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