How can we assume that basic operations on numbers take constant time?

Normally in algorithms we do not care about comparison, addition, or subtraction of numbers -- we assume they run in time $O(1)$. For example, we assume this when we say that comparison-based sorting is $O(n\log n)$, but when numbers are too big to fit into registers, we normally represent them as arrays so basic operations require extra calculations per element. Is there a proof showing that comparison of two numbers (or other primitive arithmetic functions) can be done in $O(1)$? If not why are we saying that comparison based sorting is $O(n\log n)$? I encountered this problem when I answered a SO question and I realized that my algorithm is not $O(n)$ because sooner or later I should deal with big-int, also it wasn't pseudo polynomial time algorithm, it was $P$.