To solve the problem, we need to find the number of elements in the set \( P(P(P(P(A)))) \) where \( A \) is the empty set (also known as the void set).
### Step-by-step Solution:
1. **Identify the empty set**:
Let \( A = \emptyset \). The empty set has no elements.
2. **Find the power set of \( A \)**:
The power set \( P(A) \) is the set of all subsets of \( A \). Since \( A \) is the empty set, the only subset of \( A \) is \( A \) itself:
\[
P(A) = \{\emptyset\}
\]
The number of elements in \( P(A) \) is \( 1 \) (since it contains one element, the empty set).
3. **Find the power set of \( P(A) \)**:
Now, we find \( P(P(A)) \):
\[
P(P(A)) = P(\{\emptyset\})
\]
The subsets of \( P(A) \) are:
- The empty set \( \emptyset \)
- The set \( \{\emptyset\} \)
Therefore, \( P(P(A)) = \{\emptyset, \{\emptyset\}\} \) and the number of elements in \( P(P(A)) \) is \( 2 \).
4. **Find the power set of \( P(P(A)) \)**:
Next, we find \( P(P(P(A))) \):
\[
P(P(P(A))) = P(\{\emptyset, \{\emptyset\}\})
\]
The subsets of \( P(P(A)) \) are:
- The empty set \( \emptyset \)
- The set \( \{\emptyset\} \)
- The set \( \{\{\emptyset\}\} \)
- The set \( \{\emptyset, \{\emptyset\}\} \)
Therefore, \( P(P(P(A))) \) contains \( 4 \) elements.
5. **Find the power set of \( P(P(P(A))) \)**:
Finally, we find \( P(P(P(P(A)))) \):
\[
P(P(P(P(A)))) = P(\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\})
\]
The subsets of \( P(P(P(A))) \) are:
- The empty set \( \emptyset \)
- The set \( \{\emptyset\} \)
- The set \( \{\{\emptyset\}\} \)
- The set \( \{\{\emptyset\}\} \)
- The set \( \{\emptyset, \{\emptyset\}\} \)
- The set \( \{\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\} \)
- The set \( \{\emptyset, \{\{\emptyset\}\}\} \)
- The set \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\} \)
Therefore, the number of elements in \( P(P(P(P(A)))) \) is \( 2^4 = 16 \).
### Final Answer:
The number of elements in \( P(P(P(P(A)))) \) is \( 16 \).
