SAN vs FGH

This is SUPER STRONG SAN vs FGH. Each heading have the extension range, and each subheading I will analyze each levels.

Pre-Analysis

If you want to see smaller ordinals, go to the introduction of the ordinals.

First, W will be cardinal that works in psi(_), then W_N will be cardinal that works in psi_(N-1)(_).

hyp cos already analyzed SAN up to using 'easy to understand' standard ordinal notations. So I will continue analyzing up to the limit.

pDAN Range

Up to {1,,1,,1,,2}

Now we define further ordinals.

• psi_chi(0)(0)[0] = 1
• psi_chi(a)(0)[0] = chi(a-1)+1
• psi_chi(a)(b+1)[0] = psi_chi(a)(b)+1
• psi_chi(a)(b+1)[n] = W_(psi_chi(a)(b+1)[n-1])
• psi_chi(a)(b)[n] = psi_chi(a)(b[n])
• chi(a) will be cardinal that works in psi_chi(a)(_)
• psi_chi(0) is a function that generates W_A.
• psi_chi(n) is a function that works like psi_chi(0) using W_(chi(n-1)+A) instead of W_A.

Then let get started at {1,,1{1,,1,,2}2}.

• {1,,1{1,,1,,2}2} have level psi_chi(0)(0)
• {1,,2{1,,1,,2}2} have level W_(psi_chi(0)(0)+1)
• {1,,A{1,,1,,2}2} have level W_(psi_chi(0)(0)+A)
• {1,,1{1,,2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)+W)
• {1,,1{1,,A}2{1,,1,,2}2} have level W_(psi_chi(0)(0)+W_A)
• {1,,1{1,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)2)
• {1,,A{1,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)2+A)
• {1,,1{1,,1{1,,1,,2}2}3{1,,1,,2}2} have level W_(psi_chi(0)(0)3)
• {1,,1{1,,1{1,,1,,2}2}A{1,,1,,2}2} have level W_(psi_chi(0)(0)A)
• {1,,1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)^2)
• {1,,1{2,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)^w)
• {1,,1{A,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)^A)
• {1,,1{1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)^psi_chi(0)(0))
• {1,,1{1{1,,1{1,,1,,2}2}1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)^psi_chi(0)(0)^psi_chi(0)(0))
• {1,,1{1{1{1,,1{1,,1,,2}2}2,,1{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(psi_chi(0)(0)^psi_chi(0)(0)^psi_chi(0)(0)^psi_chi(0)(0))
• {1,,1{1{1,,2{1,,1,,2}2}2,,1{1,,1,,2}2}2{1,,1,,2}2} have level W_(e(psi_chi(0)(0)+1))

Now let △ as {1,,1,,2}.

