In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> But if there are more than countably many real numbers, then there are > most of them so called moonlightr numbers. We do not know anything > about them. How can they obey trichotomy?
In every sufficiently complex system there are propositions that are true but which cannot be proven true within that system and propositions than are false but cannot be proven false within that system.
So that there is no problem in real mathematics in having real numbers x and y and being unable to determine whether x < y, or x = y, or x > y, even though it is known that one of them must be true and the other two false.
WM mistakenly believes that his mathematic rules form the only mathematics games possible, but we find the rules to his game silly enough not to want to play by them.