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# LittlePeng9

## aka Penguin!

My favorite wikis
• I live in Poland
• I was born on May 23
• My occupation is http://youtu.be/sDj72zqZakE
• I am your local penguin
• ## Formalizing googology - naming schemes

April 1, 2018 by LittlePeng9

For quite a while we have had around here users which were complaining about the low level of formality when it comes to many things on the wiki (I admit I was one of them, though perhaps not the most notorious one). Today I wanted to propose a long-term project whose goal would be to formalize most (perhaps all) of googology in order to put it on par with other branches of mathematics.

In this post I would like to start off this project by outlining a formal foundation to one of the more popular parts of googology, especially among newcomers - the theory of naming schemes. It turns out that it can be explained in the language of algebraic geometry, a widely accepted branch of mathematics.

Algebraic geometry has its own notion of a scheme, a…

• ## Long hierarchies of functions (on my other blog)

October 31, 2017 by LittlePeng9

I thought this might be of interest to some of you, so I'm going to mention it here.

Yesterday I've put up a new blog post on my personal blog, which you can find here. It shows existence of hierarchies of functions somewhat akin to SGH which however have length exceeding $$\omega_1$$. To be more precise, I show existence of hierarchies of any length below $$\omega_2$$ in which functions higher up eventually dominate functions below them. There is also a section about computable functions containing such hierarchies of any length below $$\omega_1$$.

If you have any questions or comments feel free to post them here or on my blog.

• ## Hard-coding long binary strings

October 29, 2017 by LittlePeng9

Today I have discovered a way of hard-coding binary strings into Turing machines with asymptotically fewer states. More precisely, I've found the following:

For any string of $$2^k\cdot l$$ bits, there is a Turing machine with three colors (0, 1 and blank) and $$2^k+k+4\cdot 2^l+1$$ states which, on blank input, writes that string of bits and halts with its head on the first character of the string.

By simulating a three-color Turing machine with a two-color one and adding a few more steps to take care of deciphering the string, we can show the following (I don't bother with establishing the constants):

For any $$l$$ and large enough $$k$$ (depending on $$l$$), for any string of $$2^k\cdot l$$ there is a Turing machine with at most $$2^k$$ st…
Suppose we have chosen a system of fundamental sequences for countable limit ordinals up to some bound. We now construct a sequence $$f_\alpha:\mathbb C\rightarrow\mathbb C$$ which satisfies the following propertes:
1. $$f_\alpha$$ is an ,