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An ordinal notation is, generally, a method for systematically naming ordinals (usually countable ones). More specifically, it is a partial function from finite strings in a finite alphabet to ordinals, although in practice the term is applied to more general cases than this.

Iterated Cantor normal form Edit

Cantor normal form (CNF) expresses an ordinal \(\alpha\) as a sum of a finite decreasing sequence of ordinals of the form \(\omega^\beta\). If we require each \(\beta\) to be in CNF and restrict the system to finite levels of nesting, then we have an ordinal notation -- the "iterated Cantor normal form" -- that uniquely describes all ordinals \(< \varepsilon_0\).

Veblen's \(\varphi\) Edit

Oswald Veblen's \(\varphi\) function is not only an early ordinal notation, but it could also be considered the first-ever array notation, preceding BEAF by more than 90 years.[1] Due to the time when the paper was written, Veblen's definitions of the rules of his function are very cumbersome. Here, a conservative amount of modernization has been applied in the interest of brevity. Note that although Veblen used the dated convention where ordinals start with 1, we instead start ordinals with 0 and thus all letters (Greek or Latin) below are \(\geq 0\).

Before we can discuss the function itself, we first define a well-ordering \(\prec\) of arrays. An array is a function \(A : \eta \mapsto \omega_1\) for some countable ordinal \(\eta\). Veblen denotes its arguments by using \(x_\alpha\) to indicate the value of \(A(\alpha)\), and, say, \(0_\alpha\) to indicate that \(A(\alpha) = 0\). We use \(\sup_\prec\) to denote a supremum under the ordering \(\prec\). It is very important to note that we are not defining ordinals, but an ordering of arrays.

A. \(\varphi(0_0, 0_1, \ldots, 1_{\alpha + 1}) = \sup_\prec\{\varphi(0_0, 0_1, \ldots, x_\alpha): x \in \text{On}\}\).
B. \(\varphi(0_0, 0_1, \ldots, 1_\alpha) = \sup_\prec\{\varphi(0_0, 0_1, \ldots, 1_\beta): \beta < \alpha\}\) where \(\alpha \in \text{Lim}\)
C. \(\varphi((x + 1)_0, y_1, \ldots, z_\beta) \succ \varphi(x_0, \ldots, z_\beta)\), and there is no array between them
D. \(\varphi(0_0, \ldots, (x + 1)_{\alpha + 1}, \ldots, y_\beta) = \sup_\prec\{\varphi(0_0, \ldots, (x + 1)_\alpha, x_{\alpha + 1}, \ldots, y_\beta)\}\)
E. \(\varphi(0_0, \ldots, x_{\alpha + 1}, y_{\alpha + 2}, \ldots, z_\beta) = \sup_\prec\{\varphi(0_0, \ldots, w_{\alpha + 1}, y_{\alpha + 2}, \ldots, z_\beta) : w < x\}\) where \(x \in \text{Lim}\)
F. \(\varphi(0_0, \ldots, (x + 1)_\alpha, \ldots, y_\beta) = \sup_\prec\{\varphi(0_0, \ldots, 1_\gamma, \ldots, x_\alpha, \ldots, y_\beta) : \gamma < \alpha\}\) where \(\alpha \in \text{Lim}\)
G. \(\varphi(0_0, \ldots, x_\alpha, \ldots, y_\beta) = \sup_\prec\{\varphi(0_0, \ldots, z_\alpha, \ldots, y_\beta) : z < x\}\) where \(x, \alpha \in \text{Lim}\)

Now we can create an actual ordinal notation. Define \(\varphi(x)\) as a continuous (under the order topology) increasing function with \(\varphi(0) > 0\); conventionally this is chosen to be \(\varphi(\alpha) = \omega^\alpha\), although Veblen focused on the case \(\varphi(\alpha) = 1 + \alpha\). Then:

  • \(\varphi(x_0, 0_1, \ldots, 1_\beta) = \) the \(x_0\)th element of the set \(\{y: \forall \gamma < \beta: \varphi(0_0, 0_1, \ldots, y_\gamma) = y\}\)
  • for \(y > 1\), \(\varphi(x_0, 0_1, \ldots, y_\alpha, \ldots, z_\beta) = \) the \(x_0\)th element of the set \(\{v: \forall \gamma < \alpha: \forall w < y: \varphi(0_0, 0_1, \ldots, v_\gamma, 0_{\gamma+1}, \ldots, w_\alpha, z_{\alpha+1}, \ldots, z_\beta) = v\}\)

where "the \(x_0\)th element" is according to the ordering defined by the seven rules above.

