> I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it,

Yes, that is a good way to put it.

Unconstructible reals are a major and interesting "what if". What if there were numbers that had no finite relation to other numbers?

It is a great idea, from a mathematical boundary pushing way. So abstract we can never do anything not abstract with it!

> even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?

But unconstructible structures can never be reformulated as constructible by definition. That would mean they were constructible.

We can never define a specific unconstructible real.

But anyone who manages to create an interesting systems theory that uses them, with dynamics that constructible math can't match, would have created a major work of mathematical art!