Well, I was hand-waving a bit when I said, “we can regard names as …”, since the question of what a name is exactly is also a philosophical question. However, I think for all practical purposes we can assume names are strings of characters from a countable alphabet (I would guess finite too, but I am less certain about that). I have never seen anyone use an uncountable alphabet. I would be amazed to find one in use.
I sometimes use F sharp when I need a letter in between F and G.
I think I can almost visualize i apples. It involves an apple tree with square roots, which isn’t there because the apple orchard was cut down to build tract housing.
I think I better go back to bed, don’t you?
For a real world calculation with complex numbers, try this:
On a table, fill a square with apples. Make the square 4i by 4i, and put an apple in each “cell”.
Now pile two dozen more apples next to the square.
Now gather them all together, and you have 8 apples.
It’s no good saying “Show me one of the 4i apples”. Suppose you made a filled rectangle of 5 by 6 apples, and got 30 apples as a result - can you show me one of the 5 apples? Not that one, that’s one of the 30. The 5 apples aren’t real, either, by this test, so it’s not a good test.
I don’t think any of these number systems are more real than any others, they just make different kinds of sense and solve different kinds of problems.
Oh, I doubt there’s any in use, but we’re talking about countable vs. uncountable infinities here, so let’s not get bogged down in practicalities. Formal languages are always taught with a finite alphabet, so it’s interesting to see which results carry over if you violate that assumption.