I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions.

Integer number counting is essentially a successor function; take N, add 1, output N+1. One input, one output.

Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. So we get .10, .11, .12, etc. This function gives more outputs than inputs, thus an increase in cardinality. Forcing just seems like changing the function to increase the cardinality.

In my mind, all infinities are equal, it's how you generate them that matters.