The supremum of the range of Veblen's function is the large Veblen ordinal. If arrays are restricted to finite sizes (i.e. the domain of the array is \(< \omega\)), the supremum of the range is the small Veblen ordinal.

Modern conventions Edit

In practice, "Veblen \(\varphi\) function" typically refers to a far simpler two-argument form, defined as \(\varphi_0(\alpha) = \omega^\alpha\) (or some other normal function), and \(\varphi_\beta\) as the enumerating function for all the common fixed points of \(\varphi_\gamma\) for all \(\gamma < \beta\). More concisely, \(\varphi_\alpha\) is the \(\alpha\)'th derivative of \(\varphi_0\). The original is thus often called the "extended Veblen function."

Veblen's dated convention of starting ordinals at 1 is often corrected for in modern usage, so instead of, Veblen writing, say, \(\varphi(2, 1, 1)\), we would write \(\varphi(1, 0, 0)\); this modern convention is used in the above section. Using the zero-indexed version, we have the even simpler equality \(\varphi_\beta(\alpha) = \varphi(\alpha, \beta)\).

Ordinal collapsing function Edit

An ordinal collapsing function (OCF) is a method of naming large ordinals using even larger ones. More specifically, ordinal collapsing functions take the structures found in large (often uncountable) ordinals and mirror those structures onto smaller ordinals. OCFs are employed as notations for large recursive ordinals, for which they have the most relevance to googology.

There are many OCFs in use, often similar to each other and easily confused (some even use the same symbols).

Bachmann's \(\psi\) Edit

Heinz Bachmann's \(\psi\) function was the first true ordinal collapsing function.[2] It is somewhat cumbersome as it depends on fundamental sequences for all limit ordinals.

Rathjen suggests a "recast" of the system[3] as follows. Let \(\Omega\) be an uncountable ordinal such as \(\aleph_1\). Then define \(C^\Omega(\alpha, \beta)\) as the closure of \(\beta \cup \{0, \Omega\}\) under \(+, (\xi \mapsto \omega^\xi), (\xi \mapsto \psi_\Omega(\xi))_{\xi < \alpha}\), and \(\psi_\Omega(\alpha) = \min \{\rho < \Omega : C^\Omega(\alpha, \rho) \cap \Omega = \rho\}\).

\(\psi_\Omega(\varepsilon_{\Omega + 1})\) is the Bachmann-Howard ordinal, the proof-theoretic ordinal of Kripke-Platek set theory.

Feferman's \(\theta\) Edit

Feferman's \(\theta\)-functions constitute a hierarchy of single-argument functions \(\theta_\alpha: \text{On} \mapsto \text{On}\) for \(\alpha \in \text{On}\).[4] It is often considered a two-argument function with \(\theta_\alpha(\beta)\) written as \(\theta\alpha\beta\). It is defined like so:

\begin{eqnarray*} C_0(\alpha, \beta) &=& \beta \cup \{0, \omega_1, \omega_2, \ldots, \omega_\omega\}\\ C_{n+1}(\alpha, \beta) &=& \{\gamma + \delta, \theta_\xi(\eta) | \gamma, \delta, \xi, \eta \in C_n(\alpha, \beta); \xi < \alpha\} \\ C(\alpha, \beta) &=& \bigcup_{n < \omega} C_n (\alpha, \beta) \\ \theta_\alpha(\beta) &=& \min\{\gamma | \gamma \not\in C(\alpha, \gamma) \wedge \forall \delta < \beta: \theta_\alpha(\delta) < \gamma\} \\ \end{eqnarray*}

Informally:

  • An ordinal \(\beta\) is considered \(\alpha\)-critical iff it cannot be constructed with the following elements:
    • all ordinals less than \(\beta\),
    • all ordinals in the set \(\{0, \omega_1, \omega_2, \ldots, \omega_\omega\}\),
    • the operation \(+\),
    • applications of \(\theta_\xi\) for \(\xi < \alpha\).
  • \(\theta_\alpha\) is the enumerating function for all \(\alpha\)-critical ordinals.

