# Ordinal notations (part 2) – θ function

θ function is a binary function. It’s defined as follows:

• ${C_0(\alpha,\beta)=\{\gamma|\gamma<\beta\}\cup\{0\}}$
• ${C_{n+1}(\alpha,\beta)=\{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}\cup\{\theta(\gamma,\delta)|\gamma<\alpha\land\gamma,\delta\in C_n(\alpha,\beta)\}\cup\{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\}}$
• ${C(\alpha,\beta)=\bigcup_{n<\omega}C_n(\alpha,\beta)}$
• ${\theta(\alpha,\beta)=min\{\gamma|\gamma\notin C(\alpha,\gamma)\land(\forall\delta<\beta:\gamma>\theta(\alpha,\delta))\}}$

Where ${\Omega_0=0}$ and ${\Omega_\alpha}$ means the αth uncountable cardinal.
${\Omega_1=\Omega}$ is the first uncountable ordinal and has cardinality ${\aleph_1}$${\Omega_2}$ is the smallest ordinal that has cardinality ${\aleph_2}$. Generally ${\Omega_\alpha}$ is the smallest ordinal that has cardinality ${\aleph_\alpha}$. They’re so large that ${\omega^{\Omega_\alpha}=\Omega_\alpha,\ \varepsilon_{\Omega_\alpha}=\Omega_\alpha,\ \varphi(\Omega_\alpha,0)=\Omega_\alpha}$, and for all ${\beta<\Omega_\alpha,\ \varphi(\beta,\Omega_\alpha)=\Omega_\alpha}$.

It means that, θ(α,β) is the (1+β)-th ordinal such that it cannot be built from ordinals less than it by addition, applying θ(δ,_) where δ < α and getting an uncountable cardinal.

# Explanation

To get ${\theta(0,\beta)}$ we need to deal with C(0,α). C0(0,0) already contains 0, so let’s begin from 1. For C(0,1), C0(0,1) only contains 0, adding and getting uncountable cardinals cannot get 1, so ${\theta(0,0)=1}$. For C(0,2), 1 is in it, so 2=1+1 is in it. And so on, 3 is in C(0,3), 4 is in C(0,4) ,etc. Until ω, it cannot be built from natural numbers by addition and getting uncountable cardinals, so ${\theta(0,1)=\omega}$. Further, ${\theta(0,\beta)=\omega^\beta}$.

${\theta(1,0)}$ is the least ordinal that can’t be built from ordinals less than it by addition and applying ${\theta(0,\beta)=\omega^\beta}$ (and getting uncountable cardinals) – that’s where the Cantor’s normal form cannot get. So ${\theta(1,0)=\varepsilon_0}$. Further, ${\theta(1,\beta)=\varepsilon_\beta}$.

It seems that ${\theta(\alpha,\beta)=\varphi(\alpha,\beta)}$ below ${\Gamma_0}$, making θ function an extension of φ function. Even ${\theta(\Gamma_0,\beta)=\varphi(\Gamma_0,\beta)}$ is true.

Beyond Feferman–Schütte ordinal

However, things suddenly go wrong. In ${C(\Gamma_0+1,\Gamma_0)}$, we cannot use ${\theta(\Gamma_0,\beta)}$, we can just use ${\theta(\alpha,\beta)}$ for ${\alpha<\Gamma_0}$, so ${\theta(\Gamma_0+1,0)=\Gamma_0}$.
From ${\Gamma_0+1}$ on, we can use ${\theta(\Gamma_0,\beta)}$ in ${C(\Gamma_0+1,\alpha)}$, so ${\theta(\Gamma_0+1,1)=\varphi(\Gamma_0+1,0)}$. And further, ${\theta(\Gamma_0+1,1+\beta)=\varphi(\Gamma_0+1,\beta)}$.

