Ordinal


 * Not to be confused with the (meaning words such as "first", "second", etc.).

In set theory, an ordinal number or ordinal is an equivalence class of well-ordered sets under the relation of order isomorphism. Intuitively speaking, the ordinals form a number system that can be viewed as an extension of the natural numbers into infinite values. They are important to googology since they describe the growth rates of functions via the fast-growing hierarchy and other ordinal hierarchies, as well as their appearances in other meeting points between googology and set theory.

Definition
Here we will describe a few formal and informal ways to introduce the ordinal number system.

Order type
Order types are the most general way to define ordinals. Under this definition, an ordinal is an equivalence class of well-ordered sets under the relation of order isomorphism.

A binary relation \(\leq\) is a well-ordering of the set \(S\) iff all the following are true for all \(a,b,c\) in \(S\):


 * \(a \leq b \vee b \leq a\)
 * \(a \leq b \wedge b \leq c \Rightarrow a \leq c\)
 * \(a \leq b \wedge b \leq a \Rightarrow a = b\)
 * Every subset of \(S\) has a minimum element.

A well-ordered set a consists of a set \(S\) and a relation \(\leq_S\) that is a well-ordering of \(S\).

An order isomorphism between two well-ordered sets \((S, \leq_S)\) and \((T, \leq_T)\) is a bijection \(f : S \mapsto T\) such that \(\forall x,y \in S : x \leq_S y \Leftrightarrow f(x) \leq_T f(y)\). The relation of order isomorphism asks whether an order isomorphism exists. It is not difficult to show that this relation is an equivalence relation, and thus partitions all well-ordered sets into equivalence classes. Each of these equivalence classes is represented by a distinct ordinal.

Von Neumann definition
The Von Neumann definition gives us a specific way to define ordinals within the framework of set theory. It is handy as it can represent ordinals as actual objects within a set theory. Under the Von Neumann definition, an ordinal number is the set of all smaller ordinals. Formally, an ordinal number is a transitive set well-ordered by \(\in\).

Whereas the above definition is indisputable, this particular definition is not always applicable and it is a good idea to clarify when it is being used. It is convenient since it lets us easily apply set descriptors to ordinals with no explanation needed at all: recursive ordinal, countable ordinal, etc.

As an extension of the natural numbers
While the above definitions may be comprehensible to set theorists, it is easiest to think intuitively of the ordinals as an extension of the nonnegative integers. The nonnegative integers are 0, 1, 2, 3, 4, ..., but the ordinal system extends this with the first transfinite ordinal \(\omega\). \(\omega\) is the first ordinal larger than all the natural numbers.

Next we can continue with the ordinal \(\omega + 1\). Strangely, the definition of ordinal addition tells us that this is different \(1 + \omega\), which is simply equal to \(\omega\). \(\omega + 1\), however, is larger than \(\omega\). Naturally there are further ordinals such as \(\omega + 2\), \(\omega + 3\), etc.

Introduction
An ordinal number is defined as the set of all smaller ordinal numbers. Formally, an ordinal \(\alpha\) is \(\{\beta: \beta < \alpha\}\), and the successor to an ordinal is defined as \(\alpha + 1 = \alpha \cup \{\alpha\}\).

We start with \(0 := \{\}\) &mdash; zero is the empty set, since no ordinal number is smaller than zero. We continue with \(1 := \{0\} = \{\{\}\}\), \(2 := \{0, 1\} = \{\{\}, \{\{\}\}\}\), \(3 := \{0, 1, 2\} = \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}\), etc.

Small transfinite ordinals
The next ordinal after all these is \(\{0, 1, 2, 3, \ldots\}\), the set of all nonnegative integers. We represent this with the Greek letter \(\omega\) (omega), the least transfinite ordinal. Next, we can define \(\omega + 1 = \{0, 1, 2, 3, \ldots, \omega\}\), \(\omega + 2 = \{0, 1, 2, 3, \ldots, \omega, \omega + 1\}\), etc.

