Is it true that Any infinite set contains a subset of the power continuum

Yes, it is true that any infinite set contains a subset whose cardinality is the power of the continuum, \(\mathfrak{c}\). This fact can be understood through some fundamental results from set theory and the study of cardinalities.

### Explanation:
1. **Power of the Continuum**: The power of the continuum, denoted by \(\mathfrak{c}\), is the cardinality of the set of real numbers \(\mathbb{R}\) or, equivalently, the set of all subsets of the natural numbers \(\mathcal{P}(\mathbb{N})\). It is equal to \(2^{\aleph_0}\), where \(\aleph_0\) is the cardinality of the natural numbers.

2. **Cantor's Theorem**: For any set \(S\), the power set of \(S\), \(\mathcal{P}(S)\), has strictly greater cardinality than \(S\). Therefore, \(2^{|S|} > |S|\) for any set \(S\). For an infinite set \(S\), if \(|S| = \aleph_0\), then \(|\mathcal{P}(S)| = \mathfrak{c}\).

3. **Infinite Sets and Subsets**: Any infinite set \(A\) has a countably infinite subset \(B\) with \(|B| = \aleph_0\). The power set \(\mathcal{P}(B)\) will then have the cardinality \(2^{\aleph_0} = \mathfrak{c}\).

4. **Conclusion**: This implies that any infinite set contains a subset whose cardinality is the power of the continuum \(\mathfrak{c}\).

This result is a consequence of the general properties of cardinal arithmetic and the structure of infinite sets.

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### Related Questions:
1. What is the significance of Cantor's theorem in set theory?
2. How is the cardinality of infinite sets compared and determined?
3. Can you explain the difference between countable and uncountable infinities?
4. What are some examples of sets with cardinality \(\mathfrak{c}\)?
5. How does the Continuum Hypothesis relate to the power of the continuum?

**Tip:** Remember that in set theory, the size (cardinality) of sets is more about their comparative "sizes" rather than numerical values. Understanding the distinctions between different types of infinities is crucial for grasping advanced concepts.

Is it true that any infinite set contains a subset of the power continuum?

Explore the relationship between infinite sets and the power of the continuum. Learn how Cantor's theorem and cardinal arithmetic show that any infinite set contains a subset with the cardinality of the continuum (\mathfrak{c}). Ideal for students studying set theory and infinity.

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