User blog:Fejfo/Meta-thinking about large numbers

So let's forget every thing we've learnt and do some thinking about how we can create a huge number, we might find a new angle from which to look at things (instead of the standard array notations, ordinals and ordinal notations, largest number named by system X, ect). I won't get into any specifics here, none of this is formal or well defined in any way. Use it if you're ever short on inspiration.

So lets start. We need a large number. A good way to create one might be to take a "fast" growing function \( f \), like exponentiation with a large input. This way we can keep generating larger and large numbers.

So we've reduced the problem to finding fast growing functions. Wouldn't it be conveniant if we had a "second order function" which would take one of our large numbers and output a faster growing function. Like \( n to f^n \).

We can keep going this way, 3ʳᵈ order functions which output better second order functions,

Nth order- functions

If I'm not mistaken this is exactly the approach taken by a Fish number, so I'm not original after all...

1,0 - order functions, which output n-th order functions

1,n+1 - order, which output better 1,n- order

1,0,0 - order

...

Lets just, skip the lists of natural numbers and go straight to ordinals

\( 1,0 \to \omega \)

\( 1,0,0 \to \omega^2 \)

Ect.

\( \alpha+1 \) order functions ouput better \( \alpha \) ordered ones.

\( \alpha \) order ouput \( \alpha[n] \) for countable limit ordinals

Now it's clear it's all about ordinals, so we should expand our starting functions to work on ordinals:

\( f \) the succesor function

second order: transfinite interation

ect (I'm not sure how to generalise these ideas but I'm sure it's possible)

Next up should eventually be things like

\( \Omega \) order which would then output \( \alpha \) ordered functions for countable \( \alpha \)

I'm sure there is a way to walk up the large cardinal chain in some way here, but ultimately I don't think it'll be of much use as the \( \alpha\) orderd functions won't be much stronger than \( f_\alpha \) in the fast growing hierachy, so it would just be a lot of efford for not much gain.

What do you think about this kind of "lower quality" less thought out post? Are the ideas still interesting? Hope, you learned somthing (or learn me something)