minimum cardinal for a set of sets

Question

Let A be a set of sets. Has A an element with minimum cardinality? For example, if we consider A a finite set, the A has an element (which is not necessarily unique) with minimum cardinal. With many thanks.

Answer 1

Assuming $A$ is not empty, a general answer will depend on whether you accept the Axiom of Choice (AoC), in the form of the Well-Ordering Principle.

If you do not accept AoC, then that implies that you are open to the possibility of non-comparable cardinalities. One example is an infinite, Dedekind-finite set (Dedekind-finite means it is not bijective with any proper subset of itself), which will be incomparable to $|\Bbb N|$, if it exists.

If you do accept the AoC, then any set has the cardinality of some ordinal number, and the set of all ordinals with the same cardinality as some element of $A$ will have a least element. You may not be able to tell which one, though. One common example is $\Bbb R$ and any uncountable cardinal (as long as it's reasonably small, but not the smallest one). While we know $\Bbb R$ has the cardinality of some ordinal (as a consequence of AoC), it's impossible to tell which.

If you do not accept the AoC, but weaken your question to ask about whether there is an element of $A$ which is not strictly larger than any other element of $A$, rather than asking about whether there is an element of $A$ which is smaller than any other element of $A$, then I don't know what the answer is.