I have noticed that whenever the scope of a problem is pushed to infinity, problems in NP have an uncountably infinite decision space whereas problems in P seem to have a countably infinite decision space. I'm curious if this is always true, and if there is any value to this observation.
Consider a case of 3-sat (which is NP complete) in which there are countably infinite variables to assign values to. There are uncountably infinitely many assignments to variables because an assignment corresponds to an infinite length binary string. The decision space in this case in uncountably infinite. Knapsack is clearly the same - consider a case with an infinite number of items. Which items are chosen corresponds to an infinite length binary string.
Consider the problem of finding the greatest common factor of two numbers x and y (a problem known to be solvable in polynomial time). As x and y both are pushed to infinity, it is not hard to see that the set of possible solutions as we approach this limit becomes countably infinite.