User blog:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals

I introduce a very easy technique to define an ordinal notation associated to a proper class of ordinal numbers. This blog post is just for beginners of googology who knows usual mathematics, and hence deals with a countable ordinals smaller than SVO.

Throughout this blog post, I work in ZFC set theory. Although the notion of a class is not a term in ZFC set theory, I use the traditional covention on a class in ZFC set theory.

= Convention =

I denote by \(\textrm{ON}\) the proper class of ordinal numbers. Let \(X \subset \textrm{ON}\) be a proper class. I will construct an ordinal notation associated to \(X\) and a class \(F(X) \subset X\).

For a class \(Y \subset \textrm{ON}\), I denote by \(\textrm{Lim}(Y) \subset Y\) the class of limit points of \(Y\), and put \(\textrm{cl}(Y) := Y \cup \textrm{Lim}(Y)\). I say that \(Y\) is closed if \(\textrm{cl}(Y) = Y\) holds.

= Enumeration Function =

I denote by \begin{eqnarray*} \textrm{enum}(X) \colon \textrm{ON} & \twoheadrightarrow & X \\ \alpha & \mapsto & \textrm{enum}(X)[\alpha] \end{eqnarray*} the surjective function recursively defined by the following transcendental induction: \begin{eqnarray*} \textrm{enum}(X)[\alpha] := \min \left( X \setminus (\{0\} \cup \{ \textrm{enum}(X)[\alpha'] \mid \alpha' \in \alpha \}) \right) \end{eqnarray*} Then \(\textrm{enum}(X)) is a strictly increasing function. Moreover, it is Scott continuous if and only if \(X\) is closed.

= Iterated Enumeration Function =

I denote by \begin{eqnarray*} C_X \colon \textrm{ON} \times \textrm{ON} & \twoheadrightarrow & X \\ (\alpha,\beta) & \mapsto & C_X(\alpha,\beta) \end{eqnarray*} the surjective function recursively defined by the following transcendental induction: \begin{eqnarray*} C_X(\alpha,beta) := \textrm{enum}(\{ \gamma \in X \mid \forall \alpha' \in \alpha, C_X(\alpha',\gamma) = \gamma \})[\beta] \end{eqnarray*} Then it is obvious by definition that for any \(\alpha,\beta,\gamma,\delta \in \textrm{ON}\), the equality \(C_X(\alpha,\beta) = C_X(\gamma,\delta)\) holds if and only if either one of the following holds: - \(alpha < \gamma\) and \(\beta = C_X(\gamma,\delta)\) - \(alpha = \gamma\) and \(\beta = \delta\) - \(alpha > \gamma\) and \(C_X(\alpha,\beta) = \delta\)

= Fixed Points =

I denote by \(F(X) \subset X\) the class \(\{ \alpha \in X \mid C_X(\alpha,0) = \alpha\}\), and by \begin{eqnarray*} D_X \colon \textrm{ON} & \to & X \\ \alpha & \mapsto & D_X(\alpha) \end{eqnarray*} the strictly increasing function defined as \begin{eqnarray*} D_X(\alpha) := C_X(\alpha,0). \end{eqnarray*} If \(X\) is closed, then \(D_X\) is Scott continuous, and hence \(F(X)\) is a proper class by a straightforward extension of Kleene's fixed point theorem.

Assume that \(X\) is closed in the following. Then by the argument above, \(F(X)\) is a proper class, and hence all of the constructions for \(X\) above are also valid for \(F(X)\). (If you want to iterate \(F\) in a transfinite way, then you need to be carful about the fact that \(X\) is a proper class.)

= Reduction =

I put \(\kappa_X := \textrm{enum}_{F(X)}(0) \in X\). Then for any \(\alpha \in \kappa_X \setminus \{0\}\), one has \(D_X(0) \subset \alpha \in D_X(\alpha)\). Therefore \(\{ \alpha' \in \textrm{ON} \mid D_X(\alpha') \subset \alpha \}\) is a non-empty closed subset of \(\textrm{ON}\), and its maximum \(r_1(\alpha)}\) satisfies \(r_1(\alpha) \in \alpha).

By the definition of \(r_1(\alpha)\), \(\{\beta \in \textrm{ON} \mid C_X(r_1(\alpha),\beta) \subset \alpha \}\) is a non-empty closed subset of \(\textrm{ON}\), and its maximum \(r_2(\alpha)\) satisfies \(C_X(r_1(\alpha),r_2(\alpha)) \subset \alpha).

I denote by \(r_3(\alpha) \in \alpha\) the unique element with \(\alpha = C_X(r_1(\alpha),r_2(\alpha)) + r_3(\alpha)\). Then \(r_1\), \(r_2\), and \(r_3\) are strictly decreasing function \(\kappa_X \setminus \{0\} \to \kappa_X\), and hence the well foundedness of \(\kappa_X\) ensures the termination of the presentation of \(\alpha\) using the symbols \(C_X\), \(+\), and \(0\).

