Following simple statement came to my mind when I was thinking about infinite sets.
Statement: There is no set $X\subset\mathbb{N}$ that has cardinality strictly between any finite set $S\subset\mathbb{N}$ and the set $\mathbb{N}$.
I have three question that I want to ask:
If the statement is true, then does it imply that there is no set $X\subset\mathbb{N}$ so that $|\mathbb{N}|^{|S|}<|\mathbb{N}|^{|X|}<|\mathbb{N}|^{|\mathbb{N}|} \space$?
If the statement is not true, then does it imply that there is a set $X\subset\mathbb{N}$ so that $|\mathbb{N}|^{|S|}<|\mathbb{N}|^{|X|}<|\mathbb{N}|^{|\mathbb{N}|}\space$?
If the statement is not defined, then does it imply that there is a set $X\subset\mathbb{N}$ that has no definite cardinality?
If the Statement is meant to say that there is no infinite subset of $\Bbb N$ which has strictly smaller cardinality than $\Bbb N$, then the statement is indeed true.
The three questions are really "all the options" and exactly one can have a positive answer. And indeed the first one is true. The reason is that if $X$ is infinite, then $|X|=|\Bbb N|$, and therefore $|\Bbb N|^{|X|}=|\Bbb N|^{|\Bbb N|}$. If $X$ is not infinite, then $|\Bbb N|^{|S|}=|\Bbb N|^{|X|}=|\Bbb N|$ (or $X=\varnothing$, and then $|\Bbb N|^{|X|}=1$).
Let me also add that cardinality is some sort of measurement of size of a set which is defined by an equivalence relation. This means that every set has a definite cardinality. It just might happen that we have a definite set, but we cannot determine its cardinal in terms of other sets. But every set has a definite cardinality.