- FINAL -
Ill-defined expermental notation. Created 11/5/2022.
For non-negative integers:
a(0)b = a[b] in Copy notation
a(#)1 = a for # > 0
a(1)b = a(0)[b(1)a]
a(c)b = a(c-1)[b(c)a]
a(c, 0)b = a(c)b
a(0, 1)b = a(b)b
a(c, 1)b = a(c-1, 1)[b(c, 1)a]
a(0, 2)b = a(a, 1)b
a(0, 3)b = a(a, 2)b
a(0, c)b = a(a c-1)b
a(0, 0, 1)b = a(a, b)b
a(0, 0, c)b = a(a, b, c-1)b
a(0, 0, 0, 1)b = a(a, a, b)b
a(0, 0, 0, 0, 1)b = a(a, a, a, b)b
a(0<1>1)b = a(a, a, a, ..., a, a, a)b w/ b entries of a
a(1<1>1)b = a(0<1>1)[b(1<1>1)a]
a(0<1>2)b = a(a, a, ..., a, a <1> 1)b w/ b entries of a
a(0<1>c)b = a(a, a, ..., a, a <1> c-1)b w/ b entries of a
a(0<1>0, 1)b = a(a, a, ..., a, a <1> b)b w/ b entries of a
a(0<1>0, c)b = a(a, a, ..., a, a <1> b, c-1)b w/ b entries of a
a(0<1>0, 0, 1)b = a(a, a, ..., a, a <1> a, b)b w/ b entries of a in the left side of <1>
a(0<1>0<1>1)b = a(a, a, ..., a, a <1> a, a, ..., a, a)b w/ b entries of a on each rows
a(0<1>0<1>0<1>1)b = a(a, a, ..., a, a <1> a, a, ..., a, a <1> a, a, ..., a, a)b w/ b entries on each rows
a(0<2>1)b = a(a <1> a <1> ... <1> a <1> a)b w/ b rows
a(0<3>1)b = a(a <2> a <2> ... <2> a <2> a)b w/ b rows
a(0<c>1)b = a(a <c-1> a <c-1> ... <c-1> a <c-1> a)b w/ b rows
a(0<0, 1>1)b = a(a <b> a <b> ... <b> a <b> a)b w/ a rows
a(0<c, 1>1)b = a(a <c-1, 1> a <c-1, 1> ... <c-1, 1> a <c-1, 1> a)b w/ b rows
a(0<0, c>1)b = a(a <b, c-1> a <b, c-1> ... <b, c-1> a <b, c-1> a)b w/ a rows
a(0<0, 0, 1>1)b = a(a <a, b> a <a, b> ... <a, b> a <a, b> a)b w/ a rows
a(0<0 <1> 1>1)b = a(a <a, a, a, ..., a, a, a> a <a, a, a, ..., a, a, a> ... <a, a, a, ..., a, a, a> a <a, a, a, ..., a, a, a> a)b w/ b entries of a in <> and a entries of a between <a, a, a, ..., a, a, a>
a(0<0 <0, 1> 1>1)b = a(a <a <b> a <b> ... <b> a <b> a> a <a <b> a <b> ... <b> a <b> a> ... <a <b> a <b> ... <b> a <b> a> a <a <b> a <b> ... <b> a <b> a> a)b w/ a rows of a separated by <b>, and a rows of a separated by <a <b> a <b> ... <b> a <b> a>
Let's define the function:
aQ0 = a(a)a
aQ1 = a(a <a> a)a
aQ2 = a(a <a <a> a> a)a
aQn = a(a <... a <a> a ...> a)a w/ n pairs of <>
R(1) = 1Q1 = 1(1)1 = 1
R(2) = 2Q2 = 2(2 <2> 2)2 = 2(1 <2> 2)[2(2 <2> 2)2]
R(n) = nQn
R(1, 2) = R(10)
R(2, 2) = R(R(1, 2))
R(a, 2) = R(R(a-1, 2))
R(1, 3) = R(10, 2)
R(1, b) = R(10, b-1)
R(1, 1, 2) = R(10, 10)
R(1, 1, 1, 2) = R(10, 10, 10)
R(1 <2> 2) = R(10, 10, 10, 10, 10, 10, 10, 10, 10, 10)
R(2 <2> 2) = R(10, 10, 10, ..., 10, 10, 10) w/ R(1 <2> 2) 10's
R(1, 2 <2> 2) = R(10 <2> 2)
R(1 <2> 3) = R(10, 10, ..., 10, 10 <2> 2) w/ R(1 <2> 2) 10's
R(1 <2> 1, 2) = R(10 <2> 10)
R(1 <2> 1 <2> 2) = R(10 <2> 10, 10, 10, ..., 10, 10, 10) w/ 10 10's after <2>
R(1 <3> 2) = R(10 <2> 10 <2> ... <2> 10 <2> 10) w/ 10 10's
R(1 <4> 2) = R(10 <3> 10 <3> ... <3> 10 <3> 10) w/ 10 10's
R(1 <1, 2> 2) = R(10 <10> 10 <10> ... <10> 10 <10> 10) w/ 10 rows
R(1 <1, 1, 2> 2) = R(10 <10, 10> 10 <10, 10> ... <10, 10> 10 <10, 10> 10) w/ 10 rows
R(1 <1 <2> 2> 2) = R(10 <10, 10, ..., 10, 10> 10 <10, 10, ..., 10, 10> ... <10, 10, ..., 10, 10> 10 <10, 10, ..., 10, 10> 10) w/ 10 10's in <> and 10 rows
S(1) = R(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
S(2) = R(10 <... 10 <10> 10 ...> 10) w/ S(1) pairs of <>
S(3) = R(10 <... 10 <10> 10 ...> 10) w/ S(2) pairs of <>
S(n) = R(10 <... 10 <10> 10 ...> 10) w/ S(n-1) pairs of <>
S(1, 2) = S(10)
T(1) = S(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
U(1) = T(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
V(1) = U(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
Now, let's define yet another function:
J(1) = R(1) = 1
J(2) = S(1)
J(3) = T(1)
J(4) = U(1)
J(5) = V(1)
J(1, 2) = J(J(...J(J(2))...)) w/ S(1) J's
K(1) = J(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
<3(1) = J(69)
<3(2) = K(69)
If we keep defining functions like this, and then combine them all into one enormous function, we can create much more enormous numbers.
<3(1, 2) = <3(<3(...<3(<3(69))...)) w/ <3(1) <3's
<4(1) = <3(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
<5(1) = <4(10 <... 10 <10> 10 ...> 10) w/ 10 pairs of <>
We can even do <1337(1), and that, is the limit of my array notation. That limit is approximately Inaccessible (ψ(ψ_I(0))) level.