October 2021

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 $\mathrm{cof}(x)=\mathrm{cof}(\mathrm{reduce}(x))$ and $x[\alpha ]=\mathrm{reduce}(x)[\alpha ]$. If $y$ is a limit, then $\mathrm{cof}(xy)=\mathrm{cof}(y)$ 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 $${\mathrm{\Omega }}_{\nu }=\{\begin{array}{ll}1& \text{if }n=0\\ {\mathrm{\aleph }}_{\nu }& \text{otherwise}\end{array}$$ And let $\mathrm{\Omega }={\mathrm{\Omega }}_1$. Then we define the function ${\psi }_{\nu }(\alpha )$ as the least ordinal that cannot be constructed by: All ordinals