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Notation using the \(\Phi\) function

To go beyond \(\theta (\Omega_{\Omega_{\Omega_\ldots}}, 0)\), we must create a notation for ordinals beyond \(\Omega_{\Omega_{\Omega_\ldots}}\).  We can define a Veblen hierarchy based on \(\Omega_\alpha\):

\(\Phi(0, \beta) = \Omega_{\beta}\)

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

For \(\alpha\) a limit ordinal, \(\Phi(\alpha, \beta) = \) the \(\beta\)th ordinal in the intersection of the sets \(\lbrace \gamma | \Phi(\delta, \gamma) = \gamma \rbrace \) for \(\delta < \alpha\)

We can then extend our notation:

\(C_0 (\alpha, \beta) = \beta \)

\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Phi(\gamma, \delta), \theta(\eta, \gamma) | \gamma, \delta, \eta \in C_n (\alpha, \beta); \eta < \alpha \rbrace \)

\(C (\alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n (\alpha, \beta) \)

\(In (\alpha) = \lbrace \beta | \beta \notin C (\alpha, \beta) \rbrace \)

\(\theta (\alpha, \beta) = \) the \(\beta\)th ordinal in \(In (\alpha)\)

Notation using an inaccessible

We can extend our notation by defining an Extended Veblen function or Schutte Klammersymbolen function starting from the function \(f(\alpha) = \Omega_\alpha\). But the Bachmann-Howard hierarchy is stronger than those functions, so we would like a similar version here. To achieve this, we need a "big" ordinal; just as \(\Omega\) was useful to diagonalize over countable ordinals because it was larger than any recursive extension of \(\alpha \mapsto \omega^\alpha\), we need an ordinal larger than any recursive extension of \(\alpha \mapsto \Omega_\alpha\) so we can diagonalize over it. This is where the notion of weakly inaccessible comes in.

We have previously defined the cofinality of an ordinal \(\alpha\) as the cardinality of the smallest set of ordinals less than \(\alpha\) such the greatest lower bound of the set of ordinals is \(\alpha\). A regular ordinal is an ordinal whose cofinality is itself; otherwise, the ordinal is called singular. A limit cardinal is a cardinal of the form \(\Omega_\alpha\) where \(\alpha\) is a limit ordinal; a successor cardinal is a cardinal of the form \(\Omega_\alpha\) where \(\alpha\) is a successor ordinal. It turns out that, in ZFC, a successor cardinal is always regular. A limit cardinal is almost always singular; in fact, it is consistent with ZFC that every limit cardinal is singular. But it is possible that there exist cardinals that are both regular and limit; such cardinals are called weakly inaccessible. Weakly inaccessible cardinals are larger than any cardinal constructible in ZFC, and a weakly inaccessible cardinal is larger than any cardinal constructible in ZFC using smaller weakly inaccessible cardinals.

So, if we let I be the smallest weakly inaccessible cardinal, we are guaranteed that I will be "large enough" for our purposes. We define the following version of \(\psi\), as defined by Michael Rathjen:

\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0, I \rbrace\)

\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, \psi_\pi(\eta) | \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace \)

\(C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) \)

\(\psi_\pi (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace) \)

This is more similar to the \(\vartheta\) function than the \(\psi\) function of Pohlers, but Rathjen uses \(\psi\). Note the slight change in definition; whereas before \(\vartheta_\nu\) collapsed ordinals to ordinals of cardinality \(\Omega_\nu\) or less, \(\psi_\pi\) collapses ordinals to ordinals of cardinality less than \(\pi\). This allows us to set \(\pi = I\), and generate large ordinals less than I.

\(\psi_I(0)\) is defined as the smallest \(\beta\) such that \(C(0, \beta)\) does not generate any ordinals between \(\beta\) and I. Since we can repeatedly apply \(\alpha \mapsto \Omega_\alpha\), \(\beta\) must be larger than \(0, \Omega_0, \Omega_{\Omega_0}, \Omega_{\Omega_{\Omega_0}}, \ldots\). So \(\beta\) must be at least \(\Omega_{\Omega_{\Omega_\ldots}}\). On the other hand, if we apply \(\gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}\) to ordinals less than \(\Omega_{\Omega_{\Omega_\ldots}}\), we'll still get ordinals less than \(\Omega_{\Omega_{\Omega_\ldots}}\), so in fact \(\psi_I(0) = \Omega_{\Omega_{\Omega_\ldots}}\).

\(\psi_I(1)\) is defined the same way as \(\psi_I(0)\) except that \(C(1, \beta)\) contains \(\psi_I(0)\). So we need to go to the second fixed point of \(\alpha \mapsto \Omega_\alpha\). Similarly, \(\psi_I(\alpha)\) is the \(1+\alpha\)th fixed point of \(\alpha \mapsto \Omega_\alpha\). \(\psi_I(I)\) diagonalizes over \(\psi_I(\alpha)\), so \(\psi_I(I + \alpha)\) is the \(1 + \alpha\)th fixed point of \(f(\beta) = \psi_I(\beta)\). And the hierarchy continues for larger and larger ordinals based on I. An important ordinal is the proof theoretic ordinal of the theory KPI, which is \(\psi_{\Omega_1} (\varepsilon_{I + 1})\).

To generate larger functions of I, we can go to \(\psi_{\Omega_{I+1}}(\alpha)\) and higher; \(\psi_{\Omega_{I+1}}(\alpha) = \Gamma_{I+1+\alpha}\), then \(\psi_{\Omega_{I+1}}(\Omega_{I+1})\) diagonalizes over that function, and so on.