Then ${\theta(\alpha,0)=\Gamma_0}$ for ${\Gamma_0<\alpha<\Omega}$, and ${\theta(\alpha,1+\beta)=\varphi(\alpha,\beta)}$ for ${\Gamma_0<\alpha<\Gamma_1}$ – It meets another problem at ${\Gamma_1}$.

${\theta(\Gamma_1,0)=\Gamma_0,\theta(\Gamma_1,1)=\Gamma_1}$, and ${\theta(\Gamma_1,1+\beta)=\varphi(\Gamma_1,\beta)}$. Next, ${\theta(\Gamma_1+1,0)=\Gamma_0,\theta(\Gamma_1+1,1)=\Gamma_1,\theta(\Gamma_1+1,2)=\varphi(\Gamma_1+1,0)}$, and ${\theta(\Gamma_1+1,2+\beta)=\varphi(\Gamma_1+1,\beta)}$. Now this pattern ${\theta(\alpha,1+\gamma+\beta)=\varphi(\alpha,\beta)}$ where ${\Gamma_\gamma<\alpha\leq\Gamma_{\gamma+1}}$ will continue up to Ω, so it’s enough to define larger ordinals. But we cannot name, say, what’s the first fixed point of ${\alpha\mapsto\Gamma_\alpha}$, what’s LVO, what’s BHO, etc.

To express large ordinals only using θ function (and Cantor’s normal form), we need to deal with something beyond ${\theta(\Omega,0)}$. Using the result above, ${\theta(\Omega,\alpha)=\Gamma_\alpha}$ for all α < Ω. Now notice that C(Ω+1,α) contains Ω and can apply θ(Ω,α) in it, so ${\Gamma_0}$ is in ${C(\Omega+1,\Gamma_0)}$. In fact, ${\theta(\Omega+1,\beta)}$ is the (1+β)-th fixed point of ${\alpha\mapsto\Gamma_\alpha}$.

It seems that, if we write θ function in this way: ${\alpha=\theta(\gamma_1,\beta)}$, where ${\gamma_i=\omega^{\gamma_{i,1}}+\omega^{\gamma_{i,2}}+\cdots+\omega^{\gamma_{i,n_i}}+\omega^{\gamma_{i+1}}}$ with ${\gamma_{i,1}\geq\gamma_{i,2}\geq\cdots\gamma_{i,n_i}\geq\gamma_{i+1}}$ and ${\gamma_k=\Omega}$, then it’s the (1+β)-th fixed point of ${\alpha\mapsto\theta(\delta_1,0)}$ where ${\delta_i=\omega^{\gamma_{i,1}}+\omega^{\gamma_{i,2}}+\cdots+\omega^{\gamma_{i,n_i}}+\omega^{\delta_{i+1}}}$ and ${\delta_k=\alpha}$ itself. This pattern holds up to ${\theta(\Gamma_{\Omega+1},0)}$ – even beyond Bachmann-Howard ordinal. For example, ${\theta(\Omega^\Omega,0)=\theta(\omega^{\omega^{\Omega2}},0)}$ is the first fixed point of ${\alpha\mapsto\theta(\omega^{\omega^{\Omega+\alpha}},0)}$, which is called large Veblen ordinal. The small Veblen ordinal is ${\theta(\Omega^\omega,0)=\theta(\omega^{\omega^{\Omega+1}},0)}$, and the Bachmann-Howard ordinal is ${\theta(\varepsilon_{\Omega+1},0)}$.

Beyond Bachmann-Howard ordinal

The (1+β)-th fixed point of ${\alpha\mapsto\theta(\varphi(\alpha,\Omega+1),0)}$ is ${\theta(\varphi(\Omega,1),\beta)}$. For ${\alpha<\Omega,\ \theta(\Omega,\alpha)=\Gamma_\alpha}$, but ${\theta(\Omega,\Omega+\alpha)=\varphi(\Omega,1+\alpha)}$ – The same thing at ${\Gamma_0}$ happens at Ω now.

