This is a list of functions and numbers defined by 霽霄 (formally 小魚七). Note that this page contains Chinese.
All functions are ill-defined!
EP(n) is defined as 'the least number of steps needed to get 0'.
EP(1) = (1) = (0, 0) = (0) = 0, EP(1) = 4
EP(2) = (2) = (2, 2) = (1, 1, 1, 1) = (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1) = ..., EP(2) = ?
EP^{n}(a) = EP(EP(EP(... (a) ...))) (n iterations)
EP^{10^100}(2^65,536) = EP's number
不要跟中層增長的層次混淆
h_{0}(n) = n + 1
h_{m + 1}(n) = h_{m}^{h_{m}^{h_{m}^ ... ^h_{m}(n) ... (n)}(n)}(n) (h_{m}(n) {n} n times)
h_{α}(n) = h_{α[n]}(n) (如果α是極限序號)
h_{10^100}(2^65,536) = H's number 1
不要跟中層增長的層次混淆
h_{0}(n) = n + 1
h_{m + 1}(n) = h_{m}^{h_{m}^{h_{m}^ ... ^h_{m}(n) ... (n)}(n)}(n) (h_{m}(n) {h_{m}(n) {h_{m}(n) {... {h_{m}(n) {n} n} ...}} n} n (h_{m}(n) times) times)
h_{a}(n) = h_{a[n]}(n) (如果a是極限序號)
h_{10^100}(2^65,536) = New H's number
[0] = 1
[1] = [0][0] = [0] 1 = 2
[2] = [1][1][1][1] = 16
[3] = [2][2][2][2][2][2][2][2][2][2][2][2][2][2][2][2][2][2] = 4 {15} 16
[n] > f_{ω}(n)
[1, 1] = [2]
[1, 2] = [3]
[2, 2] = [[[2]]]
[n, n] > f_{ω + 1}(n)
[n, n, n, ..., n] (n entries) > f_{ω^2}(n)
[1], [1] = [1]
[1], [2] = [1, 1]
[2], [2] = [2, 2]
[n], [n] = [n, n, n, ..., n] (n entries) > f_{ω^2}(n)
[n], [n], [n] = [n, n, n, ..., n], [n, n, n, ..., n] (n entries for each array)
[n], [n], [n], ..., [n] (n [n]'s) = h_{ω}(n)
[n](m) = [n], [n], [n], ..., [n] (m n's) = h_{m}(n)
[n](a)(b) = h_{φ(φ(φ(... φ(b, 0) ..., 0), 0), 0)}(n)
[n](a)(b)(c) = h_{φ(φ(φ(... φ(a, b) ..., b), b), b)}(n)
[10^100](2^65,536)(2^65,536)(2^65,536) ... (2^65,536) (2^65,536 (2^65,536)'s) = 序列數組數
AOSG(1) = (1){1}[1] = (0, 0){1, 0, 0}[2] = (0){1, 0, 0, 0}[4]= 0{1, 0, 0, 0, 0}[8] = H^{16}_{φ(1, 0, 0, 0, 0)}(16) >>> G(64)
AOSG(2) = (2){2}[2] = (1, 1, 1, 1){2, 1, 1}[4] = (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1){2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1}[8] = (1, 1, 1){2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1}[2^{3+8}] = (1, 1, 1){2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1}[2^11] = (1, 1, 1){2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1}[2048] = ...
