+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Hyperoperations and Related Studies (https://tetrationforum.org/forumdisplay.php?fid=11)
+--- Thread: the logical hierarchy (/showthread.php?tid=231)
the logical hierarchy - tommy1729 - 02/08/2009
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
RE: the logical hierarchy - bo198214 - 02/08/2009
Ya this hierarchy was already considered.
The main observation is that the real numbers with operations \( a+b \) and \( a*b \) are isomorphic to the positive real numbers with \( a*b \) and \( a^{\log(b)} \) (The isomorphism is \( \exp \)). I.e. we dont add really something new. Each two consecutive operations are isomorphic (i.e. behave completely the same as) to + and *.
RE: the logical hierarchy - tommy1729 - 02/08/2009
bo198214 Wrote:Ya this hierarchy was already considered.
The main observation is that the real numbers with operations \( a+b \) and \( a*b \) are isomorphic to the positive real numbers with \( a*b \) and \( a^{\log(b)} \) (The isomorphism is \( \exp \)). I.e. we dont add really something new. Each two consecutive operations are isomorphic (i.e. behave completely the same as) to + and *.
right.
but some people always insist on
a + b
a * b
a ^ b
which is wrong.
glad we agree.
thread closed ?
regards
tommy1729
RE: the logical hierarchy - bo198214 - 02/08/2009
tommy1729 Wrote:but some people always insist on
a + b
a * b
a ^ b
which is wrong.
Its not wrong its a different hierarchy.
It follows the pattern: a[n+1](b+1)=a[n](a[n+1]b)
where [n] is the the nth operation.
For example:
a[2](b+1)=a[1](a[2]b) which corresponds to a*(b+1)=a+a*b
a[3](b+1)=a[2](a[3]b) which corresponds to a^(b+1)=a*(a^b)
and so tetration [4] satisfies:
a[4](b+1)=a[3](a[4]b)