03/14/2008, 10:44 AM
During the introduction of zeration we saw already that there are some inconsistencies with the initial values.
The addition as the successor operation of the successor operation
(I consider here a[0]x=x+1 according to the rule a[n+1](x+1)=a[n](a[n+1]x) ) differs in its initial value from all the higher operations:
a [1] 1 = a + 1
a [2] 1 = a
a [3] 1 = a
a [4] 1 = a
....
If we would choose a[1]1=a we would get
a[1]x=a+(x-1)
I see a quite similar phenomenon when considering a[3L]x, the lower super operation of exponentiation, defined by a[3L](x+1)=(a[3L]x)[3]a.
If we chose the initial condition in the used to a[3L]1=a manner then we get a[3L]x=a^(a^(x-1)). Especially in comparision with the similarly wrong defined addition a[1]x=a+(x-1) I would prefer to be
a[3L]x=a^a^x
a[3L]1=a^a
Then we had unified
a[3L]1=a[3]a
a[1]1=a[0]a
So does anyone see a strictly describable pattern when to use
a ** 1 = a and when to use a ** 1 = a * a, where * is the sub operation of **.
The addition as the successor operation of the successor operation
(I consider here a[0]x=x+1 according to the rule a[n+1](x+1)=a[n](a[n+1]x) ) differs in its initial value from all the higher operations:a [1] 1 = a + 1
a [2] 1 = a
a [3] 1 = a
a [4] 1 = a
....
If we would choose a[1]1=a we would get
a[1]x=a+(x-1)
I see a quite similar phenomenon when considering a[3L]x, the lower super operation of exponentiation, defined by a[3L](x+1)=(a[3L]x)[3]a.
If we chose the initial condition in the used to a[3L]1=a manner then we get a[3L]x=a^(a^(x-1)). Especially in comparision with the similarly wrong defined addition a[1]x=a+(x-1) I would prefer to be
a[3L]x=a^a^x
a[3L]1=a^a
Then we had unified
a[3L]1=a[3]a
a[1]1=a[0]a
So does anyone see a strictly describable pattern when to use
a ** 1 = a and when to use a ** 1 = a * a, where * is the sub operation of **.
