Is there any value $n$ such that the number of sets of size $n$ is neither $0$ nor $1$ nor $\infty$?

Question

Please let me know if you think that my question is not well-defined.

Is there any value $n$ such that the number of sets of size $n$ is neither $\color\red0$ nor $\color\red1$ nor $\color\red\infty$?

A few examples:

• For size $n=\frac12$, there are $\color\red0$ sets of this size
• For size $n=0$, there is exactly $\color\red1$ set of this size: $\emptyset$
• For size $n=\infty$, there are $\color\red\infty$ly many sets of this size:
  • $\mathbb{N}$
  • $\mathbb{R}$
  • $\mathbb{R}\backslash\mathbb{N}$
  • $\mathbb{R}\times\mathbb{N}$
  • $\{2n\mid n\in\mathbb{N}\}$
  • $\{2^n\mid n\in\mathbb{N}\}$
  • $\dots$

Can it be proved that no value of $n$ yields a different number of sets of size $n$?

Answer 1

Your question is not entirely defined.

For starters, "size" is only relevant when you consider cardinals. $\frac12$ is no more a cardinal, than a is a letter of the Hebrew alphabet. You could argue that there are no words in Hebrew which start with the letter $\pi$, as well. And you'd be technically right, but the question comes off as odd. Simliarly, $\infty$ is not a cardinal either, and what you might mean is "infinite".

To your question, if I understand it correctly: Yes, there is exactly one empty set; and for every other cardinal there is a proper class of different sets: if $A$ is a set of certain size, then $A\times\{0\}\neq A$ and they are equipotent. So for any non-empty cardinal there are so many sets of that cardinality, they don't even form a collection we can measure its size (technically, anyway).