Terminology about ordinals

Question

I came across the phrase "Consider a free ultrafilter on $\omega$". I'm not too familiar with ordinals but a quick read tells me that ordinals describes the position of an element in a set and $\omega$ is the smallest infinite ordinal.

On the other hand, ultrafilters are defined on sets. So what does it mean to consider an ultrafilter on a "number" $\omega$? It's like saying "consider an ultrafilter on 42". What does that mean?

Answer 1

In set theory, ordinals are sets. Natural numbers are also sets, as are real numbers, etc. In set theory, basically everything is composed of sets.

An ordinal is a set that is both transitive (under the $\in$ relation, i.e. $x$ is transitive if $a\in b\in x$ implies $a\in x$) and well-ordered by $\in$ (so every nonempty subset of an ordinal number has an $\in$-minimal element, and $\in$ is a linear order on the set). There is a transfinitely recursive way of generating all the ordinals:

• $\varnothing$ is an ordinal
• If $\alpha$ is an ordinal, then $\alpha\cup\{\alpha\}$ is an ordinal (which we call $\alpha+1$)
• If $A$ is a set of ordinals, then $\bigcup A$ is an ordinal (which is the supremum of the ordinals in $A$).

The ordinals are then ordered by $\in$: An ordinal $\alpha$ is less than an ordinal $\beta$ iff $\alpha\in\beta$. Since $\in$ is a well-order on ordinals, this implies that the class of all ordinals is itself well-ordered by $\in$.

In turn, we can use the ordinals to model the natural numbers, by setting \begin{align} 0&=\varnothing\\ 1&=0\cup\{0\}=\{0\}, \\ 2&=1\cup\{1\}=\{0,1\},\\ 3&=2\cup\{2\}=\{0,1,2\},\\ &\text{etc.} \end{align}

The least infinite ordinal is then called $\omega=\{0,1,2,3,\dots\}$ and contains all natural numbers. Since $\omega$ is an ordinal, we can keep counting onwards to get $\omega+1=\omega\cup\{\omega\}=\{0,1,2,\dots,\omega\}$, and so on.

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So, in conclusion, an ultrafilter on $42$ would be an ultrafilter on the set $42=\{0,1,2,\dots,40,41\}$ of all natural numbers less than $42$. For example the ultrafilter $F=\{X\subseteq 42\mid 13\in X\}$ containing all subsets $X$ of $\{0,1,\dots,41\}$ that have the number $13$ as an element.

An ultrafilter on $\omega$ would be an ultrafilter on the natural numbers $\omega=\{0,1,2,3,\dots\}$. A free ultrafilter is hard to describe explicitly, since a weak form of the Axiom of Choice is needed to prove the existence of free ultrafilters. One can start with a free filter, such as the Fréchet filter $F=\{X\subseteq \omega\mid \omega\setminus X\text{ is finite}\}$ of all cofinite subsets of $\omega$, and then use Zorn's lemma to extend $F$ to a maximal filter (= ultrafilter).