How does boiling water affect its hardness? Does boiling water reduce its hardness, or on the contrary, increase it?

It seems there are ambivalent phenomena at play here:


boiling will evaporate pure water, thus increasing the concentration of minerals including calcium, thereby increasing hardness
boiling will remove $\ce{CO2}$, making apparently $\ce{CaCO3}$ precipitate, thereby reducing hardness


This is based on rather uncertain sources, as I didn't find any better, and maybe I missed other aspects.

I would appreciate any clarification, but I understand that the answer might depend on the context and type of water considered.
You may need to think outside chemistry a little bit and pay a little bit attention to your everyday life. This is what happened when you boil water in a container:


Most hardness in water, calcium/magnesium, will precipitate onto the hot surface and is removed from water. Now if you can get more something by removing it, is it against the very fundamentals of science?