As cardinal numbers involve the notion of ever increasing multitudes of things, is there a mathematically useful concept of ever decreasing multitudes?
We already have rationals tending to zero, and there's seemingly nothing meaningful "beyond zero", so intuitively it seems absurd to try to "invert cardinals".
I'm asking this because I'm worried about that asymmetry, whereby qualitative differences between successive cardinals are generated only through increments in the size of infinite sets, and not through decrease. After all, zero, i.e. "nothingness" is no less elusive than "infinity".
The various comments pointing out that there are no negative or infinitesimal cardinal numbers do not really address the OP's actual question: "As cardinal numbers involve the notion of ever increasing multitudes of things, is there a mathematically useful concept of ever decreasing multitudes?"
I think the best answer to this would be John Conway's surreal numbers. The surreal numbers include (isomorphic copies of) the real numbers, the cardinal numbers that the OP was interested in (actually all ordinal numbers are there, not just cardinals), and much more. And you can define natural operations of addition and multiplication in such a way that the field axioms are satisfied. As a result, you can have "ever decreasing multitudes" by taking $\frac1{\lambda}$ for larger and larger infinite $\lambda.$
The collection of surreal numbers is a proper class (it's too large to be a set), but there's no avoiding that with OP's question, since the same thing is true of the class of cardinal numbers. (By the way, this is why I said above that the surreal numbers satisfy the field axioms, rather than saying that the surreal numbers form a field. Strictly speaking, a field has to be a set, not a proper class.)
See the Wikipedia page on surreal numbers for more info, or get John Conway's book On Numbers and Games. There are also Knuth's How two ex-students turned on to pure mathematics and found total happiness and Berlekamp, Conway, and Guy's Winning Ways for Your Mathematical Plays, but I'm not as familiar with those books.