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  • 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…

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  • 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.

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  • 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…
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  • LittlePeng9

    Extending the hyperoperators to all real or complex values is a known open problem. Some time ago I had a somewhat converse idea on how to approach this - instead of starting with a fast growing function(s) on natural numbers and extending it to more inputs, we can directly construct a sequence of fast growing functions defined for all complex values. I do not claim this is of as much interest as extending hyperoperators, but might nevertheless be interesting for some.

    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 ,
    2. \(f_\alpha(z)\in(0,\infty)\) for \(z\in\m…
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  • LittlePeng9

    Because of Emlightened's recent blog post I was pushed to think again about FOOT, truth predicates and related. The conclusion I came to is:

    FOOT is not as strong as I thought it is. Indeed, it is about as strong as FOST with a single truth predicate added. However, it still seems that at the time of definition BIG FOOT was the largest number explicitly defined (in particular, larger than Fish number 7), and possibly still is.

    I will address the three parts separately below.

    When defining FOOT, I first defined \(\text{Ord}\), on order to be able to define the truth predicate with help of it to be able to define Rayo's function. As Emlightened points out, we can't get the full truth predicate this way - we only get it for formulas with paramet…

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