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

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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

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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,

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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

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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

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Binomial Theorem Corollaries

Here are some simple proofs based on the binomial theorem. They also demonstrate how amazing mathematical proofs can be. 1. The hockey stick identity By the formula for the sum of a geometric series, \(\frac{(1+r)^{n+1}-1}{(1+r)-1}=\displaystyle\sum_{i=0}^n (1+r)^i\). Now expanding both sides by the binomial theorem and simplifying, $$\displaystyle\sum_{i=0}^n \binom{n+1}{i+1}r^i=\displaystyle\sum_{i=0}^n \displaystyle\sum_{j=0}^i \binom{i}{j}r^j$$ Now rearrange the sum on

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Some Math Jokes

A \(K_5\) graph walks into a bar. A \(K_4\) graph follows it. The bartender points to the \(K_4\) and says, “Sorry, we don’t serve minors here.” A man goes to a yard sale. He sees a nice mug. He asks the seller, “How much does this cost?” The seller replies, “\$2” He then eyes a

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What is a Log?

Scientist: Log is the basis of the pH and Richter scales Mathematician: Log is the integral of $\frac{1}{x}$ Programmer: A log is generated by a program to help in debugging Lumberjack: Logs are what keep my buisness running IT professor: There are two types of log: log in and log out Camper: Logs are used

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