I was wondering whether non-computable numbers are ever of "direct" use ? I understand they are immensely useful indirectly, because we need them to do analysis in the real numbers for instance. However what I mean by "direct" is some use, where one would for instance want to actually give some non-computable number a name ?
I don't really understand non-computable numbers well enough yet to make my question any clearer, but I think there are instances of mathematical objects that we don't construct directly, but we can show their existence, and may want to give names to specific instances (take sigma-algebras for instance right ?)
So in a similar way, even though we cannot directly get at a non-computable number, is it conceivable to show existence of a specific non-computable, and to actually want to use it in a specific context ?