05/12/2009, 01:00 AM

(06/04/2008, 07:20 AM)Ivars Wrote:bo198214 Wrote:Sorry, but I dont see any meaning for hyper operations in defining a fucked up function that is so crude that you can not define them on the reals.

andydude Wrote:You need to learn the difference between divergent and false.

The other point is who cares about functions on reals if they are so fucked up in principle that they can not deal with simple reduction of tetration to simpler analysis. Disctinction between true and false in tetration and above can not be determined by the properties of function on reals.

But of course this is just uneducated intuitive opinion. Though I do not see a problem to define a function like f(xL,x)= 0, f(x, xR)=1. xL,

Where xL, xR are Left and Right surreal numbers between any 2 real values of x.Function is just a unique relationship between 2 sets, be it reals and surreals or what ever. Once defined, it exists. Next step is to study if it has any reasonable properties and does it fit the purpose of understanding hyperoperations.

Ivars

Let 0 = {|}, 1 = {0|}, -1 = {|0}, as usual.

Then {-1 | 1} = 0.

So 1 = f({|0}, 0) = f(-1, 0) = f(-1, {-1 | 1}) = 0.

The function is not well-defined. (It would be well-defined on combinatorial games, except that it is of course not defined on all of them.)

Also, if f(x^y) = f(x) f(y) on the surreals, then either f is identically 1, f is identically zero, or there is some y with 1^y not equal to 1, or some x with x^1 not equal to x. This follows by bo198214's proof. I daresay all the standard definitions of exponentiation on the surreals have the latter two properties, so if you still want this setup, you'll need a new definition of exponentiation. I would recommend making it commutative, since surreal multiplication is.