I know ultrafinitists want to require not only that mathematical objects be constructible, but be constructible given finite resources (such as time).
So I wonder about something like the famous "Graham's number"
It's used as (non-binding) upper constraint so—are ultrafinitists OK with it? If I said some combinatorial number is less than $\infty$ they would maybe agree with this also? Unless $\infty$ "doesn't make sense" because it would "take too long" to "make" the upper bound one is talking about.
A related question which is probably better attached here: What about the operator $\forall$? $\forall x$ can be translated into English in two ways:
- "for all $x$", which paints a picture of "going around and getting each and every $x$",
- or "for any $x$", which paints the picture of a "gate" and if something comes up to your gate, and it is an $x$, then let it through—no "gathering" or "rounding up" is necessary. Just "leave the instructions with the guard" and the problem is solved passively.
(The second interpretation also seems more consistent with requiring $\exists x$ in some "resource-constrained" sense. If $\not \exists x$, the gate guard would stand there his whole life and never let anyone through, whereas in the "for all" sense of $\forall$, we would wander around the universe forever, check every object, and never find an $x$.)