[FOM] Re: Sharp mathematical distinction between potential and actual infinity?

[Timothy Y. Chow]

On Sun, 28 Sep 2003, Dean Buckner wrote:
> The distinction between potential & actual is traditional and
> distinguishes that which could or can exist, from that which is, i.e.
> that which "actually" exists.

O.K., I like this.  So perhaps there is a way of formulating mathematics
in terms of modal logic, rather than classical first-order logic?  What
are some examples of what this might look like?

1. "For every prime, there actually/potentially exists a larger prime."
   Which of these does Euclid's proof establish?

2. "There actually/potentially exists a set of all integers."  Does it
   make sense to use "potentially" here?  Can a completed infinity
   "potentially" exist?

3. "A rule tells you what to do in all actually/potentially existing
   scenarios."  "Potentially" feels right here, which I think is a
   good sign.

I don't want to give more examples because I'm groping in the dark here.
Can someone else help out?

Tim