Finitism and primitive recursive arithmetic

[Timothy Y. Chow]

Joe Shipman wrote:

> In the context of ZFC, which is a fine context for foundational 
> discussions, it seems to me that "finitist result" ought to mean 
> "provable without using the axiom of infinity", which gives us, for 
> arithmetic, the same theorems that PA does.

"Provable without using the axiom of infinity" would certainly seem to be 
a necessary condition to be finitist, but it's not clear that it is 
sufficient.

At issue is the meaningfulness of properties or statements involving an 
arbitrary number of alternations of quantifiers.  These make sense if the 
set of integers is available as a completed totality, or if you are happy 
with interpreting "for all" as an infinite conjunction and "there exists" 
as an infinite disjunction, but they might not make sense if you are a 
finitist.

Tim