Consider the set \(\mathcal P(\mathbb N)\) of the subsets of the natural integers \(\mathbb N\). \(\mathcal P(\mathbb N)\) is endowed with the strict order \(\subset\). Let’s have a look to the chains of \((\mathcal P(\mathbb N),\subset)\), i.e. to the totally ordered subsets \(S \subset \mathcal P(\mathbb N)\).

### Some finite chains

It is easy to produce some finite chains like \(\{\{1\}, \{1,2\},\{1,2,3\}\}\) or one with a length of size \(n\) where \(n\) is any natural number like \[

\{\{1\}, \{1,2\}, \dots, \{1,2, \dots, n\}\}\] or \[

\{\{1\}, \{1,2^2\}, \dots, \{1,2^2, \dots, n^2\}\}\]

### Some infinite countable chains

It’s not much complicated to produce some countable infinite chains like \[

\{\{1 \},\{1,2 \},\{1,2,3\},…,\mathbb{N}\}\] or \[

\{\{5 \},\{5,6 \},\{5,6,7\},…,\mathbb N \setminus \{1,2,3,4\} \}\]

Let’s go further and define a one-to-one map from the real interval \([0,1)\) into the set of countable chains of \((\mathcal P(\mathbb N),\subset)\). For \(x \in [0,1)\) let \(\displaystyle x = \sum_{i=1}^\infty x_i 2^{-i}\) be its binary representation. For \(n \in \mathbb N\) we define \(S_n(x) = \{k \in \mathbb N \ ; \ k \le n \text{ and } x_k = 1\}\). It is easy to verify that \(\left(S_n(x))_{n \in \mathbb N}\right)\) is a countable chain of \((\mathcal P(\mathbb N),\subset)\) and that \(\left(S_n(x))\right) \neq \left(S_n(x^\prime))\right)\) for \(x \neq x^\prime\).

**What about defining an uncountable chain?**

### A chain having the cardinality of the continuum

Let’s first recall that the set of the rational numbers \(\mathbb Q\) is countable. Hence one can find a bijection \(\varphi : \mathbb Q \to \mathbb N\) (see following article).

Now consider the set of real numbers \(\mathbb R\) which has the cardinality of the continuum \(\mathfrak c = 2^{\aleph_0}\). For any real \(x \in \mathbb R\), we can define the set of rationals \(\psi(x) = \{q \in \mathbb Q \ ; \ q \lt x\}\). Then \(\varphi [\psi(x)]\) is a subset of \(\mathbb N\).

\(\left(\varphi [\psi(x)]\right)_{x \in \mathbb R}\) is a chain as for \(x \lt y\) we have \(\psi(x) \subset \psi(y)\). It has the cardinality of the continuum as \(\psi\) is one-to-one.