The Feferman theta function is considered an extension of the two-argument Veblen function — for countable arguments, \(\theta_\alpha(\beta) = \varphi_\alpha(\beta)\). For this reason, \(\varphi\) may be used interchangeably with \(\theta\) even for uncountable arguments, so for example \(\varphi_\Omega(0) = \theta_\Omega(0) = \Gamma_0\). The supremum of the range of the function is the Takeuti-Feferman-Buchholz ordinal \(\theta_{\varepsilon_{\Omega_\omega + 1}}(0)\).

Buchholz discusses a set he calls \(\theta(\omega + 1)\), which is the set of all ordinals describable with \(\{0, \omega_1, \omega_2, \ldots, \omega_\omega\}\) and finite applications of \(+\) and \(\theta\).

Buchholz's \(\psi\) Edit

Main article: Buchholz's function

Buchholz's \(\psi\) is a hierarchy of single-argument functions \(\psi_v: \text{On} \mapsto \text{On}\) for \(v \leq \omega\), with \(\psi_v(\alpha)\) abbreviated as \(\psi_v\alpha\).[5] Define \(\Omega_0 = 1\) and \(\Omega_v = \aleph_v\) for \(v > 0\), and let \(P(\alpha)\) be the set of distinct terms in the Cantor normal form of \(\alpha\) (with each term of the form \(\omega^\xi\) for \(\xi \in \text{On}\)):

\begin{eqnarray*} C_v^0(\alpha) &=& \Omega_v\\ C_v^{n+1}(\alpha) &=& C_v^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_v^n(\alpha)\} \cup \{\psi_v\xi | \xi \in \alpha \cap C_v^n(\alpha) \wedge \xi \in C_u(\xi) \wedge u \leq \omega\} \\ C_v(\alpha) &=& \bigcup_{n < \omega} C_v^n (\alpha) \\ \psi_v\alpha &=& \min\{\gamma | \gamma \not\in C_v(\alpha)\} \\ \end{eqnarray*}

The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega + 1})\), the Takeuti-Feferman-Buchholz ordinal.

Madore's \(\psi\) Edit

David Madore (under "Gro-Tsen" on Wikipedia) defined the following simpler variant of Buchholz's function as a demonstration of how ordinal collapsing functions work.[6] The popularity of the article led to widespread use of the modified function.

\begin{eqnarray*} C_0(\alpha) &=& \{0, 1, \omega, \Omega\}\\ C_{n+1}(\alpha) &=& \{\gamma + \delta, \gamma\delta, \gamma^{\delta}, \psi(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\} \\ C(\alpha) &=& \bigcup_{n < \omega} C_n (\alpha) \\ \psi(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\ \end{eqnarray*}

Informally:

  • \(C(\alpha)\) is the set of all ordinals constructible using only \(0\), \(1\), \(\omega\), \(\Omega\), and finite applications of the following functions: addition, multiplication, exponentiation, and \(\kappa \mapsto \psi(\kappa)\) (the latter only if \(\psi(\kappa)\) has yet been defined).
  • \(\psi(\alpha)\) is the smallest countable ordinal not in \(C(\alpha)\).

Chris Bird uses this function throughout his googological papers.[7]

Bird's \(\theta\) Edit

Chris Bird devised the following shorthand for the extended (zero-indexed) Veblen function \(\phi\):[8]

\[\theta(\Omega^{n - 1}a_{n - 1} + \cdots + \Omega^2a_2 + \Omega a_1 + a_0, b) = \phi(a_{n - 1}, \ldots, a_2, a_1, a_0, b)\]

\(\theta(\alpha, 0)\) is abbreviated as \(\theta(\alpha)\). This function is only defined for arguments less than \(\Omega^\omega\), and its outputs are limited by the small Veblen ordinal. In his papers, Bird erroneously uses \(\theta(\Omega^\omega)\) and higher values without properly defining the function.