= Ordinal Notation =

I denote by \(\textrm{T}_X\) the smallest set satisfying the following properties: - For any elements \(\alpha\) and \(\beta\) of \(\textrm{T}_X\), the vertical array \(\letf\langle \begin{array}{c} \alpha & \beta \end{array} \right\rangle\) lies in \(\textrm{T}_X\). I call an element of this type a primitive term of \(\textrm{T}_X\). - For any sequence \((\alpha_i)_{i \in n}\) of primitive terms of \(\textrm{T}_X\) of length \(n \in \omega\), the horizontal array \((\alpha_0,\ldots,\alpha_{n-1}\)\) lies in \(\textrm{T}_X\). I note that in the second condition, \((\alpha_0,\ldots,\alpha_{n-1})\) in the case \(n = 0\) means the empty sequence \(\).

I define a strict partial order \(<_X\) on \(\textrm{T}_X\) by the following property: - \(\letf\langle \begin{array}{c} \alpha & \beta \end{array} \right\rangle <_X \letf\langle \begin{array}{c} \gamma & \delta \end{array} \right\rangle\) holds if and only if either one of the following holds: -- \(\alpha <_X \gamma\) -- \(\alpha = \gamma\) and \(\beta <_X \delta\) - \((\alpha_0,\ldots,\alpha_{n-1}) <_X \letf\langle \begin{array}{c} \gamma & \delta \end{array} \right\rangle) holds if either one of the following holds: -- \(n = 0\) -- \((n > 0) \wegde \left( \alpha_0 <_X \letf\langle \begin{array}{c} \gamma & \delta \end{array} \right\rangle \right)\) - \(\letf\langle \begin{array}{c} \alpha & \beta \end{array} \right\rangle <_X \gamma\) holds if and only if either one of the followig holds: -- \((n > 0) \wedge \left( \letf\langle \begin{array}{c} \alpha & \beta \end{array} \right\rangle <_X \alpha_0 \right)\) -- \((n > 1) \wedge \left( \letf\langle \begin{array}{c} \alpha & \beta \end{array} \right\rangle = \alpha_0 \right)\) - \((\alpha_0,\ldots,\alpha_{n-1}) <_X (\beta_0,\ldots,\beta_{m-1})\) holds if and only if either one of the following holds: -- \((n < m) \wedge (\forall i \in n, \alpha_i = \beta_i)\) -- \(\exists k \in n \cap m, (\forall i \in k, \alpha_i = \beta_i) \wedge (\alpha_k <_X \beta_k)\)

It is easy to define a standard form \(\textrm{OT}_X \subset \textrm{T}_X\) in the same way as in the section 2 of Buchholz's paper, and hence I omit it.

I denote by \begin{eqnarray*} o \colon \textr{T}_X & \to & \kappa_X \\ \alpha & \mapsto & o(\alpha) \end{eqnarray*} the map defined in the following way: - \(o \left( \letf\langle \begin{array}{c} \alpha & \beta \end{array} \right\rangle \right) := C_X(o(\alpha),o(\beta))\) - \( o(\alpha_0,\ldots,\alpha_{n-1}) := o(\alpha_0) + \cdots + o(\alpha_{n-1})\) Then \(o\) is an order-preserving surjective map (and its restriction to \(\textrm{OT}_X\) is bijective.)).

= Example =

- \(X = \Lim(\textrm{ON})\) -- \(C_X(\alpha,\beta) = \omega^{\omega^{\alpha}} \times (1 + \beta) \) -- \(D_X(\alpha) = \varepsilon_{\alpha} \) -- \(\kappa_X = \varepsilon_0\) -- \(F(X) = \{ \varepsilon_{\alpha} \mid \alpha \in \textrm{ON} \}\) - \(X = \{ \varepsilon_{\alpha} \mid \alpha \in \textrm{ON} \}\) -- \(C_X(0,\beta) = \varepsilon_{\beta}\) -- \(C_X(1,\beta) = \zeta_{\beta}\) -- \(C_X(\alpha,\beta) = \psi_{1 + \alpha}(\beta)\) (Veblen function) -- \(D_X(\alpha) = \Gamma_{\alpha} = \psi(1,0,\alpha)) (multi-variable Veblen function) -- \(\kappa_X = \Gamma_0\) -- \(F(X) = \{ \Gamma_{\alpha} \mid \alpha \in \textrm{ON} \}\) - \(X = \{ \Gamma_{\alpha} \mid \alpha \in \textrm{ON} \}\) -- \(C_X(\alpha,\beta) = \psi(1,\alpha,\beta)\) -- \(D_X(\alpha) = \psi(2,0,\alpha)\) -- \(\kappa_X = \psi(\gamma,0,0)\) -- \(F(X) = \{ \psi(\gamma,0,\alpha) \mid \alpha \in \textrm{ON} \}\) - \(X = \{ \psi(\gamma,0,\alpha) \mid \alpha \in \textrm{ON} \}\) -- \(C_X(\alpha,\beta) = \psi(\gamma,\alpha,\beta)\) -- \(D_X(\alpha) = \psi(\gamma + 1,0,\alpha)\) -- \(\kappa_X = \psi(\gamma + 1,0,0)\) -- \(F(X) = \{ \psi(\gamma + 1,0,\alpha) \mid \alpha \in \textrm{ON} \}\)

= Refereces =