User blog:Deedlit11/Ordinal Notations I: Up to the Schutte Klammersymbolen

Before I go to stronger ordinal notations, I'll step back a bit a go over the smaller notations leading up to the Small Veblen Ordinal.

Cantor Normal Form
Every ordinal \(\alpha\) can be expressed uniquely in the form

\(\alpha = \omega^{\alpha_1} + \omega^{\alpha_2} + \ldots + \omega^{\alpha_n}\), where \(\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n\).

This expression is known as Cantor Normal Form.

The ordinal \(\varepsilon_\alpha\) is the \(\alpha\)th critical ordinal, where an ordinal is critical if it is in the set \(\lbrace \beta | \omega^\beta = \beta \rbrace \). By expressing an ordinal \(\alpha\) in Cantor Normal Form, with each of the \(\alpha_i\) expressed in Cantor Normal Form, etc., we can express any ordinal in terms of \(0, \omega,\) and critical ordinals. If \(\alpha\) is less than \(\varepsilon_0 = \omega^{\omega^{\omega^{\cdots}}}\), than \(\alpha\) can be expressed in terms of 0 and \(\omega\).

The Veblen Function
To go beyond the \(\varepsilon\)-numbers, we define the Veblen \(\varphi\) function.

\(\varphi(0, \beta) = \omega^\beta\)

\(\varphi(\alpha + 1, \beta) =\) the \(\beta\)th ordinal in the set \(\lbrace \gamma | \varphi(\alpha, \gamma) = \gamma \rbrace \)

If \(\alpha\) is a limit ordinal, \(\varphi(\alpha, \beta) =\) the \(\beta\)th ordinal that is in the intersection of the sets \(\lbrace \gamma | \varphi(\delta, \gamma) = \gamma \rbrace \) for all \(\delta < \alpha\)

We can then define a "Veblen Normal Form" by expressing any ordinal \(\alpha\) in the form

\(\alpha = \varphi(\beta_1, \gamma_1) + \varphi(\beta_2, \gamma_2) + \ldots + \varphi(\beta_n, \gamma_n)\), where the terms are decreasing and \(\gamma_i < \varphi(\beta_i, \gamma_i)\) for all \(i\).

Let the ordinal \(\Gamma_\alpha\) be the \(\alpha\)th strongly critical ordinal, where an ordinal is strongly critical if it is in the set \(\lbrace \beta | \varphi(\beta, 0) = \beta \rbrace \). By expessing an ordinal \(\alpha\) in Veblen Normal Form, then expressing each of the subterms in Veblen Normal Form, etc., we can express any ordinal in terms of 0 and strongly critical ordinals. If \(\alpha < \Gamma_0\), then \(\alpha\) can be expressed in terms of 0 and \(\varphi\).

The Extended Veblen Function
We can extend the Veblen \(\varphi\) function by using more variables. For example, we can let \(\varphi(1, 0, \alpha) = \Gamma_\alpha\). In general, the Extended Veblen Function is defined using the following rules:

Let \(S\) be an arbitrary string of ordinals, and \(O\) be a string of 0's.

\(\varphi(\alpha) = \omega^\alpha\)

\(\varphi(O, S) = \varphi (S)\)

\(\varphi(S, \alpha+1, O, \beta)\) is the \(\beta\)th ordinal in the set \(\lbrace \gamma | \varphi(S, \alpha, \gamma, O) = \gamma \rbrace\)

If \(alpha\) is a limit ordinal, \(\varphi(S, \alpha, O, \beta)\) is the \(\beta\)th ordinal in the intersection of the sets \(\lbrace \gamma | \varphi (S, \delta, \gamma, O) = \gamma \rbrace\) for all \(\delta < \alpha\)

As before, we can define an "Extended Veblen Normal Form" by expressing an ordinal \(\alpha\) in the form

