I learned that in thermodynamics, pressure is regarded as an intensive variable, while properties that you can add up such as mass or volume, are considered extensive.
However, to me, Dalton's law seems to contradict this idea. Dalton's law states that the total pressure of a mixture of gases is the sum of the partial pressures of each gas.
Given this, my question is: if the pressure can be added up, why is pressure intensive? What am I understanding wrong? Why is Dalton's law not a valid argument for pressure being an extensive property?
To give a simple example, let's say that I have a "mixture" of two same gases (i.e. oxygen gas), at a total pressure of 1 bar. Now, if I "divide" the gas into two mole fractions and determine their partial pressures, the sum of those partial pressures would also be 1 bar. So clearly, in this example, I'm adding the two pressures to get a larger, total pressure. So, to me, this makes it appear that pressure can be an extensive property, since it can be added up. However, I know that I'm wrong somewhere but don't know where, so can someone explain why my example would not prove that pressure is extensive?