How many Subsets does a Null set has? 2 or 1

Solution 1:

(1) The null set is a subset of every set; so the null set certainly has at least one subset, namely itself.

(2) Now suppose that S is a second subset of the null set. in that case, we have :

$(a)\Large S\subseteq \emptyset$

and

$(b)\Large S\neq \emptyset $

(3) The extensionnality axiom tells us that two sets must differ by at least one element in order to be different. So $(b)$ implies that : either $\emptyset$ has an element that does not belong to $S$ , or $S$ has an element that does not belong to $\emptyset$. But the first option is impossible ( by the definition of the empty set), so only the second remains.

(4) The second option says : there is at least some object $x$ such that $\Large x\in S$ and $\Large x\notin \emptyset$, meaning ( by &-elimination) that there is at least some object $x$ such that $\Large x\in S$. But , by $(2.a)$ above , $\Large S\subseteq \emptyset$, which means that all elements of $S$ are also elements of $\emptyset$. This holds in particular for object $x$. So, the second option leads to : $\Large x\in \emptyset$ , another impossibility.

(5) Conclusion : the empty set cannot have any subset in addition to itself; the number of its subsets is $1$.