However, this function is equivalent to Feferman's \(\theta\) function (defined above) where both are defined, so it may be that he was merely using an extension of this \(\theta\) function.

Wilken's \(\vartheta\) Edit

Wilken's \(\vartheta\) is more generic than other OCFs. Let \(\Omega_0\) be either \(1\) or a number of the form \(\varepsilon_\alpha\), let \(\Omega_1 > \Omega_0\) be an uncountable regular cardinal and for \(0 < i < \omega\) let \(\Omega_{i+1}\) be the successor cardinal to \(\Omega_i\). Then, for \(0 < n < \omega\) and \(0 \leq m < n\), define the following for \(\beta < \Omega_{m+1}\):[9]

\begin{eqnarray*} \Omega_m \cup \beta &\subseteq& C_m^n(\alpha, \beta) \\ \xi, \eta \in C_m^n(\alpha, \beta) &\Rightarrow& \xi + \eta \in C_m^n(\alpha, \beta) \\ \eta \in C_m^n(\alpha, \beta) \cap \Omega_{k + 2} &\Rightarrow& \vartheta_k^n(\xi) \in C_m^n(\alpha, \beta) \text{ for } m < k < n \\ \eta \in C_m^n(\alpha, \beta) \cap \alpha &\Rightarrow& \vartheta_m^n(\xi) \in C_m^n(\alpha, \beta) \\ \vartheta_m^n(\alpha) &=& \min(\{\xi < \Omega_{m+1}|C_m^n(\alpha, \xi) \cap \Omega_{m+1} \subseteq \xi \wedge \alpha \in C_m^n(\alpha, \xi)\} \cup \{\Omega_{m+1}\}) \\ \end{eqnarray*}

\(n\) is needed to define the function, but the actual value of \(n\) does not affect the function's behavior. So, for example, \(\vartheta_0^1(\alpha) = \vartheta_0^2(\alpha) = \vartheta_0^3(\alpha) = \cdots\) So we can safely eliminate \(n\) and express the function using only two arguments \(\vartheta_m(\alpha)\).

Wilken and Weiermann's \(\bar\vartheta\) Edit

Wilken and Weiermann's \(\bar\vartheta\) is closely related to Wilken's \(\vartheta\), and their paper closely analyzes the relationship between the two.[10] As before, let \(\Omega_0\) be either \(1\) or a number of the form \(\varepsilon_\alpha\), let \(\Omega_1 > \Omega_0\) be an uncountable regular cardinal and for \(0 < i < \omega\) let \(\Omega_{i+1}\) be the successor cardinal to \(\Omega_i\), and finally let \(\Omega_\omega = \sup_{i < \omega} \Omega_i\). For all \(\beta \leq \Omega_{i+1}\) define the following:

\begin{eqnarray*} \Omega_i \cup \beta &\subseteq& \bar{C}_i(\alpha, \beta) \\ \xi, \eta \in \bar{C}_i (\alpha, \beta) &\Rightarrow& \xi + \eta \in \bar{C}_i(\alpha, \beta) \\ j \leq i < \omega \wedge \xi \in \bar{C}_j(\xi, \Omega_{j + 1}) \cap \bar{C}_i(\alpha, \beta) \cap \alpha &\Rightarrow& \bar{\vartheta}_j(\xi) \in \bar{C}_i(\alpha, \beta) \\ \bar{\vartheta}_i(\alpha) &=& \min(\{\xi < \Omega_{i + 1} | \alpha \in \bar{C}_i(\alpha, \xi) \wedge \bar{C}_i(\alpha, \xi) \cap \Omega_{i + 1} \subseteq \xi\} \cup \{\Omega_{i + 1}\}) \\ \end{eqnarray*}

