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I feel like I should introduce myself, but the user page already kinda does that.

Intro Stuff

Generally, I'm interested in generalised googology. Nothing below \varepsilon_0 has ever really interested me, and I like looking at novel or simple ways to reach limits for notations (via ordinals), or sometimes looking at how strong it's possible to make a notation.

A few of the things I plan doing/showcasing on this blog are:

  • An analysis of HAN (hopefully).
  • A new notation. I've designed it so that each level fits in well with a particular ordinal, and it's fairly easy to analyse (most levels are independently made, not some giant stack), but also reaches some very large ordinals with minimal effort.
  • A few posts on some interesting unary functions, that use methods of representing natural numbers as ordinals as a way to create fast growing unary functions.
  • A look at trees, and how they are useful in defining very fast-growing notations, and easy ways to collapse them.
  • Looking at comparing Goodstein sequences to an inverse-SGH, and using it to create a very fast growing, but easily understandable, function.
  • A quick look at the Veblen hierarchy, including a way to reach very large recursive ordinals (beyond \psi(\psi_I(0))) using only one collapsing ordinal \Omega.



Primitive Recursive Competition (not open)

The competition was to create the fastest growing function G, such that the functions F and G only use primitive recursive functions* and F in their definitions. It did not start.

I've not had any interest (completely unsurprising; I'm new here and I don't know how many people would be interested in it anyway), so it's withdrawn. By the way, It is possible to beat \(TREE(n)\approx. f_{\theta(\Omega^\omega\omega)}(n-2)\) using this. If anyone's still interested in this stuff, then go for it. I've kept the rest of the post for interest of anyone who happens to read it.

At the moment, I have plans on doing a few blog posts, and one of them in-particular is submitable to the competition, so I don't want to post it yet, in case people are interested in the competition. Submissions will probably be by email. I'm also going to award bonus points based on any interesting properties of F, such as only using the successor function and constant value functions in the definition, or only having one or two arguments.

*Primitive recursive functions are functions that have a set number of iterations to calculate, that can't be modified within the function and don't nest using themselves. Individual hyperoperators (tetration etc.) are primitive recursive, but generalised hyperoperators (where the number of up arrows is not fixed), and the Ackermann function (and anything that grows faster) are not.

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