I think that's basically fair. If we were dealing with the reals I'd assume that there was some uncountable number of pathological examples that just happen to

I think that's basically fair. If we were dealing with the reals I'd assume that there was some uncountable number of pathological examples that just happen to exclude all the numbers we really care about (since God ran out of good numbers and had to scrape the bottom of the barrel to fill in the gaps), but the integers seem safe.