Anyway going back to the -1, -1/2, -1/3, ..., 1/3, 1/2, 1 "string".
Let's extend the definition of a string.
>a sequence of digits such that every digit after the first has a successor and a predecessor
From that, it follows that you can divide a string at any point to get substrings of any length. All substrings are strings themselves.
That also means that strings *have* to have a first and last element. Which means that all of their substrings also do.
Let's assume -1, -1/2, -1/3, ..., 1/3, 1/2, 1 is a string. Because it's a string you can divide it in half into a string -1, -1/2, -1/3, ... and a string ..., 1/3, 1/2, 1. Neither of them has both a first and a last element meaning they cannot be strings. If they aren't strings they cannot be substrings of the original "string". The conclusion is that -1, -1/2, -1/3, ..., 1/3, 1/2, 1 is in fact not a string.
