[FOM] How much of math is logic?

[praatika@mappi.helsinki.fi]

Here is a setting I like: 

Let us take as our logic two-sorted FO logic, with separate variables for 
numbers and classes. We have introduction and elimination rules for both 
sorts of quantifiers. If one then adds only the very elementary defining 
axioms of successor, addition and multiplication (no induction or 
anything), one gets a system equivalent to ACA_0. 

And we know, thanks to the work of Friedman, Simpson and others, that 
practically all ordinary mathematics can be developed in this system.   

Presumably the same would hold for the axioms of extensionality, null set, 
and adjunction, with the above background logic (right?).

This is as close to logicism I can get - though I still think that even 
the axioms of successor are not really logically true, for they are false 
in all finite models.

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