Weiermann's \(\vartheta\) Edit

The \(\vartheta\) function has the advantage of having only a single argument, at the cost of some added complexity.[11]

\begin{eqnarray*} C_0(\alpha, \beta) &=& \beta \cup \{0, \Omega\}\\ C_{n+1}(\alpha, \beta) &=& \{\gamma + \delta, \omega^{\gamma}, \vartheta(\eta) | \gamma, \delta, \eta \in C_n (\alpha, \beta); \eta < \alpha\} \\ C(\alpha, \beta) &=& \bigcup_{n < \omega} C_n (\alpha, \beta) \\ \vartheta(\alpha) &=& \min \{\beta < \Omega | C(\alpha, \beta) \cap \Omega \subseteq \beta \wedge \alpha \in C(\alpha, \beta)\} \\ \end{eqnarray*}

Rathjen and Weiermann showed that \(\vartheta(\alpha)\) is defined for all \(\alpha < \varepsilon_{\Omega+1}\), but do not discuss higher values.

\(\vartheta\) follows the archetype of many ordinal collapsing functions — it is defined inductively with a "marriage" to the \(C\) function. Interpreting the equations:

  • \(C(\alpha, \beta)\) is the set of all ordinals constructible using only the following:
    • Zero, all ordinals less than \(\beta\), and \(\Omega\).
    • Finite applications of addition, \(\kappa \mapsto \omega^\kappa\), \(\kappa \mapsto \vartheta(\kappa)\) (the latter only if \(\vartheta(\kappa)\) has yet been defined).
  • \(\vartheta(\alpha)\) is the smallest ordinal \(\beta\) so that \(\alpha \in C(\alpha, \beta)\), and \(\beta\) is greater than all the countable ordinals in \(C(\alpha, \beta)\).


Rathjen's \(\psi\) Edit

Rathjen's \(\psi\) function is based on the least weakly Mahlo cardinal to create large countable ordinals.[12] A weakly Mahlo cardinal can be defined as a cardinal \(\text M\) such that all normal functions closed in \(\text M\) have a regular fixed point. He uses this to diagonalise over the inaccessible hierarchy.

Due to the complexity of the original notation, the one provided below has been simplified slightly, but still emulates the main properties of the original notation.

We restrict \(\pi\) to uncountable regular ordinals.

\(\text{enum}(X)\) is the unique increasing function \(f\) such that the range of \(f\) is exactly \(X\).

\(\text{cl}(X)\) is the closure of \(X\); that is \(\text{cl}(X):=X\cup\{\beta:\sup(X\cap\beta)=\beta\}\)

\(\text B_0(\alpha,\beta):=\beta\cup\{0,\text M\}\)

\(\text B_{n+1}(\alpha,\beta):=\{\gamma+\delta,\varphi_\gamma(\delta),\chi_\mu(\delta):\gamma,\delta,\mu\in\text B_n(\alpha,\beta)\wedge\mu<\alpha\}\)

\(\text B(\alpha,\beta):=\cup_{n<\omega}\text B_n(\alpha,\beta)\)

\(\chi_\alpha:=\text{enum}(\text{cl}(\{\pi:\text B(\alpha,\pi)\cap\text M\subseteq \pi\wedge\alpha\in\text B(\alpha,\pi)\}))\)

\(\text C_0(\alpha,\beta):=\beta\cup\{0,\text M\}\)

\(\text C_{n+1}(\alpha,\beta):=\{\gamma+\delta,\varphi_\gamma(\delta),\chi_\gamma(\delta),\psi_\pi(\mu):\gamma,\delta,\mu,\pi\in\text C_n(\alpha,\beta)\wedge\mu<\alpha\}\)

\(\text C(\alpha,\beta):=\cup_{n<\omega}\text C_n(\alpha,\beta)\)

\(\psi_\pi(\alpha):=\min\{\beta:\text C(\alpha,\beta)\cap\pi\subseteq\beta\wedge\alpha\in\text C(\alpha,\beta)\}\)

Rathjen's ordinal is \(\psi_{\omega_1}(\chi_{\varepsilon_{\text M+1}}(0))\), the proof-theoretic ordinal of KPM, where KPM is Kripke–Platek set theory augmented by the axiom schema "every normal function has an admissible ordinal as a fixed point".

