1) The field axioms of the real number system are inconsistent; Felix Brouwer and this blogger provided counterexamples to the trichotomy axiom and Banach-Tarski to the completeness axiom, a variant of the axiom of choice. Therefore, the real number system is ill-defined and FLT being formulated in it is also ill-defined. What it took to resolve this conjecture was to first free the real number system from contradiction by reconstructing it as the new real number system on three simple consistent axioms and reformulating FLT in it. With this rectification of the real number system, FLT is well-defined and resolved by counterexamples proving that it is false. (Main reference: Escultura, E. E., The new real new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 17 (2009), 59 – 84).

2) The other fatal defect is that the complex number system that Wiles used in the proof being based on the vacuous concept i is also inconsistent. The element i is the vacuous concept: the root of the equation x^2 + 1 = 0 which does not exist and is denoted by the symbol i = sqrt(-1) from which follows that,

i = sqrt(1/-1) = sqrt 1/sqrt(-1) = 1/i = i/i^2 = -i   or

1 = -1 (division of both sides by i),

2 = 0,  1 = 0, i = 0, and, for any real number x, x = 0,

and the entire real and complex number systems collapse. The remedy is in the appendix to the paper, The generalized integral as dual to Schwarz distribution, in press, Nonlinear Studies.

Another example of a vacuous concept is the greatest integer. Let N be the greatest integer. By the trichotomy axiom one and only one of the following axioms holds: N < 1, N = 1, N > 1. The first inequality is clearly false. If N > 1, then N^2 > N, contradicting the choice of N. therefore N = 1. This is the original statement of the Perron paradox and it is blamed on the vacuous concept N. In general, any vacuous concept yields a contradiction.