Can a machine decide the statement 0>0?

Question

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...

Answer 1

(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.)

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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 $0\not<0$.

Answer 2

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.