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Section 5.3 Unions

For any set \(X\text{,}\) we also permit ourselves to form the union \(\cup X\text{.}\) This is the set whose elements are the elements of the elements of \(X\text{;}\) that is, \(A\in\cup X\) if and only if there exists an element \(S\in X\) such that \(A\in S\text{.}\) If \(X=\{A,B\}\text{,}\) then we write

\(A\cup B\) for \(\cup X\text{.}\)

We can start building the finite von Neumann ordinals according to the following recipe: first, \(0\coloneq\varnothing\text{.}\) Then, for every von Neumann ordinal \(n\text{,}\) one can create its successor von Neumann ordinal

\begin{equation*} n+1\coloneq n\cup\{n\}\text{.} \end{equation*}

So the first few von Neumann ordinals look like this:

\begin{align*} 0 \amp \coloneq \varnothing;\\ 1 \amp \coloneq \{0\}=\{\varnothing\};\\ 2 \amp \coloneq \{0,1\}=\{\varnothing,\{\varnothing\}\};\\ 3 \amp \coloneq \{0,1,2\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\};\\ 4 \amp \coloneq \{0,1,2,3\}=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\},\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}\};\\ \amp \textit{etc.} \end{align*}

If \(n\) is a finite von Neumann ordinal, then for every \(\alpha,\beta\in n\text{,}\) exactly one of the following is the case:

\begin{equation*} \alpha\in\beta ; \qquad \alpha=\beta ; \textit{or} \alpha \ni \beta\text{.} \end{equation*}

The axioms then let you build the first infinite von Neumann ordinal \(\omega\text{,}\) which is the set of all the finite von Neumann ordinals. Once you have that, you can build the successor to \(\omega\text{:}\)

\begin{equation*} \omega+1\coloneq\omega\cup\{\omega\}\text{.} \end{equation*}

Wait, isn't that like \(\infty+1\text{?}\) Isn't that \(\infty\) again? That sort of thing is true if you're talking about cardinals. Here, we're talking about ordinals, and the distinction is important. We'll get into this more when we talk about number systems in the next section. But we absolutely can construct

\begin{equation*} \omega+1,\omega+2,\dots,2\omega=\omega+\omega\text{,} \end{equation*}

where \(2\omega=\omega+\omega\) is the set

\begin{equation*} \{0,1,2,\dots,\omega,\omega+1,\omega+2,\dots\}\text{.} \end{equation*}

Likewise, you can build \(3\omega,4\omega,\dots\text{,}\) and then even \(\omega^2,\omega^3,\dots\text{,}\) and even \(\omega^\omega\text{.}\) The key point is that each ordinal is the set of all the ordinals smaller than it. We'll investigate this more deeply in the next section.

A key construction for building “big” sets is the power set construction. To explain, a subset of a set \(X\) is a set \(S\) such that for any \(s\in S\text{,}\) one has \(s\in X\) as well; in this case, we write \(S\subseteq X\text{.}\) The axioms of set theory permit us to form the power set \(\PP(X)\text{,}\) which is the set of all subsets of \(X\text{,}\) so that \(S\in\PP(X)\) if and only if \(S\subseteq X\text{.}\)

The set \(\PP(\varnothing)\) is \(\{\varnothing\}\text{.}\) The set \(\PP(\PP(\varnothing))\) is \(\{\varnothing, \{\varnothing\}\}\text{.}\)