Ideal OCF attempt

Ideal OCF \(\kappa\) is hereditarily \(\Pi_n\)-reflecting if and only if it is \(\Pi_n\)-reflecting or \(\Pi_m\)-reflecting on the hereditarily \(\Pi_n\)-reflecting ordinals. \(C_\kappa(\alpha)\) is the closure of \(\kappa\cup{\text{least }\Pi_n\text{-reflecting}\mid n>1}\) under \(\beta,\gamma\mapsto\beta+\gamma\) and \(\beta,\gamma,\delta\mapsto\chi_\beta(\gamma,\delta)\) provided \(\gamma<\alpha\). Suppose \(\kappa\) is \(\Pi_{n+1}\)-reflecting (\(n>0\)) but not \(\Pi_{n+2}\)-reflecting. Then set \(\mathcal A_\kappa^\alpha\) to be the set of \(\pi<\kappa\) such that: If […]

Ideal OCF attempt Read More »

Simplified Reflection OCF

I’m back. Surprise! I’m making a simplified collapsing function reaching $\Pi_\omega$-ref. $K$ denotes the least $+1$-stable, First, let’s set up our reflection configurations. We define them by stating that for ordinal $\kappa$, integer $n>0$, ordinal $\alpha$, and reflection configurations $\mathbb{X},\ldots$, $(\kappa,n,\alpha,\mathbb{X},\ldots)$ is a reflection configuration, provided $\kappa$ is $\Pi_{n+1}$-reflecting on $\mathcal A\mathbb X\cap\ldots$. The variables

Simplified Reflection OCF Read More »

Beyond the stability OCFs

The post is meant to extend https://googology.fandom.com/wiki/User_blog:Hyp_cos/Attempt_of_OCF_up_to_Stability. Firstly, I’ll verify that it’s worth extending. I compared it with the other OCFs (Stegert, Duchhardt). It is virtually identical to Stegert, if we associate the second reflection instance element be the position in the hierarchy. The only worrying difference is that Stegert’s hierarchies require their elements to

Beyond the stability OCFs Read More »

Fixed arrow OCF

The arrow OCF has some issues, so I give a fixed version. Let \(B_0(\beta)=\{0,\omega_1,\omega_2,\ldots,\omega_\omega\}\). \(B_{n+1}(\beta)=\{x,x+y,xy,x\uparrow^zy:x,y,z\in B_n(\beta)\land z<\beta\}\). \(B(\beta)=\beta\cap\bigcup_{n<\omega} B_n(\beta)\). and \(C_0(\alpha,\beta)=\alpha+1\). \(C_{\epsilon+1}(\alpha,\beta)=\{x,x+y,xy,x\uparrow^zy:x,y\in C_\epsilon(\alpha,\beta)\land z\in B(\beta)\}\). If \(\epsilon\) is a limit, \(C_\epsilon(\alpha,\beta)=S\cup\operatorname{limits}(S)\), where \(S=\bigcup_{\zeta<\epsilon}C_\zeta(\alpha,\beta)\). Then \(\alpha\uparrow^0\beta=1\) and \(\alpha\uparrow^{1+\gamma}\beta=\sup(C_\gamma(\alpha,\beta))\).

Fixed arrow OCF Read More »

How to make an OCF

Making an OCF is easy. The idea of pre-inaccessible OCFs (OCFs without inaccessibles) are easy to grasp. Let’s define a function \(\psi(\alpha)\) as the smallest ordinal you can’t name with 0, addition, and \(\psi\) to smaller arguments. The idea is when we run out of names, we just give a name to the next ordinal

How to make an OCF Read More »

The arrow OCF

Not-so-recently the Knuth arrow notation was extended to infinite ordinals. The definition is here. It’s pretty straightforward, and the precise details are unimportant for this post. Anyways, we have an aordinal function. (Conjecturally on par with the two-argument Veblen function.) So, we can pull a Solomon Feferman and define an OCF which extends it. Conjecturally,

The arrow OCF Read More »

Transfinite combinators and admissible ordinals

I’m back with another topic concerning ordinals! The transfinite combinator calculus The transfinite combinator calculus is an extension of normal combinator calculus. We define several related concepts as follows: If an expression \(x\) is beta-reducible, then \(\operatorname{cof}(x)=\operatorname{cof}(\operatorname{reduce}(x))\) and \(x[\alpha]=\operatorname{reduce}(x)[\alpha]\). If \(y\) is a limit, then \(\operatorname{cof}(xy)=\operatorname{cof}(y)\) and \((xy)[\alpha]=x(y[\alpha])\). If \(x\) is a limit, \(y\) is

Transfinite combinators and admissible ordinals Read More »


Hello, I’m back (who could have expected?) with my new OCF (Not an ordinal notation.). We follow several conventions from the world of OCFs. First, let $$\Omega_\nu=\begin{cases} 1 & \mbox{if }n=0 \\ \aleph_\nu & \mbox{otherwise}\end{cases}$$ And let \(\Omega=\Omega_1\). Then we define the function \(\psi_\nu(\alpha)\) as the least ordinal that cannot be constructed by: All ordinals

An OCF Read More »