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.


   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.


One thought on “Ordinal notations (part 2) – θ function

  1. Aarex Tiaokhiao says:

    For collapsing weakly inaccessible cardinals:
    psi_I(a) = a+1-th fixed point of W_W_….
    I is the cardinals that used in psi_I() or over.
    I_n is the next weakly inaccessible cardinal than I_n-1.



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