• {1,,1{1{1,,2△2}2,,1△2}2△2} have level W_(e(psi_chi(0)(0)+1))
• {1,,1{1{1,,2△2}2,,1△2}A△2} have level W_(e(psi_chi(0)(0)+1)A)
• {1,,1{A{1,,2△2}2,,1△2}2△2} have level W_(e(psi_chi(0)(0)+1)^A)
• {1,,1{1{1{1,,2△2}2,,1△2}2{1,,2△2}2,,1△2}2△2} have level W_(e(psi_chi(0)(0)+1)^e(psi_chi(0)(0)+1))
• {1,,1{1{1,,2△2}3,,1△2}2△2} have level W_(e(psi_chi(0)(0)+2))
• {1,,1{1{1,,2△2}1{1,,2△2}2,,1△2}2△2} have level W_(z(psi_chi(0)(0)+1))
• {1,,1{1{1{1,,2△2}2,,2△2}2,,1△2}2△2} have level W_(psi_(psi_chi(0)(0)+1)(W_(psi_chi(0)(0)+1)^W_(psi_chi(0)(0)+1)))
• {1,,1{1{1,,3△2}2,,1△2}2△2} have level W_(psi_(psi_chi(0)(0)+1)(e(W_(psi_chi(0)(0)+1)+1)))
• {1,,1{1{1,,A△2}2,,1△2}2△2} have level W_(psi_(psi_chi(0)(0)+1)(W_(psi_chi(0)(0)+A)))
• {1,,1{1,,2△2}2△2} have level W_W_(psi_chi(0)(0)+1)
• {1,,1{1,,2△2}A△2} have level W_(W_(psi_chi(0)(0)+1)A)
• {1,,1{A,,2△2}2△2} have level W_(W_(psi_chi(0)(0)+1)^A)
• {1,,1{1{1,,2△2}2,,2△2}2△2} have level W_(W_(psi_chi(0)(0)+1)^W_(psi_chi(0)(0)+1))
• {1,,1{1{1,,3△2}2,,2△2}2△2} have level W_(e(psi_chi(0)(0)+1))
• {1,,1{1,,3△2}2△2} have level W_W_(psi_chi(0)(0)+2)
• {1,,1{1,,A△2}2△2} have level W_W_(psi_chi(0)(0)+A)
• {1,,1{1,,1{1,,1△2}2△2}2△2} have level W_W_(psi_chi(0)(0)2)
• {1,,1{1,,1{1,,A△2}2△2}2△2} have level W_W_W_(psi_chi(0)(0)+1)
• {1,,1{1,,1{1,,1{1,,1△2}2△2}2△2}2△2} have level W_W_W_(psi_chi(0)(0)2)
• {1,,1△3} have level psi_chi(0)(1)
• {1,,A△3} have level W_(psi_chi(0)(1)+A)
• {1,,1{1,,1△3}2△3} have level W_(psi_chi(0)(1)2)
• {1,,1{1,,1{1,,1△3}2△3}2△3} have level W_W_(psi_chi(0)(1)2)
• {1,,1△4} have level psi_chi(0)(2)
• {1,,1△A} have level psi_chi(0)(A)
• {1,,1△1{1,,1△2}2} have level psi_chi(0)(psi_chi(0)(0))
• {1,,1△1{1,,1△A}2} have level psi_chi(0)(psi_chi(0)(A))
• {1,,1△1{1,,1△1{1,,1△2}2}2} have level psi_chi(0)(psi_chi(0)(psi_chi(0)(0)))
• {1,,1△1△2} have level psi_chi(0)(chi(0))
• {1,,1△A△2} have level psi_chi(0)(chi(0)+A)
• {1,,1△1△A} have level psi_chi(0)(chi(0)A)
• {1,,1{A,,1,,2}2} have level psi_chi(0)(chi(0)^A)
• {1,,1{1{1,,1△2}2,,1,,2}2} have level psi_chi(0)(chi(0)^psi_chi(0)(0))
• {1,,1{1{1,,1{1{1,,1△2}2,,1,,2}2}2,,1,,2}2} have level psi_chi(0)(chi(0)^psi_chi(0)(psi_chi(0)(chi(0)^psi_chi(0)(0))))
• {1,,1{1△2,,1,,2}2} have level psi_chi(0)(chi(0)^chi(0))
• △ have level chi(0)
• {1{1,,2,,2}2,,1,,2} have level e(chi(0)+1)
• {1{1{1,,2,,2}2,,1,,2}2{1,,2,,2}2,,1,,2} have level e(chi(0)+1)2
• {1{1,,2,,2}3,,1,,2} have level e(chi(0)+2)
• {1{1,,2,,2}1{1,,2,,2}2,,1,,2} have level z(chi(0)+1)
• {1,,2,,2} have level W_(chi(0)+1)
• {1,,A,,2} have level W_(chi(0)+A)
• {1,,1△2,,2} have level W_(chi(0)2)
• {1,,1{1,,A,,2}2,,2} have level W_W_(chi(0)+A)
• {1,,1{1,,1△2,,2}2,,2} have level W_W_(chi(0)2)

We can see a pattern: {1,,A} have level W_A and {1,,A,,2} have level W_(chi(0)+A).

Then we can go further:

• {1,,1,,3} have level chi(1)
• {1,,A,,3} have level W_(chi(1)+A)
• {1,,1,,4} have level chi(2)
• {1,,1,,A} have level chi(A)
• {1,,1,,1△2} have level chi(chi(0))
• {1,,1,,1{1,,1,,1△2}2} have level chi(chi(chi(0)))

Let f(n) will be s(n,n{1,,1,,1{1,,1,,1{...{1,,1,,1,2}...}2}2}2) with n '1,,1,,1's have level psi(psi_chi(M)(0)).

Up to {1{1`,,2^,,}2}

Let chi(0) will be a cardinal that works like psi() in W_(A). Then let chi(#+1) will be a cardinal that works like psi() in W_(chi(#)+A) instead of W_(A) and chi(#+M) will be a cardinal that works like psi() in chi(#+A) instead of W_(A).

