5 ms·
Depends what you mean by 'indexed'. If you accept the axiom of choice then you can index the real numbers with some uncountable ordinal.
by contravariant 4y ago
Depends what you mean by 'indexed'. If you accept the axiom of choice then you can index the real numbers with some uncountable ordinal.
- roarcher 4y agoI meant it in the sense that I assume GP meant it, i.e., the index of an element in an ordered set is an integer corresponding to its position in that set, starting at 0 or 1 for the first element (depending on your indexing scheme of choice).
- Dylan16807 4y agoDoes the axiom of choice let you repeatedly pick the same real numbers in the same order out of a set? Or do you need uncountably infinite storage to keep your list of indexes?
- contravariant 4y agoThat's an interesting question, but kind of hard to answer. Programming is inherently constructive so approaching a non-constructive proof as a programming problem tends to be problematic. I don't think you run into any problems if you let the axiom of choice pick the same numbers every time. It would be a bit problematic if you were to add that as an extra axiom (suddenly every proof let x,y in X then ... therefore x=y would become trivial).