The Universe (matthias)

[Timothy Y. Chow]

Oystein Linnebo wrote:
> Second, consider a potentialist who is *strict*, roughly in the sense that
> she insists not only that *every object* is generated at some stage or
> other, but also that *every truth* is accounted for at some stage or other.
> On the analysis that Shapiro and I provide, a strict potentialist is only
> entitled to intuitionistic logic, not classical -- with obvious
> consequences already for first-order arithmetic.

Thanks for your response.  I take it that "Paul" (the potentialist) 
accepts all statements of first-order arithmetic as meaningful?  In 
particular, Paul would accept "Given a Turing machine M and an input x, M 
either halts or doesn't halt on input x" as a meaningful statement, but 
wouldn't be able to prove it.  Can Paul prove all theorems in Heyting 
arithmetic, or are there further restrictions?

> First, while the actualist is entitled to full second-order arithmetic, 
> the potentialist is not, at least not on a combinatorial conception of 
> the classes of numbers (in which case every such class that is ever 
> generated is finite).

Could you please clarify what you mean when you say that the potentialist 
is not "entitled" to full second-order arithmetic?  Does this mean that 
there are statements in full second-order arithmetic that the potentialist 
does not even accept as *meaningful*?  Are there examples of statements in 
second-order arithmetic that Paul accepts as meaningful but cannot prove 
(other than the first-order examples already mentioned above)?

Tim