AOSG^{n}(a) = AOSG(AOSG(AOSG(... (a) ...))) (n iterations)
AOSG^{10^100}(2^65,536) = AOSG's number
C_{0}(N) = N + 1
C_{a}(n) = h_{h_{h_{... h_{a}(N) ...}(N)}(N)}(N) (a times)
C_{ω}(n) = h_{h_{h_{... h_{a}(N) ...}(N)}(N)}(N) (N times)
C_{10^100}(2^65,536) = C's number
a(b) = a^b
a(b)(c) = a {b} c
a(b)(c)(d) = {{L, L, a}, {L, L, b}, {L, L, c} ({L, L, d}) 2}
a(b)(c)(d)(e) = (a, a, a ..., a (b times))(a + 1, a + 1, a + 1(b + 1 times) … (a + b, a + b, a + b (b+b times))(c times)[d {e} d]
[10^100](2^65,536)(2^65,536)(2^65,536) ... (2^65,536) (2^65,536 (2^65,536)'s) = 無限代數套娃數
T(n) = H^{T(n - 1)}_{Σ^{φ(ω_1^{CK}, ..., ω_{T(n - 1)}^{CK})}}(T(n - 1))
T(0) = H^{10^100}_{Σ^{φ(ω_1^{CK}, ..., ω_{10^100}^{CK})}}(10^100)
T(n, a) = T(T(T(... (n) ...))) (H^{T(n - 1, a)}_{T(n - 1, a)}(T(n - 1, a)) iterations)
T(0, 0) = T(T(T(... (n) ...))) (H^{T(0)}_{T(0)}(T(0)) iterations)
H_{0}(n) = n + 1
H^{n + 1}_{a}(b) = H^{n}_{a}(H^{n}_{a}(H^{n}_{a}(... H^{n}_{a}(n) ...))) (H^{n}_{a}(T(b, n)) iterations)
H_{m + 1}(n) = T(H_{m}^{H_{m}^{H_{m}^ ... ^{H_{m}(T(n, m)) ...}(T(n, m))}(T(n, m))}(T(n, m)), m) (T(H_{m} {T(H_{m} {T(H_{m} {... {T(H_{m} {T(n, m)} T(n, m)} ...} T(n, m)} T(n, m)} T(n, m) (T(H_{m}(T(n, m)), m) times) iterations)
M(a + 1, n) = {a, ..., a} (H^{a}_{T(n, a)}(T(n, a)) entries)
If a = 0, {0}[n] = H^{T(n, n)}_{T(n, n)}(T(n, n))
If a != 0, {a}[n] = {M(a, n)}[H_{T(a, n)}^{T(a, n)}(T(a, n))]
X(n) = {H^{T(X(n - 1), X(n - 1))}_{T(X(n - 1), X(n - 1))}(T(X(n-1), X(n-1))), …, H^{T(X(n - 1), X(n - 1))}_{T(X(n - 1), X(n - 1))}(T(X(n - 1) ,X(n - 1)))}[H^{T(X(n - 1),X(n - 1))}{T(X(n - 1), X(n - 1))}(T(X(n - 1), X(n - 1))] (H^{T(X(n - 1), X(n - 1))}_{T(X(n - 1), X(n - 1))}(T(X(n - 1), X(n - 1))) entries)
X(0) = {T(0, 0), ..., T(0, 0)} (H^{T(0, 0)}_{T(0, 0)}(T(0, 0)) entries) [H^{T(0, 0)}_{T(0, 0)}(T(0, 0))] = ...
X^{n}(a) = X(X(X(... (H^{T(a, n)}_{T(a, n)}(T(a, n))) ...))) (H^{T(0, 0)}_{T(0, 0)}(T(0, 0)) iterations)
X^{10^100}(2^65536) = X's number >>> BB_{∞}(n)
T(0) = 2
T(1) = T(0) * T(0) = 2 * 2 = 4
T(2) = (T(0) + T(1)) * (T(0) + T(1)) = (2 + 4) * (2 + 4) = 6 * 6 = 36
T(n) = (T(0) + T(1) + ... + T(n - 1))^2
T(-0) = -2
T(1) = T(-0) * T(-0) = (-2) * (-2) = 4
T(2) = (T(-0) + T(-1)) * (T(-0) + T(-1)) = ((-2) + (-4)) * ((-2) + (-4)) = (-6) * -6 = (36)
T(n) = (T(-0) + T(-1) + ... + T(-n - 1))^2
T^{n}(a) = T(T(T(... (a) ...))) (n iterations)
T^{10^100)(2^65536) = 龜龜數
T^{10^100)(-2^65536) = 異類龜龜數
K(n) = 用n個字符所能定義出的最大非無窮數字
K(0) = 1
K(1) = 9
K(2) = 99
K(n + 64) > f^{9^9,999}(9^9,999)
K(84) > f^{9^9,999}(9^9,999)
D_{0}(n) = f(n + 1)
D_{n + 1}(x) = f^{x}_{n}(n)
s(n, x) = D^{n}_{I0}(x)
f(n) = s(n, n)
f^{9^9,999}(9^9,999) = f's number
K^{n}(a) = K(K(K(... (a) ...))) (n iterations)
K^{10^100}(2^65,536) = K's number
AT(n) = input 1 ... 1^ ... ^1 ... 1 (n 1's on both sides and n ^'s) in the Turing machine http://morphett.info/turing/turing.html?7cc49f42ad3b91d5e37699fdd9cb6acb
e. g. (1) AT(1) = input 1^1
e. g. (2) AT(2) = input 11^^11
AT(1) = 8
AT(2) = 94
AT(3) = 2,784
AT(4) = ???
