# Secret Passage found to achieve Epsilon 0 in SPS (Now Hyper N-ary Sequence)

Analysis (The system is called Snap Primitive System / Hyper N-ary Sequences and this is version 2, for version 1, here it is, and its level is $$\omega^{\omega^{8}}$$):

Snap Project with the expansions:
I realized if we change the last rule of the system to instead subtracting everything after the 1 in the bad part subtracted by 1, expanding that, and then adding it by one again, we can achieve $$\varepsilon_0$$! Although it's just a small change, it achieves what I originally wanted! Epsilon 0 without nesting lists! Note that some of the arrays that were valid in V1 aren't valid in V2. For example, 1,3 can't be expanded because then you have to expand the "3" part, and it can't expand. This means that in a sequence, each next number can't be added by a number more than one, but can be subtracted by a number equal to or greater than zero.1,2,3,4,5,6,7,8,9,10,... is $$\varepsilon_0$$. The new script:

WE DID IT, WE ACHIEVED EPSILON 0!
WELL, CRAP. YOU'RE RIGHT. YOU WILL BE NEXT, ZETA 0!

... Aren't eta 0 and zeta 0 the same thing?

What?

I thought we were just trying to get higher on the greek alphabet.

... Oh. I was talking about going higher in transfinite ordinals.

$$\eta_0=\sup\left\lbrace\alpha\mapsto\zeta_\alpha\right\rbrace$$

Oh. My. Word. You know the saying, "A picture is worth 1000 words"? Well, this

is worth 10^6 words and quite a few more equations.

How do you make this LaTeX math text?

$$TeX here$$

Eg:
$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

$${\color{Teal}\textbf{Thank you!}}$$

I just wanted to ask because I know LaTeX supports coloring but this version of Discourse doesn't, so I asked to make some colored text.

it's just a few words. It is simply that eta 0 is the fixed point that alpha can be mapped to zeta-alpha.

$$\eta_0[0]=0$$
$$\eta_0[1]=\zeta_0$$
$$\eta_0[2]=\zeta_{\zeta_0}$$
$$\eta_0[3]=\zeta_{\zeta_{\zeta_0}}$$
And so on...

Oh. Ok. So which one's bigger, eta 0, or zeta 0?

eta 0. It's that $$\eta_0 = \zeta_{\zeta_{\zeta_\ldots}}$$

there's also a sequence.
$$\varphi(0,0)=\omega$$
$$\varphi(1,0)=\varepsilon_0$$
$$\varphi(2,0)=\zeta_0$$
$$\varphi(3,0)=\eta_0$$
$$\varphi(4,0)=\eta_{\eta_\ldots}$$
...
$$\varphi(\omega,0)$$
$$\Gamma_0=\varphi(\varphi(\ldots,0),0)$$

WHAT!? Now this full sequence has to be made! Why stop at Epsilon 0?

Because i don't know how. However, I'm thinking of making a pair sequence version, which Pair sequence system itself (the googology one by BashicuHydora) is $$\psi(\Omega_\omega)$$, beyond any of these ordinals possible. In fact, the Small veblen ordinal, which in FGH is about TREE(n), is just $$\psi(\Omega^{\Omega^\omega})$$!

! So we would be going BEYOND THE SMALL VEBLEN ORDINAL!? Wow, that's a huge jump from epsilon 0.

I couldn't make SVO, but I made beyond your goal Zeta 0, and even reached phi(omega,0)!!!!!!! Implementation of Large Primitive Sequence system in Snap!: φ(ω,0)