The smallest infinite ordinal, often denoted $\omega$ (omega), has the order type of the natural numbers. As a [](Ordinal.md), $\omega$ is in fact equal to the set of natural numbers. Since $\omega$ is infinite, it is not equinumerous with any smaller ordinal, and so it is an <a href="index.php?title=Initial_ordinal&amp;action=edit&amp;redlink=1" class="new" title="Initial ordinal (page does not exist)">initial ordinal</a>, that is, a [cardinal](Cardinal.md "Cardinal"). When considered as a cardinal, the ordinal $\omega$ is denoted $\aleph_0$. So while these two notations are intensionally different---we use the term $\omega$ when using this number as an ordinal and $\aleph_0$ when using it as a cardinal---nevertheless in the contemporary treatment of cardinals in ZFC as initial ordinals, they are extensionally the same and refer to the same object. ## Countable sets A set is *countable* if it can be put into bijective correspondence with a subset of $\omega$. This includes all finite sets, and a set is *countably infinite* if it is countable and also infinite. Some famous examples of countable sets include: - The natural numbers $\mathbb{N}=\{0,1,2,\ldots\}$. - The integers $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ - The rational numbers $\mathbb{Q}=\{\frac{p}{q}\mid p,q\in\mathbb{Z}, q\neq 0\}$ - The real algebraic numbers $\mathbb{A}$, consisting of all zeros of nontrivial polynomials over $\mathbb{Q}$ The union of countably many countable sets remains countable, although in the general case this fact requires the <a href="Axiom_of_choice" class="mw-redirect" title="Axiom of choice">axiom of choice</a>. A set is <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> if it is not countable.