Although I guess the actual surface could be made up of two rubber sheets (and some connections between them) and you could inflate the air between the two sheets.
@SimplyBeautifulArt Hello! I think my internet has been physically damaged - there was a van trying to fix it but it hasn't helped with the frequent dropping out.
@wizzwizz4 x'D why do you think they should other than "they are a natural continuation of basic arithmetic operations, natural in a mathematical sense."
Oh, btw, there's a neat property I want to tell you about the Hardy Hierarchy.
H(n,a+b) = H(H(n,b),a)
From this you can prove that: H(n,a∙ω) = H(n,a∙n) = H(n,a+a+...+a) = H(H(H(...),a),a)
So given that H(n,ω) = 2∙n = multiplication, we can see that H(n,ω²) = recursions of multiplication = exponentiation and in general, H(n,ω^k) ≈ (k+1)th hyperoperation.
Assuming the 1st hyperoperation is addition, and the second being multiplication.
From there you can get the idea that H(n,ω^ω) ≈ (n+1)th hyperoperation
Apply this n times and you get ≈ Graham's sequence.
You could troll them with something like "why the Busy Beaver function should be considered computable" and hand-wave about being able to apply a similar algorithm to Hash-Life to Turing Machines to reduce their computation complexity.
Actually, my program may still surpass TREE(3), but I want to be safe than sorry.
@fejfo More or less the issue is that while ψ'(Ω_1^Ω_1^Ω_1^...) would certainly surpass the level of TREE(3), it is difficult to recreate uncountable exponents using only recursive addition.
Especially with ordinal less than Ω_ω
And oh god this looks messy xD
I figured I could surpass this problem by allowing greater variety of subscripts.
But I now realize that's a lot more coding, and probably far from optimal.
I'm gonna go back to doing my homework and ponder on the problem.
gdaymath.com/courses/exploding-dots (Skip down to the "Lessons" sections.) When I started reading about this "exploding dots" thing, I was like "yeah, this is just typical number bases stuff". Then I was surprised by how easy division was. And then polynomials were easy too, which was a bit of a shock. The lessons ended with some really juicy problems.
@El'endiaStarman Hm, can I take a guess that these exploding dots are related to series? (I totally didn't open the link, but even if I did, I didn't play the video, just looked at the preview image.)
The basic idea is that you start with some dots in some boxes, and A <- B means that you can convert B dots in one box to A dots in the box immediately to its left.
