most of us can easily
accept.) 

All corrections are still warmly welcome. 

Best, Panu



Panu Raatikainen

Academy Research Fellow, The Academy of Finland
Docent in Theoretical Philosophy, University of Helsinki

Department of Philosophy
P.O.Box 9
FIN-00014 University of Helsinki
Finland

e-mail: panu.raatikainen at helsinki.fi