The \(\chi\) functions are defined to be normal functions that enumerate over layers of the weakly inaccessible hierarchy. The \(\chi_0\) function enumerates the uncountable cardinals less than \(\text M\). The \(\chi_1\) function enumerates the weakly inaccessible cardinals less than \(\text M\), and their limits. Similarly, for each \(\alpha<\text M\), \(\chi_{1+\alpha}\) enumerates the weakly \(\alpha\)-inaccessible cardinals less than \(\text M\) (and their limits, because the function is normal), and the function \(\chi_{\text M}\) diagonalises over these, and enumerates the weakly hyper-inaccessible cardinals and their limits less than \(\text M\). \(\chi_{\text M2}\) enumerates the weakly hyper-hyper-inaccessible cardinals and their limits less than \(\text M\), and so on and so fourth.

Although it may not be obvious why, the limits of these large cardinals are added to the \(\chi\) function to make it much easier to notate, say, the suprenum of \(\text I_0, \text I_1, \text I_2\cdots\) (where \(\text I_\alpha\) is the \(\alpha\)th weakly inaccessible cardinal), as values like these don't otherwise have any way of being notated easily.

Note that it is possible to define the same ordinal using only a single \(\text C\) function instead of both \(\text B\) and \(\text C\), however this is not done in the original text, as it is more transparent and easier to manage values when the functions are defined separately.

Stegert's notation Edit

As part of his 2010 Ph.D. dissertation, Jan-Carl Stegert developed a powerful ordinal notation based on the work of Rathjen.[13] To our knowledge, it is the current state of the art in the field of ordinal analysis.

Taranovsky's C Edit

Dmytro Taranovsky, much more recently, described the following meta-ordinal-notation system in a self-published web page.[14] Let \(\eta\) be an ordinal, and let \(0^S\) and let \(\text{Ld}(a, b)\) be the statement that \(a\) is a limit of ordinals \(c\) such that \((c, b) \in D\). Let \(D\) be the following binary relation over \(\eta\):

  • \(\forall a < \eta: (a, 0) \in D\)
  • \(\forall a < \eta: a \neq 0 \Rightarrow (0, a) \notin D\)
  • \(\forall b \in \text{Lim} \cup \eta: (a, b) \in D \Leftrightarrow \forall c < b: (a, c) \in D\).
  • \(\forall b: (a, b) \in D \Leftrightarrow \text{Ld}(a, b + 1)\)
  • \(\forall b: \exists d \in \text{Lim} \cup \eta: d \leq b \Rightarrow \forall c: (c, a + 1) \in D \Leftrightarrow (c \leq d \vee \text{Ld}(c, b))\)

Then \(C(a, b) = \min\{c : c \in \eta \wedge c > b \wedge (c, a) \in D\}\).

Define a partial ordinal notation system \(O\) as a partial mapping from ordinals to finite strings comprised of symbols and ordinals such that \(O(\alpha)\) is undefined if \(\alpha\) occurs in a string in the range of \(O\). Given a partial ordinal notation system \(O\), we define Taranovsky's notation for \(a\):

  • If \(O(a)\) is defined, then we simply use the string \(O(a)\) to notate \(a\).
  • Otherwise, let \(b,c\) be ordinals so that \(a = C(b, c) \wedge b'>b \Rightarrow C(b', c) > C(b, c) \wedge c' < c \Rightarrow C(b, c') < C(b, c)\). We then use the string "\(C(b,c)\)."

Second-order arithmetic Edit

One implementation Taranovsky conjectures to reach further than the proof-theoretic ordinal of second-order arithmetic, which if true would make it an exceptionally strong system.

For \(k \geq 0\), define the binary relation "\(a\) is \(k\)-built from below by \(b\)" over ordinals as follows:

  • \(a\) is \(0\)-built from below by \(b\) iff \(a < b\).
  • \(a\) is \(k+1\)-built from below by \(b\) iff the standard representation of \(a\) does not use ordinals above \(a\) except in the scope of an ordinal \(k\)-built from below by \(b\).

