User blog:Edwin Shade/How High Can You Go ?

Earlier today I decided to come up with the largest number I could. I made a resolve to myself that each successive number I wrote down would be significantly larger than the last.

Of course, I knew mashing together preexisting numbers in the hopes of coming up with something larger was impractical, so I decided to see how far I could go with the fast growing hierarchy. I first wrote down $$f_10(10)$$, then $$f_\omega+1(10)$$, after which I wrote down $$f_{\epsilon_0}(10)$$. I kept doing this with the zetas, etas, and so on until it became apparent that I needed a stronger notation.

It was then that I switched to Veblen notation and wrote down the next number as $$f_{\varphi(10,10)}(10)$$. After this I took the next big leap and wrote down $$f_{\Gamma_0}(10)$$, which denotes recursion of the Veblen functions. Using extended Veblen notation however, I was able to write down the number $$f_{\varphi(1,0,0,0,0,0,0,0,0,0)}(10)$$, which is likewise a large leap forward.

This was not enough for me though, so I used the psi-notation shown in this video and wrote down $$f_{\Psi(\Omega^{\Omega^{\Omega}})}(10)$$, or the fast-growing hierarchy with the Large Veblen Ordinal in the subscript. This is currently the highest number I can comprehend and, if given an infinite amount of ink and paper, would be able to calculate and write out in full.

This though is where I'm stuck, and I would like to know what comes next ? I would like to know the fastest growing current notations and functions for the fast-growing hierarchy and where I can find basic explanations of them.

I will update this blog post accordingly as I learn stronger notations for the fast-growing hierarchy.