AT^{a}(n) = AT(AT(AT(... (AT(n) ...))) (a iterations)
AT^{10^100}(2^65,536) = AT's number
AT's number is an uncomputable number but isn't big, it's less than G(64) (no proof)
This function based on the q(n) function and the Turing machine.
QT(1) = 12,892 (proof: http://morphett.info/turing/turing.html?48ad3d0e4a8d8067c519dbe783090914)
QT(2) = ???
QT^{a}(n) = QT(QT(QT(... (QT(n) ...))) (a iterations)
QT^{10^100}(2^65,536) = QT's number
RT(n) = input 1 … 1 (n 1's)
RT(1) = 66
RT(2) = 1,966
RT(3) >= BB(n)???
RT^{a}(n) = RT(RT(RT(... (RT(n) ...))) (a iterations)
RT^{10^100}(2^65,536) = RT's number
R(n) = (n){n}[n]<n>
R(1) = (1){1}[1]<1> = (0,0){1, 0, 0}[2]<2> = (0){1, 0, 0, 0}[4]<3> = H^{4}_{φ(1, 0, 0, 0)}(3)
R^{a}(n) = R(R(R(... (R(n) ...))) (a iterations)
R^{10^100}(2^65,536) = R's number
h_{0}(n) = n * n
h_{m + 1}(n) = h^{n}_{m}(n)
O(n) = h_{O^{O(n - 1)}(n - 1)}(O^{O(n - 1)}(n - 1))
O(0) = ω
A(n) = f_{O^{A(n - 1)}(A(n - 1))}(A(n - 1))
A(0) = f_{O^{10^100}(10^100)}(10^100)
A^{10^100}(2^65,536) = A's number
Let [ϕ] and [ψ] be Gödel-coded formulas and s and t and be variable assignments. Define Sat([ϕ], s) as follows:
∀K {
{
∀[ψ], s: K([ψ],t) ↔ ([ψ] = "xi ∈ xj" ∧ t(xi) ∈ t(xj))
∨ ([ψ] = "(θ∧ξ)" ∧ K([θ], t) ∧ K([ξ], t))
∨ ([ψ] = "(θ∨ξ)" ∧ K([θ], t) ∨ K([ξ], t))
∨ ([ψ] = "(¬θ)" ∧ ¬K([θ], t))
∨ ([ψ] = "(θ→ξ)" ∧ K([θ], t) → K([ξ], t))
∨ ([ψ] = "(θ↔ξ)" ∧ K([θ], t) ↔ K([ξ], t))
∨ ([ψ] = "(θ↦ξ)" ∧ K([θ], t)↦K([ξ], t))
∨ ([ψ] = "∀xi(θ)" ∧ ∀t′: K([θ], t’))
∨ ([ψ] = "∃xi(θ)" ∧ ∃t′: K([θ], t′))
∨ ([ψ] = "(xi,xj)" ∧ (t(xi),t(xj)))
∨ ([ψ] = "{xi,xj}" ∧ {t(xi),t(xj)})
∨ ([ψ] = "θ,ξ" ∧ K([θ], t),K([ξ], t))
∨ ([ψ] = "θ[ξ]" ∧ K([θ], t)[K([ξ], t)])
∨ ([ψ] = "xi = xj" ∧ t(xi) = t(xj))
∨ ([ψ] = "xi ≠ xj" ∧ t(xi) ≠ t(xj))
∨ ([ψ] = "xi ∈ xj" ∧ t(xi) ∈ t(xj))
∨ ([ψ] = "xi ∉ xj" ∧ t(xi) ∉ t(xj))
∨ ([ψ] = "θ>ξ" ∧ K([θ], t)>K([ξ], t))
∨ ([ψ] = "θ<ξ" ∧ K([θ], t)<K([ξ], t))
∨ ([ψ] = "θ⊂ξ" ∧ K([θ], t) ⊂ K([ξ], t))
∨ ([ψ] = "θ⊄ξ" ∧ K([θ], t) ⊄ K([ξ], t))
∨ ([ψ] = "θ⊆ξ" ∧ K([θ], t) ⊆ K([ξ], t))
