Can a machine decide the statement 0>0? My question may seem silly, but I've been having some difficulties with it... I saw that order relation in the set of computable numbers applies only if $a \neq b$, that is, we can say that $a > b$ or $b > a$ if $a \neq b$. But then I was thinking, I know $0$ is computable - because it is a natural number - then what output a computer would give if it was to decide whether $0>0$ or not...
 A: (It's not actually totally clear to me what you're asking, but I think that you're running into one of the standard confusions around equality of computable real numbers - namely, how do we reconcile the computational difficulty of comparing numbers with the obviousness of comparisons like $0\not<0$? If this doesn't address your question, let me know and I'll delete it.)

What's the input?
When we say that equality of real numbers is undecidable, or something similar, we're referring to reals as infinite bit strings. In this context, even something as $0$ is presented as an infinite object. Crucially, we are not given any additional "contextual information" about the bit string in question; so in order to tell apart the bitstrings corresponding to $0$ and to $1\over 10^{10^{10}}$ we basically have to spend $\log_2(10^{10^{10}})$-many computational steps.
By contrast, when you ask "Is $0<0$?" you've given the number $0$ as an input in a much nicer way: the input now tells me that I'm dealing with an (exact) integer, and so I can decide - without difficulty - that in fact $0\not<0$.
A: In Python, print(0>0) will print False. Just about any programming language can do the same. The trichotomy law states exactly one of $a>b, \, a=b, \, a<b$ is true.
