Let
- $G_{64}$ is a Graham Number:
https://googology.wikia.org/wiki/Graham%27s_number
- $TREE(3)$ is a particular value of a special sequence $TREE(k)$
https://googology.wikia.org/wiki/TREE_sequence
- $D^{5}(99)$ is a output of program loader.c
https://googology.wikia.org/wiki/Loader's_number
- $Rayo(googol)$ is a Rayo number
https://googology.wikia.org/wiki/Rayo%27s_number
How to prove that $1.<2.<3.<4.$ ?
First of all i know that there were several related question but in fact nobody has given a proper proof of any of these inequalities. That's why i wrote this question.
Let's take a look at all cases.
$1.<2.$
It is said that sequence that generates Graham numbers grows slower than sequence $TREE(k)$ But here are my questions:
- Where are the proofs, sources of this assertion?
- Even if some function $f$ grows faster than $g$ it doesn't prove that $f(n)>g(n)$ for certain value $n$.
$2.<3.$
Here is my idea to prove this:
First make a code that describes (not calculate) $TREE(3)$ Start it and see how much time it takes before you see a message (for example "Hello World") on the screen. The same with the code that describes $loader.c$. Compare these numbers. The number that has greater time is greater. Here are another problems.
- The code from the site i gave you in the link doesn't work on codeblocks.
- Is my reasoning even correct? If not( wich is very likely) then how to do this?
$3.<4.$
There is only one thing to know: an amount of symbols expressing $D^{5}(99)$ in language of First-Order-Set-Theory . If that number is smaller than $googol$ then proof follows.
- How we can show that?
Regards