Wait, but rational-delimited open intervals don’t form a locale, because they aren’t closed under infinite union. (For example, the union of all rational-delimited open intervals contained in (0,√2) is (0,√2) itself, which is not rational-delimited.) Of course you could talk about the locale generated by such intervals, but then it contains all open intervals and is uncountable, defeating your main point about going from countable to uncountable. Or am I missing something?
Good point; I’ve changed the wording to make it clear that the rational-delimited open intervals are the basis, not all the locale elements. Luckily, points can be defined as sets of basis elements containing them, since all other properties follow. (Making the locale itself countable requires weakening the definition by making the sets to form unions over countable, e.g. by requiring them to be recursively enumerable)
Another way to make it countable would be to instead go to the category of posets, Then the rational interval basis is a poset with a countable number of elements, and by the Alexandroff construction corresponds to the real line (or at least something very similar). But, this construction gives a full and faithful embedding of the category of posets to the category of spaces (which basically means you get all and only continuous maps from monotonic function).
I guess the ontology version in this case would be the category of prosets. (Personally, I’m not sure that ontology of the universe isn’t a type error).
I see. In that case does the procedure for defining points stay the same, or do you need to use recursively enumerable sets of opens, giving you only countably many reals?
Reals are still defined as sets of (a, b) rational intervals. The locale contains countable unions of these, but all these are determined by which (a, b) intervals contain the real number.
Wait, but rational-delimited open intervals don’t form a locale, because they aren’t closed under infinite union. (For example, the union of all rational-delimited open intervals contained in (0,√2) is (0,√2) itself, which is not rational-delimited.) Of course you could talk about the locale generated by such intervals, but then it contains all open intervals and is uncountable, defeating your main point about going from countable to uncountable. Or am I missing something?
Good point; I’ve changed the wording to make it clear that the rational-delimited open intervals are the basis, not all the locale elements. Luckily, points can be defined as sets of basis elements containing them, since all other properties follow. (Making the locale itself countable requires weakening the definition by making the sets to form unions over countable, e.g. by requiring them to be recursively enumerable)
Another way to make it countable would be to instead go to the category of posets, Then the rational interval basis is a poset with a countable number of elements, and by the Alexandroff construction corresponds to the real line (or at least something very similar). But, this construction gives a full and faithful embedding of the category of posets to the category of spaces (which basically means you get all and only continuous maps from monotonic function).
I guess the ontology version in this case would be the category of prosets. (Personally, I’m not sure that ontology of the universe isn’t a type error).
I see. In that case does the procedure for defining points stay the same, or do you need to use recursively enumerable sets of opens, giving you only countably many reals?
Reals are still defined as sets of (a, b) rational intervals. The locale contains countable unions of these, but all these are determined by which (a, b) intervals contain the real number.