The Universe (matthias)
Timothy Y. Chow
tchow at math.princeton.edu
Wed Jul 15 12:30:30 EDT 2020
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
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