> 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.

> E. E. Escultura
> Research Professor
> V. Lakshmikantham Institute for Advanced Studies
> GVP College of Engineering, JNT University
> Madurawada, Vishakhapatnam, AP, India
> http://users.tpg.com.au/pidro/