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Initial values for hyper operations
#1
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 Smile (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 **.
Reply


Messages In This Thread
Initial values for hyper operations - by bo198214 - 03/14/2008, 10:44 AM
RE: Initial values for hyper operations - by GFR - 03/19/2008, 11:03 AM

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