the logical hierarchy - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- 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)