The specifics of are not detailed here, but the general idea should be fairly clear from these examples. It is important to note that addition on the ordinals is often not commutative. \(\omega + 1 > \omega\), but \(1 + \omega = \omega\). Similarly, \(\omega + \omega = \omega \times 2\) does not equal \(2 \times \omega \) = 2 + 2 + 2... (with ω 2's) = \(\omega\).

The supremum of all the \(\omega + n\) is \(\{0, 1, 2, \ldots, \omega, \omega + 1, \omega + 2, \ldots\} = \omega + \omega = \omega \times 2\). Next we have the series \(\omega + \omega + \omega = \omega \times 3, \omega \times 4, \omega \times 5, \omega \times 6, \ldots, \omega \times \omega = \omega^2, \omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega} = {}^3\omega, \omega^{\omega^{\omega^\omega}} = {}^4\omega, \ldots\)

Eventually we reach \(^\omega\omega = \omega^{\omega^{\omega^{.^{.^.}}}}\), famously known as \(\varepsilon_0\) (epsilon-zero). (To googologists, the exponentiation seems to be a fairly arbitrary choice, but in fact it is not &mdash; it is important in induction proofs, for example.) \(\varepsilon_0\) is a fixed point such that \(\varepsilon_0 = \omega^{\varepsilon_0}\); \(\varepsilon_1\) can be defined as fixed point: \(\varepsilon_1 = \varepsilon_0^{\varepsilon_1}\). In general, \(\varepsilon_{\alpha+1}\) is a fixed point for \(\varepsilon_{\alpha+1} = \varepsilon_{\alpha}^{\varepsilon_{\alpha+1}}\). Examples of "epsilon numbers" are \(\varepsilon_0, \varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_\omega, \varepsilon_{\varepsilon_0} \ldots\).

The limit of the hierarchy of \(\varepsilon_\alpha\) leads us to \(\varepsilon_{\varepsilon_{\varepsilon_{._{._.}}}}\), sometimes called \(\zeta_0\). The "zeta numbers" are the solutions to the equation \(\zeta = \varepsilon_\zeta\). We can also have \(\eta_0 = \zeta_{\zeta_{\zeta_{._{._.}}}}\) and the "eta numbers" \(\eta = \zeta_\eta\).

Veblen hierarchy
We only have finitely many Greek letters, so the next step is to generalize these. Define the Veblen hierarchy as \(\phi_0(\alpha) = \omega^\alpha\), and define \(\phi_{\beta + 1}(\alpha)\) as the smallest solution \(\gamma\) to the equation \(\phi_\beta(\gamma) = \gamma\). This gives us \(\phi_1(\alpha) = \varepsilon_\alpha, \phi_2(\alpha) = \zeta_\alpha, \phi_3(\alpha) = \eta_\alpha, \ldots\). Further we can define \(\phi_\alpha\) for any transfinite ordinal.

The limit of the Veblen hierarchy is \(\phi_{\phi_{\phi_{._{._..}.}(0)}(0)}(0) = \Gamma_0\), the Feferman–Schütte ordinal, which cannot be reached using any smaller ordinals. It is the smallest solution to the equation \(\Gamma\) to \(\phi_\Gamma(0) = \Gamma\).

Beyond Veblen hierarchy
Some have extended the Veblen hierarchy to multiple dimensions, such as \(\phi(1, 0, 0, 0)\) (the Ackermann ordinal), but we shall skip that step to introduce a much more powerful system called an ordinal collapsing function. We will start with a system due to Buchholz. Define \(\psi(\alpha)\) as the smallest ordinal that cannot be expressed using only \(0\), \(1\), \(\omega\), \(\Omega\) (first uncountable ordinal), addition, multiplication, exponentiation, and any previously generated values of \(\psi\).