Another important point is ${\theta(\Omega_2,0)}$. What is it? ${\theta(\Gamma_{\Omega+1},0)}$ cannot be built from ordinal less than it and Ω by addition and ${\theta(\alpha,\beta)}$ where ${\alpha<\Gamma_{\Omega+1}}$, so it’s ${\theta(\Gamma_{\Omega+1},0)}$. What’s more, for ${\Gamma_{\Omega+1}\leq\alpha<\Omega_2,\ \theta(\alpha,0)=\theta(\Omega_2,0)}$.
Then ${\theta(\Gamma_{\Omega+1},1)}$ is the 2nd ordinal that cannot be built from ordinal less than it and Ω by addition and ${\theta(\alpha,\beta)}$ where ${\alpha<\Gamma_{\Omega+1}}$ – it’s ${\theta(\Omega_2,1)}$. What’s more, for β < Ω and ${\Gamma_{\Omega+1}\leq\alpha<\Omega_2,\ \theta(\alpha,\beta)=\theta(\Omega_2,\beta)}$.
${\theta(\Omega_2,\Omega)=\Gamma_{\Omega+1}}$, and ${\theta(\Omega_2,\Omega+1)=\Gamma_{\Omega+2}}$. And for ${\alpha<\Omega_2,\ \theta(\Omega_2,\Omega+\alpha)=\Gamma_{\Omega+1+\alpha}}$. Note that ${\Omega\in C(\alpha,\Omega)}$ for any α!

The next step is ${\theta(\Omega_2+\Gamma_{\Omega+1},0)}$. Since ${\Gamma_{\Omega+1}=\theta(\Omega_2,\Omega)}$, it can be built from ordinals below it and Ω and Ω2 by addition and ${\theta(\alpha,\beta)}$ where ${\alpha<\Omega_22}$, so it’s not ${\theta(\Omega_22,0)}$ – there’re many things between them, such as ${\theta(\Omega_2+\Gamma_{\Omega+2},0)}$, ${\theta(\Omega_2+\theta(\Omega_2+1,\Omega),0)}$, ${\theta(\Omega_2+\theta(\Omega_2+\Gamma_{\Omega+1},\Omega),0)}$, ${\theta(\Omega_2+\theta(\Omega_2+\theta(\Omega_2+\Gamma_{\Omega+1},\Omega),\Omega),0)}$, etc.

Then we turn to ${\theta(\Gamma_{\Omega_2+1},0)}$. For β < Ω2 and ${\Gamma_{\Omega_2+1}\leq\alpha<\Omega_3,\ \theta(\alpha,\beta)=\theta(\Omega_3,\beta)}$, but there’re many things between ${\theta(\Omega_3+\Gamma_{\Omega_2+1},0)}$ and ${\theta(\Omega_32,0)}$.

Beyond θ(Ωω,0)

The “getting an uncountable cardinal” in definition start to works now. ${\theta(\Omega_\omega,0)}$ is the supremum of all the ${\theta(\Omega_n,0)}$ for natural number n.

${\theta(\Omega_\Omega,0)}$ is not ${\theta(\Omega_{\Gamma_0},0)}$ because we have ${\Gamma_0}$, then ${\Omega_{\Gamma_0}}$, then ${\theta(\Omega_{\Gamma_0},0)}$ in ${C(\Omega_{\Omega},\theta(\Omega_{\Gamma_0},0))}$. In fact, ${\theta(\Omega_\Omega,\beta)}$ is the (1+β)-th fixed point of ${\alpha\mapsto\theta(\Omega_\alpha,0)}$.

Also, ${\theta(\Omega_{\Omega_2},0)}$ is not ${\theta(\Omega_{\Gamma_{\Omega+1}},0)}$; it’s ${\theta(\Omega_{\alpha\mapsto\theta(\Omega_\alpha,\Omega)},0)}$.

The limit of θ function is the supremum of ${\theta(\Omega,0),\ \theta(\Omega_\Omega,0),\ \theta(\Omega_{\Omega_\Omega},0)}$, etc.