TITLE: Is $\emptyset$ a finite union-closed family of finite sets, other than the family containing only the empty set or not
QUESTION [5 upvotes]: I am very weak in logic.
In Wikipedia, there is the following article:

In combinatorics, the union-closed sets conjecture is an elementary problem, posed by Péter Frankl in 1979 and still open. A family of sets is said to be union-closed if the union of any two sets from the family remains in the family. The conjecture states:
For every finite union-closed family of finite sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family.

I wonder if $\emptyset$ is a finite union-closed family of finite sets, ohter than the family containing only the empty set or not.
If so, I wonder if there exists an element that belongs to at least half of the sets in $\emptyset$ or not.

Proposition 1:
$\emptyset$ is a finite union-closed family of finite sets and is not equal to $\{\emptyset\}$.


Proposition 2:
There exists an element that belongs to at least half of the sets in $\emptyset$.

Is Proposition 1 true or false?
Is Proposition 2 true or false?
My attempt is the following:

About Proposition 1:
No object belongs to $\emptyset$, but $\emptyset$ belongs to $\{\emptyset\}$.
So, $\emptyset \neq \{\emptyset\}$.
$\emptyset$ is a finite set. (this fact is famous.)
Since $\emptyset\ni S$ is always false, so "if $\emptyset \ni S$, then $S$ is a finite set" is true.
So, $\emptyset$ is a family of finite sets.
Since $\emptyset\ni S, T$ is always false, so "if $\emptyset\ni S, T$, then $S \cup T\in\emptyset$" is true.
So, $\emptyset$ is a finite union-closed family of finite sets and is not equal to $\{\emptyset\}$.


About Proposition 2:
I guess this proposition is false.
There is no set which belongs to $\emptyset$.
So, there is no element that belongs to a set which belongs to $\emptyset$.
I cannot express Proposition 2 by using logical symbols.
How to express Proposition 2 by using logical symbols?

REPLY [2 votes]: To express proposition 2 you need a function that returns the size of a finite set and you need to use some of the language of arithmetic. If the size function is called $\#$, the statement of the proposition could be formalised as:
$$
\exists x (2 \cdot \#\{A \in S \mid x \in A\} \ge \#S)
$$