Ordinals

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 […]

There’s a 2009 article by Arai on interated recursive Mahloness. The main idea of the paper is that stronger systems of reflection can be analysed in terms of iterated $\Pi_2$-reflection. In section 5, Arai writes the following: A $\Pi_3$-reflecting ordinal $K$ is understood to be …

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

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

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 \(\alpha\uparrow^{1+\gamma}\beta=\sup(C_\gamma(\alpha,\beta))\).

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

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,

A problem I may need to rectract the results of the previous blog post because the transfinitary combinator calculus is way too strong. Let’s take a certain combinator. This maps an ordered pair \((a,b)\) where \(b\) is a representation of true or false and \(a\) is a representation of a Turing machine state. The combinator

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

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