Suppose we take the usual definition of ordinal exponentiation (sorry for the notation!):
a^0 = 1
a^s(b) = a^b x a where s(b) is the successor of b
a^b = Ua^c where c < b and b is a limit ordinal
Then for a = 0, b = 1, and c = omega we have:
0^1+c = 0^c = 1 because 0^c is U0^d for d < c and 0^0 = 1.
0^1 = 0 and 0^c = 1 so 0^1 x 0^c = 0
Thus for certain values of a, b, and c:
~(a^b+c = a^b x a^c)
Is this right? (Again an exercise says it should hold for all a, b, and c)
Any help would be great.