Taranovsky's notation, then, is an infinite family of notations indexed by positive integer \(n\) defined individually as follows:

  • The language consists of constants "\(0\)" and "\(\Omega_n\)" and a binary function "\(C\)" written in reverse Polish notation.
  • Ordering is lexicographic with "\(C\)" < "\(0\)" < "\(\Omega_n\)".
  • The strings "\(0\)" and "\(\Omega_n\)" are in standard form.
  • The string "\(abC\)" is in standard form iff all the following are true:
    • "\(a\)" and "\(b\)" are in standard form.
    • If "\(a\)" is of the form "\(cdC\)", \(b \leq d\) according to the aforementioned lexicographic ordering.
    • "\(b\)" is \(n\)-built from below by "\(abC\)".

For \(n = 1\), Taranovsky showed that the system reaches the Bachmann-Howard ordinal.

Nonrecursive notations Edit

The notations in this section are nonrecursive. Some authors do not consider them to be real ordinal notations.

Kleene's \(\mathcal{O}\) Edit

Kleene's \(\mathcal{O}\) is a notation for all recursive ordinals. Let \(f_0, f_1, f_2, \ldots\) be an enumeration of all the partial recursive functions. (One way to do this is to order all the Turing machines lexicographically.)

Let \(<_\mathcal{O}\) be an operator indicating a well-ordering of set \(K\), and let \(\mathcal{O}: K \mapsto \omega_1\). These three are defined inductively as follows:

  • \(0 \in K\) and \(\mathcal{O}(0) = 0\).
  • If \(n \in K\) and \(\mathcal{O}(n) = \alpha\), then \(2^n \in K\), \(\mathcal{O}(2^n) = \alpha + 1\), and \(n <_\mathcal{O} 2^n\).
  • If for all natural numbers \(n\), we have \(f_i(n) \in K\) and \(f_i(n) <_\mathcal{O} f_i(n + 1)\), then \(3 \cdot 5^i \in K\), \(\mathcal{O}(3 \cdot 5^i) = \lim_{k \rightarrow \omega} \mathcal{O}(f_i(k))\), and for all \(k\) \(f_i(k) <_\mathcal{O} 3 \cdot 5^i\).
  • \(a <_\mathcal{O} b\) and \(b <_\mathcal{O} c\) implies \(a <_\mathcal{O} c\).

Klev's \(\mathcal{O}^+\) Edit

Ansten Mørch Klev extended Kleene's \(\mathcal{O}\) to the set of all writable ordinals.[15] Let \(f_0, f_1, f_2, \ldots\) be an enumeration of all ITTM-computable partial functions (where the output of an ITTM is based on the tape when it halts). The rules are otherwise identical to Kleene's \(\mathcal{O}\):

  • \(0 \in K\) and \(\mathcal{O}^+(0) = 0\).
  • If \(n \in K\) and \(\mathcal{O}^+(n) = \alpha\), then \(2^n \in K\), \(\mathcal{O}^+(2^n) = \alpha + 1\), and \(n <_{\mathcal{O}^+} 2^n\).
  • If for all natural numbers \(n\), we have \(f_i(n) \in K\) and \(f_i(n) <_{\mathcal{O}^+} f_i(n + 1)\), then \(3 \cdot 5^i \in K\), \(\mathcal{O}^+(3 \cdot 5^i) = \lim_{k \rightarrow \omega} \mathcal{O}^+(f_i(k))\), and for all \(k\) \(f_i(k) <_{\mathcal{O}^+} 3 \cdot 5^i\).
  • \(a <_{\mathcal{O}^+} b\) and \(b <_{\mathcal{O}^+} c\) implies \(a <_{\mathcal{O}^+} c\).

\(\mathcal{O}^+\) is capable of expressing all ordinals \(< \lambda\) (all writable ordinals).