• {1,,1,,1{1,,1,,1,,2}2} have level psi_chi(M)(0)
• {1,,1,,2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)+1)
• {1,,1,,A{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)+A)
• {1,,1,,1{1,,1,,2}2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)+chi(0))
• {1,,1,,1{1,,1,,A}2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)+chi(A))
• {1,,1,,1{1,,1,,1{1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)+chi(chi(0)))
• {1,,1,,1{1,,1,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)2)
• {1,,1,,1{1,,1,,1{1,,1,,1,,2}2}1{1,,1,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)^2)
• {1,,1,,1{A,,1,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(psi_chi(M)(0)^A)
• {1,,1,,1{1{1,,2,,1{1,,1,,1,,2}2}2,,1,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(e(psi_chi(M)(0)+1)
• {1,,1,,1{1,,2,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(W_(psi_chi(M)(0)+1))
• {1,,1,,1{1,,1,,2{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(chi(psi_chi(M)(0)+1))
• {1,,1,,1{1,,1,,1{1,,1,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(chi(psi_chi(M)(0)2))
• {1,,1,,1{1,,1,,1{1,,1,,1{1,,1,,1{1,,1,,1,,2}2}2{1,,1,,1,,2}2}2{1,,1,,1,,2}2}2{1,,1,,1,,2}2} have level chi(chi(chi(psi_chi(M)(0)2)))
• {1,,1,,1{1,,1,,1,,2}3} have level psi_chi(M)(1)
• {1,,1,,1{1,,1,,1,,2}A} have level psi_chi(M)(A)
• {1,,1,,1,,2} have level chi(M)
• {1,,2,,1,,2} have level W_(chi(M)+1)
• {1,,A,,1,,2} have level W_(chi(M)+A)
• {1,,1{1,,1,,2,,2}2,,1,,2} have level psi_chi(M+1)(0)
• {1,,1{1,,1{1,,1,,2,,2}2,,1,,2}2{1,,1,,2,,2}2,,1,,2} have level W_(psi_chi(M+1)(0)2)
• {1,,1{1,,1,,2,,2}3,,1,,2} have level psi_chi(M+1)(1)
• {1,,1,,2,,2} have level chi(M+1)
• {1,,1,,A,,2} have level chi(M+A)
• {1,,1,,1{1,,1,,1,,3}2,,2} have level psi_chi(M*2)(0)
• {1,,1,,1,,3} have level chi(M*2)
• {1,,1,,2,,3} have level chi(M*2+1)
• {1,,1,,A,,3} have level chi(M*2+A)
• {1,,1,,1,,4} have level chi(M*3)
• {1,,1,,1,,A} have level chi(M*A)
• {1,,1,,1,,1,,2} have level chi(M^2)
• {1,,1,,1,,1,,1,,2} have level chi(M^3)
• {1{1,,2^,,}2} have level chi(M^M)
• {1{1,,1,,2^,,}2} have level chi(M^M^M)
• {1{1{1,,2^,,}2^,,}2} have level chi(M^M^M^M)

The limit of pDAN is psi(chi(e(M+1))). And ,, will corresponds into M.

sDAN Range

Up to {1,,,1,,,2}

Let chi_N acts like a function works like chi function in W_(M_N+A) instead of W_A. Let M_N works like M which uses chi_(N-1)() function instead of chi() function.

• {1{1`,,2^,,}2} ~ chi(e(M+1))
• {1{1`,,2^,,}3} ~ chi(e(M+1)2)
• {1{1`,,2^,,}1{1`,,2^,,}2} ~ chi(e(M+1)^2)
• {1{1{1`,,2^,,}2`,,2^,,}2} ~ chi(e(M+1)^e(M+1))
• {1{1`,,3^,,}2} ~ chi(e(M+2))
• {1{1`,,A^,,}2} ~ chi(e(M+A))
• {1{1`,,1`,,2^,,}2} ~ chi(z(M+1))
• `,, ~ W_(M+1)
• ``,, ~ W_(M+2)
• {1,,`1,2^,,} ~ W_(M+w)
• {1,,`A^,,} ~ W_(M+A)
• {1,,`1,,2^,,} ~ W_(M*2)
• {1,,`1{1,,`1,,2^,,}2^,,} ~ W_W_(M*2)
• {1,,`1{1,,`1,,`2^,,}2^,,} ~ psi_chi_1(0)(0)
• {1,,`1{1,,`A^,,}2{1,,`1,,`2^,,}2^,,} ~ W_(psi_chi_1(0)(0)+W_(M+A))
• {1,,`1{1,,`1{1,,`1,,`2^,,}2^,,}2{1,,`1,,`2^,,}2^,,} ~ W_(psi_chi_1(0)(0)2)
• {1,,`1{1,,`1{1,,`1{1,,`1,,`2^,,}2^,,}2{1,,`1,,`2^,,}2^,,}2{1,,`1,,`2^,,}2^,,} ~ W_W_(psi_chi_1(0)(0)2)
• {1,,`1{1,,`1,,`2^,,}1^,,} ~ psi_chi_1(0)(1)
• {1,,`1{1,,`1,,`2^,,}A^,,} ~ psi_chi_1(0)(A)
• {1,,`1,,`2^,,} ~ chi_1(0)

Up to {1{2,,,,2}2}

DAN Range

NDAN Range

wDEN Range

mwDEN Range

wDmEN Range

pDDN Range