\(\alpha = \alpha_1 + \alpha_2 + \ldots + \alpha_n\)

such that each \(\alpha_i\) is an additively principal ordinal (that is, an ordinal of the form \(\omega^\beta\).) We then express each \(\alpha_i\) in the form \(\varphi(\alpha_{i1}, \alpha_{i2}, \ldots, \alpha_{ij})\) such that \(\alpha_{ik} < \alpha_i\) for all possible \(i\) and \(k\). The ordinals for which this is not possible are the ordinals \(alpha\) such that \(\varphi(\alpha, 0, \ldots, 0) = \alpha\) for any number of zeroes. Call this set the "extended critical ordinals". Extended Veblen Normal Form can express any ordinal in terms of 0 and the extended critical ordinals using the extended \(\varphi\) function.

The Schutte Klammersymbolen
Now that we have extended the Veblen \(\varphi\) function to arbitrarily many variables, the next step is to extend it to transfinitely many variables. But this leads to an obvious problem with notation; how do we describe a function with transfinitely many places? The solution is to explicitly associate each variable with another variable indicating the place index of the first variable. For example, we can define

\(\varphi(a, 0, 0, 0, b, 0, 0, c, 0, d, 0, e) = \left( \begin{array}{ccc} a & b & c & d & e\\ 11 & 7 & 4 & 2 & 0 \end{array} \right) \)

This allows us to define ordinals such as \( \left( \begin{array} {cc} 1 & \beta \\ \omega & 0 \end{array} \right) \), which denotes the \(\beta\)th extended critical ordinal.

The general rules for the Schutte Klammersymbolen are:

\( \left( \begin{array} {c} \alpha \\ 0 \end{array} \right) = \omega^{\alpha} \).

\( \left( \begin{array} {ccc} \ldots & \alpha+1 & \gamma \\ \ldots & \beta+1 & 0 \end{array} \right) \) is the \(\gamma\)th ordinal in the set \(\left{ \delta | \left( \begin{array} {ccc} \ldots & \alpha & \delta \\ \ldots & \beta+1 & \beta \end{array} \right) = \delta \right} \).

If \(\alpha\) is a limit ordinal, \( \left( \begin{array} {ccc} \ldots & \alpha & \gamma \\ \ldots & \beta+1 & 0 \end{array} \right) \) is the \(\gamma\)th ordinal in the intersection of the sets \(\left{ \delta | \left( \begin{array} {ccc} \ldots & \epsilon & \delta \\ \ldots & \beta+1 & \beta \end{array} \right) = \delta \right} \) for all \(\epsilon < \alpha\).

If \(\beta\) is a limit ordinal, \( \left( \begin{array} {ccc} \ldots & \alpha+1 & \gamma \\ \ldots & \beta & 0 \end{array} \right) \) is the \(\gamma\)th ordinal in the intersection of the sets \(\left{ \delta | \left( \begin{array} {ccc} \ldots & \alpha & \delta \\ \ldots & \beta & \epsilon \end{array} \right) = \delta \right} \) for all \(\epsilon < \beta\).

If \(\alpha\) and \(\beta\) are limit ordinals, \( \left( \begin{array} {ccc} \ldots & \alpha & \gamma \\ \ldots & \beta & 0 \end{array} \right) \) is the \(\gamma\)th ordinal in the intersection of the sets \(\left{ \delta | \left( \begin{array} {ccc} \ldots & \epsilon & \delta \\ \ldots & \beta & \eta \end{array} \right) = \delta \right} \) for all \(\epsilon < \alpha, \eta < \beta\).

As before, we can define "Schutte critical ordinals" as the ordinals in the set \(\left{ \alpha | \left( \begin{array} {c} 1 \\ \alpha \end{array} \right) = \alpha \right}\). We can then define a "Schutte Normal Form" by expressing any ordinal as a sum of Klammersymbolen (the matrices defined above) such that the represented ordinals are weakly decreasing and each ordinal is greater than the subterms of the Klammersymbol. This will allow us to represent any ordinal using 0 and Schutte critical ordinals with Klammersymbolen.