The Universe (matthias) (Timothy Y. Chow)

[Øystein Linnebo]

Yes, the strict potentialist regards all statements of first-order
arithmetic as meaningful. And yes, on the analysis Shapiro and I provide,
the strict potentialist can prove every theorem of Heyting arithmetic. For
details, see Theorem 3 and footnote 31.

As for second-order arithmetic, I claimed that the potentialist (whether
liberal or strict) is not "entitled" to full second-order arithmetic --
even if he accepts the combinatorial conception of classes (as completely
arbitrary collections of antecedently available objects). This isn't
because some second-order statements are deemed meaningless. Rather, it is
because all the classes will be finite. After all, at any stage, there are
only finitely many natural numbers available to serve as members of a
class, understood combinatorially.

Oystein


> Date: Wed, 15 Jul 2020 12:30:30 -0400 (EDT)
> From: "Timothy Y. Chow" <tchow at math.princeton.edu>
> To: fom at cs.nyu.edu
> Subject: Re: The Universe (matthias)
> Message-ID: <alpine.LRH.2.21.2007151218190.704 at math.princeton.edu>
> Content-Type: text/plain; charset=US-ASCII; format=flowed
>
> 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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