Question

Show that there are infinite sets $$X_{0}, X_{1}, X_{2}, \ldots, X_{k}, X_{k+1}, \ldots$$ where for each \(k \in \mathbb{N},\left|X_{k}\right|<\left|X_{k+1}\right|\).

Short answer

Use power sets to construct an infinite sequence of sets with increasing cardinalities.

Step-by-step solution

Understand the Concept of Infinite Sets

Infinite sets are collections of elements that do not have a finite number of elements. A set is called infinite if there is no last element or it can be put into one-to-one correspondence with a proper subset of itself.

Understand Cardinality of Sets

Cardinality refers to the size of a set. If the cardinality of set \(X\) is less than the cardinality of set \(Y\) (denoted \( |X| < |Y| \)), it means there is no one-to-one correspondence from \(X\) to \(Y\), but there exists such a mapping from \(Y\) to \(X\).

Consider the Set of Natural Numbers

The set of natural numbers \( \mathbb{N} \) is an infinite set. The cardinality of this set is denoted as \( \aleph_0 \). This is the starting point for constructing the desired sets \(X_0, X_1, \ldots\).

Construct the Sets \(X_k\)

Define \( X_0 \) to be the set of natural numbers \( \mathbb{N} \). If \( X_k \) is defined, define \( X_{k+1} \) to be the power set of \( X_k \), denoted by \( \mathcal{P}(X_k) \).

Establish the Cardinality of Each Set

The power set \( \mathcal{P}(X_k) \) of any set \( X_k \) has a strictly greater cardinality than \( X_k \). If \( |X_k| = \aleph_0 \), then \( |\mathcal{P}(X_k)| = 2^{\aleph_0} \), and generally, \( |\mathcal{P}(X_k)| > |X_k| \).

Conclusion

By repeatedly taking power sets, \( |X_{k+1}| = |\mathcal{P}(X_k)| > |X_k| \) is maintained for each \( k \). Therefore, an infinite sequence of sets \( X_0, X_1, X_2, \ldots \) exists where \( |X_k| < |X_{k+1}| \) for all \( k \).

Key concepts

Cardinality

The concept of cardinality is central to understanding the size of a set, a crucial aspect when discussing sets that are either finite or infinite. Cardinality describes how many elements are in a set. For finite sets, it's simply the number of elements. However, things get trickier with infinite sets.
For infinite sets, cardinality helps us determine whether one infinite set is larger than another. When we say the cardinality of set \(X\) is less than the cardinality of set \(Y\) (denoted as \(|X| < |Y|\)), it implies that there is no way to make a perfect one-to-one matching between every member of \(X\) and every member of \(Y\) without some elements of \(Y\) left unmatched.
This concept becomes crucial when dealing with infinite sequences of sets, where we want to construct sets such that each successive set in the sequence is "larger" in cardinality.

Power Set

The idea of a power set is an exciting concept in mathematics, particularly when dealing with set theory and cardinality. The power set of any set \(X\), denoted as \(\mathcal{P}(X)\), is the collection of all possible subsets of \(X\). This includes the empty set and \(X\) itself.
One of the most interesting properties of power sets is their cardinality. If a set \(X\) has \(n\) elements, the power set of \(X\) has \(2^n\) elements. This exponential growth applies to infinite sets as well. For example, if you start with the set of natural numbers \(\mathbb{N}\), which has cardinality \(\aleph_0\), the power set of \(\mathbb{N}\), \(\mathcal{P}(\mathbb{N})\), has a cardinality of \(2^{\aleph_0}\).
This remarkable increase shows that each time you create a power set, you end up with a set that is "larger" in terms of cardinality, even in the realm of infinite sets.

Natural Numbers

The natural numbers, denoted \(\mathbb{N}\), are an essential building block in mathematics. This set includes all positive whole numbers, starting from 1, 2, 3, and so on.
Interestingly, the natural numbers form an infinite set that serves as a foundation for many mathematical concepts, especially in set theory and number theory. When discussing cardinality, \(\mathbb{N}\) is often where discussions of infinity begin. The cardinality of \(\mathbb{N}\) is expressed as \(\aleph_0\), symbolizing the smallest type of infinity.
In exercises involving sequence of sets, the natural numbers help illustrate how we can construct a series of sets with increasing cardinalities by using operations such as forming power sets. This shows that from a simple countable infinity, more complex and larger-sized infinites can be derived.