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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.

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