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To solve the problem, we need to find \( \mathcal{P}(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))))) \), where \( \mathcal{P}(S) \) denotes the power set of the set \( S \). Let's go through the steps carefully:

### Step 1: Start with \( \emptyset \) (the empty set)
The empty set is denoted as:
\[
\emptyset = \{\}
\]

### Step 2: Find \( \mathcal{P}(\emptyset) \)
The power set of a set is the set of all its subsets. The only subset of \( \emptyset \) is \( \emptyset \) itself. Thus, the power set of \( \emptyset \) is:
\[
\mathcal{P}(\emptyset) = \{\emptyset\}
\]

### Step 3: Find \( \mathcal{P}(\mathcal{P}(\emptyset)) \)
Now, we need to find the power set of \( \mathcal{P}(\emptyset) = \{\emptyset\} \). The subsets of \( \{\emptyset\} \) are:
- \( \emptyset \) (the empty subset)
- \( \{\emptyset\} \) (the set itself)

So, the power set is:
\[
\mathcal{P}(\mathcal{P}(\emptyset)) = \{\emptyset, \{\emptyset\}\}
\]

### Step 4: Find \( \mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) \)
Now, we find the power set of \( \mathcal{P}(\mathcal{P}(\emptyset)) = \{\emptyset, \{\emptyset\}\} \). The subsets of this set are:
- \( \emptyset \) (the empty subset)
- \( \{\emptyset\} \)
- \( \{\{\emptyset\}\} \)
- \( \{\emptyset, \{\emptyset\}\} \)

So, the power set is:
\[
\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}
\]

### Step 5: Find \( \mathcal{P}(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))) \)
Finally, we find the power set of \( \mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\} \). The subsets of this set are:
- \( \emptyset \)
- \( \{\emptyset\} \)
- \( \{\{\emptyset\}\} \)
- \( \{\emptyset, \{\emptyset\}\} \)
- \( \{\{\{\emptyset\}\}\} \)
- \( \{\emptyset, \{\{\emptyset\}\}\} \)
- \( \{\{\emptyset\}, \{\{\emptyset\}\}\} \)
- \( \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\} \)

Thus, the final power set is:
\[
\mathcal{P}(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset)))) = \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}, \{\{\{\emptyset\}\}\}, \{\emptyset, \{\{\emptyset\}\}\}, \{\{\emptyset\}, \{\{\emptyset\}\}\}, \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}\}
\]

Would you like more details or clarification on any of these steps?

Here are 5 related questions to expand on this:
1. How is the power set defined for any set?
2. What are the elements of the power set of a set with 3 elements?
3. How many elements does the power set of a set with \( n \) elements have?
4. What is the relationship between the cardinality of a set and the cardinality of its power set?
5. Can you find the power set of a non-empty set, say \( \{1, 2\} \)?

**Tip**: The cardinality of the power set \( \mathcal{P}(S) \) of a set \( S \) with \( n \) elements is always \( 2^n \), since every element of \( S \) can either be included or not included in a subset.

Find the set P(P(P(P(∅)))), where P(S) denotes the power set of the set S. Show all steps. That is, start with ∅, then find P(∅), then find P(P(∅)), etc.

Explore the detailed process of finding the set P(P(P(P(∅)))) using power sets. Learn about power sets, subsets, and the number of elements in the power set. This problem is suitable for advanced high school or university students studying set theory.

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