Klev's \(\mathcal{O}^{++}\) Edit

In the same paper, Klev introduced \(\mathcal{O}^{++}\), which has \(f_i\) enumerates all eventually computable partial functions (that is, the output of an ITTM is based on what the tape converges to). The rules are otherwise identical:

  • \(0 \in K\) and \(\mathcal{O}^{++}(0) = 0\).
  • If \(n \in K\) and \(\mathcal{O}^{++}(n) = \alpha\), then \(2^n \in K\), \(\mathcal{O}^{++}(2^n) = \alpha + 1\), and \(n <_{\mathcal{O}^{++}} 2^n\).
  • If for all natural numbers \(n\), we have \(f_i(n) \in K\) and \(f_i(n) <_{\mathcal{O}^{++}} f_i(n + 1)\), then \(3 \cdot 5^i \in K\), \(\mathcal{O}^{++}(3 \cdot 5^i) = \lim_{k \rightarrow \omega} \mathcal{O}^{++}(f_i(k))\), and for all \(k\) \(f_i(k) <_{\mathcal{O}^{++}} 3 \cdot 5^i\).
  • \(a <_{\mathcal{O}^{++}} b\) and \(b <_{\mathcal{O}^{++}} c\) implies \(a <_{\mathcal{O}^{++}} c\).

It is capable of expressing all ordinals \(< \zeta\) (all eventually writable ordinals).

It is worth noting that the concepts of \(\mathcal{O}^+\) and \(\mathcal{O}^{++}\) cannot be extended to all accidentally writable ordinals, because many ITTMs accidentally write more than one ordinal. Indeed, there exist "universal" ITTMs accidentally writing all accidentally writable ordinals. No ordinal notation is known that specifically targets all ordinals \(< \Sigma\).

Table of ordinal notations Edit

Below is a table comparing the different ordinal notations in this article.

NotationSupremum of expressible ordinals
Cantor normal form\(\varepsilon_0\)
Veblen function (two arguments)Feferman-Schütte ordinal \(\Gamma_0\)
Bird's \(\theta\)Small Veblen ordinal
Extended Veblen functionLarge Veblen ordinal
Bachmann's \(\psi\)Bachmann-Howard ordinal
Feferman's \(\theta\)Takeuti-Feferman-Buchholz ordinal
Madore's \(\psi\)Bachmann-Howard ordinal
Weiermann's \(\vartheta\)Bachmann-Howard ordinal
Buchholz's \(\psi\)Takeuti-Feferman-Buchholz ordinal
Rathjen's \(\psi\)Rathjen's ordinal (proof-theoretic ordinal of KPM)
Taranovsky's CUnknown, but computable and very large
Kleene's \(\mathcal{O}\)Church-Kleene ordinal \(\omega_1^\text{CK}\)
Klev's \(\mathcal{O}^+\)Supremum of writable ordinals \(\lambda\)
Klev's \(\mathcal{O}^{++}\)Supremum of eventually writable ordinals \(\zeta\)

Sources Edit

  1. Veblen, Oswald. "Continuous increasing functions of finite and transfinite ordinals". Retrieved 2014-10-11.
  2. Bachmann, Heinz. "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen"Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, vol. 95. Zürich: Orell Füssli Graphische Betriebe AG.. Retrieved 2014-08-29.
  3. Rathjen, Michael. "The Art of Ordinal Analysis"
  4. Buchholz, W.. "A New System of Proof-Theoretic Ordinal Functions"Annals of Pure and Applied Logic, vol. 32. Retrieved 2014-09-21.
  5. Buchholz, W.. "A New System of Proof-Theoretic Ordinal Functions"Annals of Pure and Applied Logic, vol. 32. Retrieved 2014-08-29.
  6. "Ordinal Collapsing Function". Wikipedia. Retrieved 2014-08-29.
  7. Chris Bird, pers. comm
  8. Bird, Chris. "The Fast-Growing Hierarchy in Terms of Bird's Array Notations". Retrieved 2014-08-29.
  9. Wilken, Gunnar. "Ordinal arithmetic based on Skolem hulling". Annals of Pure and Applied Logic. Retrieved 2014-09-21.
  10. Weiermann, Andreas and Wilken, Gunnar. "Ordinal arithmetic with simultaneously defined theta-functions". Wiley Online. Retrieved 2014-09-21.
  11. Rathjen, Michael and Weiermann, Andreas. "Proof Theoretic Investigations of Kruskal's Theorem"
  12. Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal"
  13. Stegert, Jan-Carl. "Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles"
  14. Taranovsky, Dmytro. Ordinal Notation. Retrieved 2014-09-26.
  15. [1]

See also Edit

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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