02/08/2009, 04:16 PM
here i present what is according to me the " logical hierarchy "
i found it important to say , because it appears often on math forums and is usually stated or ordered in a way i disagree with.
no tetration or non-commutativity.
no ' popularized ' ackermann or buck.
but plain good old logic in my humble opinion.
most imporant is the existance of a single neutral element
f_n ( a , neutral ) = f_n ( neutral , a ) = a for all a !
1) a + b
2) a * b
3) a ^ log(b)
to see how i arrived at 3 :
a ^ log(b) = b ^ log(a) = exp( log(a) * log(b) )
4) exp ( log(a) ^ log(log(b)) )
to see how i arrived at 4 :
note that 3) is used upon log(a) and log(b).
etc etc
note that the neutral elements are
1) addition -> 0
2) multiplication -> 1
3) a ^ log(b) -> e
4) -> e^e
5) -> e^e^e
6) -> e^e^e^e
etc
i found it important to say , because it appears often on math forums and is usually stated or ordered in a way i disagree with.
no tetration or non-commutativity.
no ' popularized ' ackermann or buck.
but plain good old logic in my humble opinion.
most imporant is the existance of a single neutral element
f_n ( a , neutral ) = f_n ( neutral , a ) = a for all a !
1) a + b
2) a * b
3) a ^ log(b)
to see how i arrived at 3 :
a ^ log(b) = b ^ log(a) = exp( log(a) * log(b) )
4) exp ( log(a) ^ log(log(b)) )
to see how i arrived at 4 :
note that 3) is used upon log(a) and log(b).
etc etc
note that the neutral elements are
1) addition -> 0
2) multiplication -> 1
3) a ^ log(b) -> e
4) -> e^e
5) -> e^e^e
6) -> e^e^e^e
etc
