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Not only arithmetic, but an arithmetic with arbitrary large numbers is required for the Godel's Incompleteness Theorem.
You need at least "all integers", that is an infinite set.
Maybe some incompleteness can be proved for the finite sets as well. But it's not known to be so. Nobody proved that yet.
Finite sets of numbers have obvious problems, well explored in computer science (if you integer has be be expressed in 4 bytes, you can express only a finite set of integers. Or real numbers, for that matter).
Trivially, if your finite set ends at n, what is the sum of (n-1) + (n-2)? A plausible answer is "you can't do that", but that's also an answer to any paradox whatsoever.
How do you feel about imaginary numbers, by the way?