is consistent as a metric for the two dimensional (real number) field (a,b)) with Peano's axioms. However, Fermat's expression requires that a=0 or b=0 for n>2 if c is to be an integer (for fields, division is always possible so
is always possible, so the field of positive real numbers is consistent and but not complete (Godel's proof is based on positive integers), since it requires complex numbers for subraction (
). The key issue is that one needs two fields (Cartesian coordinates) to define any function y=f(x) at all, not to mention z=g(x,y) as in the Binomial Expansion for z a single valued resultant variable.
Cartesian coordinates can express real numbers without the necessity of Dedekind cuts in a single dimension number line.
STR provides a way of generating continuous fields (without using Dedekind cuts) and for positive fields, also provides trigonometric functions (and negative fields if Dirac is included).
One liner responses from the Peanut Gallery with no intellectual content whatever in responses to posts like this (where I can provide Wiki links as necessary) are merely spam and bloating my thread's intellectual content with nothing other than complaints indicating that the Peanut Gallery have no idea WTF I am talking about.