Question

Show for the natural numbers \(\mathbb{N}\) that $$|P(N)|>|\mathbb{N}|$$

Short answer

The cardinality of the power set of natural numbers is greater than the cardinality of natural numbers.

Step-by-step solution

Understanding the Power Set

The power set of a set is the set of all its subsets. For a set of size \(n\), its power set contains \(2^n\) elements. If \(N\) is a natural number, \(|P(N)|\) is \(2^{|N|}\), where \(|N|\) is the cardinality of the set of natural numbers.

Cardinality of Natural Numbers

The set of natural numbers \(\mathbb{N}\) is infinite, and it has a cardinality known as \(\aleph_0\) (aleph-null), which is the smallest infinity.

Power Set of Natural Numbers

The power set of \(\mathbb{N}\), denoted as \(P(\mathbb{N})\), includes all possible subsets of \(\mathbb{N}\). Its cardinality is \(2^{\aleph_0}\).

Cantor's Theorem

Cantor's Theorem states that for any set \(S\), \(|P(S)| > |S|\). Applying this to natural numbers, \(|P(\mathbb{N})| > |\mathbb{N}|\). This is because no function can map \(\mathbb{N}\) onto \(P(\mathbb{N})\) one-to-one without missing some elements in \(P(\mathbb{N})\).

Conclusion

Thus, by Cantor's Theorem, we conclude that the cardinality of the power set of the natural numbers \(\mathbb{N}\) is strictly greater than the cardinality of \(\mathbb{N}\) itself.

Key concepts

Power Set

A power set is a fascinating concept in set theory. It is essentially the collection of all possible subsets that can be formed from a given set. Let's consider a smaller set to understand this better: if you have a set containing the numbers {1, 2}, its power set will include

• The empty set {}
• The set that contains just 1: {1}
• The set that contains just 2: {2}
• And the set that contains both elements: {1,2}

This tells us that for a set with 2 elements, the power set contains 4 subsets.
The number of elements in a power set is always expressed as a power of two. For a set with size n, its power set will have exactly \(2^n\) subsets, including the empty set and the set itself.
When you apply this idea to the natural numbers \(\mathbb{N}\), even though \(|\mathbb{N}|\) is infinite, Cantor's Theorem helps us understand that this concept holds true at the infinite level as well.

Cardinality

Cardinality is a term used to describe the size or "number of elements" in a set. When dealing with finite sets, it's as simple as counting how many distinct items are in the set.
However, things get a bit more mind-boggling when we talk about infinite sets, like the set of natural numbers \(\mathbb{N}\).
The cardinality of \(\mathbb{N}\) is a concept called \(\aleph_0\) (read as "aleph-null"), signifying the smallest type of infinity. This infinity comes from the fact that you can never finish counting natural numbers, as there's always a next number.
When we consider the power set of the natural numbers, Cantor's Theorem introduces us to a larger infinity, correlating to \(2^{\aleph_0}\).
In simplified terms, any power set of an infinite set has a greater cardinality than the set itself. This demonstrates an analytical beauty about infinity that defies simple counting.

Natural Numbers

Natural numbers are your basic counting numbers: 1, 2, 3, and so on. They're the numbers you first learn about in elementary school, and they are infinitely endless, often denoted by \(\mathbb{N}\).
Natural numbers are integral to mathematics, aiding in simple arithmetic to advanced theories. Each additional number can automatically be considered part of \(\mathbb{N}\), because there's no largest natural number, inadvertently making them infinite.
Natural numbers have a cardinality of \(\aleph_0\), symbolic of their smallest infinity. Their simple nature is deceptive because when explored through set theory, they reveal profound conclusions such as Cantor's Theorem.
This theorem tells us that the set of all subsets of natural numbers (its power set) possesses a greater multitude than the natural numbers themselves. Hence, even in their simplicity, natural numbers lead us to explore the mysterious realms of mathematical infinity.