∨ ([ψ] = "θ⫋ξ" ∧ K([θ], t) ⫋ K([ξ], t))
∨ ([ψ] = "θ⊃ξ" ∧ K([θ], t) ⊃ K([ξ], t))
∨ ([ψ] = "θ⊅ξ" ∧ K([θ], t) ⊅ K([ξ], t))
∨ ([ψ] = "θ⊇ξ" ∧ K([θ], t) ⊇ K([ξ], t))
∨ ([ψ] = "θ⫌ξ" ∧ K([θ], t) ⫌ K([ξ], t))
∨ ([ψ] = "θ∪ξ" ∧ K([θ], t) ∪ K([ξ], t))
∨ ([ψ] = "θ∩ξ" ∧ K([θ], t) ∩ K([ξ], t))
∨ ([ψ] = "θ\ξ" ∧ K([θ], t) \ K([ξ], t))
∨ ([ψ] = "θ/ξ" ∧ K([θ], t) / K([ξ], t))
∨ ([ψ] = "Vθ" ∧ VK([θ], t))
∨ ([ψ] = "Lθ" ∧ LK([θ], t))
∨ ([ψ] = "fxi(θ)" ∧ ft(xi)(K([θ], t)))
∨ ([ψ] = "(θ+xi)" ∧ (K([θ], t) + t(xi)))
∨ ([ψ] = "(θ-xj)" ∧ (K([θ], t) - t(xj)))
∨ ([ψ] = "{θ∣ξ}" ∧ {K([θ], t)∣K([ξ], t)})
∨ ([ψ] = "{θ:ξ}" ∧ {K([θ], t):K([ξ], t)})
∨ ([ψ] = "θ⊧ξ" ∧ K([θ], t) ⊧ K([ξ], t))
∨ ([ψ] = "θ⊢ξ" ∧ K([θ], t) ⊢ K([ξ], t))
∨ ([ψ] = "θ:ξ" ∧ K([θ], t):K([ξ], t))
∨ ([ψ] = "θ|ξ" ∧ K([θ], t)|K([ξ], t))
∨ ([ψ] = "∵θ∴ξ" ∧ ∵ K([θ], t) ∴ K([ξ], t))
∨ ([ψ] = "|θ|" ∧ |K([θ], t)|)
∨ ([ψ] = "P(θ)" ∧ P(K([θ], t)))
∨ ([ψ] = "θ≈ξ" ∧ K([θ], t) ≈ K([ξ], t))
} ⇒ K([ϕ],s)
}
K(n + 1) = 在用不K(n)超过个符号所能表示的最強力的集合论逻辑中，用不超过个K(n)符号所能表示的最大正整数
K(0) = 2^1,000
K_{0}(n) = K^{f_{Σ}(n)}(f_{Σ}(n))
K_{n + 1}(m) = K^{m}_{n}(m)
K_{α}(n) = K_{α[n]}(n) (如果α是極限序號)
ω^{φ}_{n} = K_{Π^{ω^{φ}_{n}}_{ω^{φ}_{n}}}(Π^{ω^{φ}_{n}}_{ω^{φ}_{n}})
K^{f_{Σ^{K}_{K}}(10^100)}_{Π^{ω^{φ}_{Σ}}_{ω^{φ}_{Σ}}}(2^65,536) = K's number super
K(2) >= {} = 0
K(4) >= {{}} = 1
K(9) >= {{}, {{}}} = 2
K(9 + n) > N[x] = x, N[9[n]] = 9[n] >> n
K(64) > f_{φ(1, 0, 0, 0) + 1}(99) >> Σ(128) (no proof)
K(81) > I0(99)
K(86) > I0^{999}(9)
K(338) > Rayo(999)
K(338 + 3n) > Rayo^{9[n]}(999)
我們通過定義ω = R_{ω}(ω)以定義一個大序數
K(379) > R_{ω + 1}(Rayo(999)) >> F^{63}_{7}(10^100)
K_{0}(4) = K^{107}(107)
V(n + 1) = Σ_{ψ_{K}(Ω_{ω^{CK}_{V(n)}})}
V(0) = Σ_{ψ_{K}(Ω_{Σ})}
D(n + 1) = f^{D(n) + 1}_{Σ_{ψ_{K}(Ω_{V^{D(n)}(ω)})} + 1}(2^65,536)
D(0) = f^{2^65,536 + 1}_{Σ_{ψ_{K}(Ω_{V^{2^65,536}(ω)})} + 1}(2^65,536)
D^{2^65,536}(10^100) = D's number
U(n + 1) = max{Σ(U(n)), S(U(n)), Ξ(U(n)), Σ_{∞}(U(n)), Rayo(U(n)), f(U(n))}
U(0) = 2
U(1) = 6
U(2) = 7.412 * 10^36,534
U(3) = f(U(2))
U(n + 2) = f^{n}(U(2))
YY(n + 1) > n + 1
G(-1) = ∞