I was thinking yesterday about how our intuition about certain infinite sets' sizes can be written mathematically. I came to the conclusion that cardinality isn't really useful (for obvious reasons), but rather measures from measure theory are; and particularly a relaxation of the definition of a measure which instead of `mu(X) in [0,inf]`, you have `mu(X) < mu(Y) iff X strict subset of Y` (ie: mu is (at least partially) ordered), or equivalent. You don't even need to define mu for it to be such a useful ordinal measure

For example: consider all the possible ways to arrange a couch in a room, vs all the possible ways to arrange the couch if you keep the rotation fixed. Obviously the fixed-rotation set is a subset of the free-rotation set, so the measure of the fixed-rotation set is smaller, meaning possibly the entropy is smaller (if so defined), etc