Psi function up to \(\alpha \rightarrow \phi_\alpha(0)\)
We begin with \(\psi(0)\). Call the set of all ordinals that can be generated \(C(0)\). Using the symbols allowed to us, we can generate \(0, 1, 2, \ldots, \omega, \omega + 1, \omega + 2, \ldots, \omega^2, \omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}, \ldots\) Our set also includes \(\Omega\), \(\Omega^\Omega\), \(\Omega^{\Omega^\Omega}, \ldots\) but the smallest ordinal we cannot access is \(\varepsilon_0\). So \(\psi(0) = \varepsilon_0\), first ordinal that not normal cantor form.

With \(C(1)\) we have all the ordinals we've already generated, but also ordinals based on \(\varepsilon_0\). So we also have \(\varepsilon_0, \varepsilon_0 + 1, \varepsilon_0 + \omega, \varepsilon_0^{\varepsilon_0}, \varepsilon_0^{\varepsilon_0^{\varepsilon_0}}, \ldots\) and the limit is clearly \(\varepsilon_1\), second ordinal that not normal cantor form. Similarly, for all \(\alpha < \omega\) we have \(\psi(\alpha) = \varepsilon_\alpha\), \(1 + \alpha\)-th ordinal that not normal cantor form.

This rule even works for ordinals beyond \(\omega\), but something interesting happens later on. Since by definition \(\zeta_0 = \varepsilon_{\zeta_0}\), \(\psi(\zeta_0) = \varepsilon_{\zeta_0} = \zeta_0\). If we try to evaluate \(\psi(\zeta_0 + 1)\), the largest ordinal we can plug into \(\varepsilon_\alpha\) is \(\zeta_0\), which just produces itself. So \(\psi(\zeta_0 + 1)\) is just \(\zeta_0\) again! In fact, nothing happens until we reach \(\Omega\).

As before, \(\psi(\Omega) = \zeta_0\). But when we reach \(\psi(\Omega + 1)\), we are allowed to use \(\psi(\Omega)\) and thus \(\zeta_0\). So \(\psi(\Omega + 1) = \zeta_0^{\zeta_0^{\zeta_0^{.^{.^.}}}} = \varepsilon_{\zeta_0 + 1}\). This extends to \(\psi(\Omega + 2) = \varepsilon_{\zeta_0 + 1}^{\varepsilon_{\zeta_0 + 1}^{\varepsilon_{\zeta_0 + 1}^{.^{.^.}}}} = \varepsilon_{\zeta_0 + 2}\), and in general \(\psi(\Omega + \alpha) = \varepsilon_{\zeta_0 + \alpha}\). But when we reach \(\psi(\Omega + \zeta_1) = \varepsilon_{\zeta_0 + \zeta_1} = \zeta_1\), the function halts again because we can't construct \(\zeta_1\) yet.

So \(\psi(\Omega 2) = \zeta_1\) or \(\phi_2(1)\). We can't construct \(\zeta_2\), because \(\psi(\Omega 2 + \zeta_2) = \zeta_2\). So \(\psi(\Omega 3) = \zeta_2\) or \(\phi_2(2)\). Continue with \(\zeta_3 = \psi(\Omega 4)\), \(\zeta_4 = \psi(\Omega 5)\), etc. The ordinal \(\varepsilon_{\zeta_{\beta - 1} + \alpha}\), is equal to \(\psi(\Omega \beta + \alpha)\).

Next, \(\psi(\Omega^2) = \eta_0\), because \(\eta_0\) is NOT equal to \(\psi(\Omega \eta_0)\). Then \(\psi(\Omega^2 2) = \eta_1\), \(\psi(\Omega^2 3) = \eta_2\), etc. The ordinal \(\varepsilon_{\zeta_{\eta_{\delta - 1} + \beta} + \alpha}\), is equal to \(\psi(\Omega^2 \delta + \Omega \beta + \alpha)\).

In general, \(\psi(\Omega^\alpha (\beta + 1))\), is equal to \(\varphi(1 + \alpha,\beta)\).

\(\alpha \mapsto \varphi_\alpha(0)\) is equal to \(\alpha \mapsto \psi(\Omega^\alpha)\).