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.

Yes, people like this are allowed on the street.

Right?

i = sqrt(1/-1) = sqrt 1/sqrt(-1) = 1/i,

therefore

i^2 = 1,

therefore -1 = 1?

I tried a few examples like

sqrt(9/4) = sqrt(9)/sqrt(4) = 3/2 = 1.5

OR

sqrt(9/4) = sqrt(2.25) = 1.5.

So what is my molasses mind missing?

When defining sqrt as a function, you have to choose which roots to
take for the nonzero values.  With reals the most useful choice is the
set of all positive roots, as they are closed under multiplication.
However in the complex field there is no possible choice that is
closed under multiplication.  At best you can say that

  sqrt(x) sqrt(y) = +/- sqrt(x y).

So the original "paradox" becomes:

 i = sqrt(1/-1) = +/- sqrt(1) / sqrt(-i) = +/- 1/i,

which is quite true.

You can also define C by the set of ordered pairs of real numbers with
specific definitions for addition and multiplication for ordered pairs
of real numbers.  Again, nothing ill-defined about that.

So, it seems you are denying the most pure essence about algebraic
concepts.  

The bottomline is that the field axioms that supposedly define the real numbers are inconsistent. In particular, the trichotomy and completeness axioms are false. The counterexamples to them are noted elsewhere on this thread and the referecences are cited. Consequently, the real number system is neither complete nor a field nor ordered by "<". In fact, it is ill-defined and all those results you cited fall through.

The objects that satisfy x^2 + 1 = 0 are ill-defined. Hamilton tried to build the complex plane as ordered pairs but he did not identify the right consistent axioms that well define them. The remedy is in the appendix to my paper, The generalized integral as dual of Schwartz distribution, in press, Nonlinear Studies.  

Cheers,

You can also define C by the set of ordered pairs of real numbers with and multiplication for ordered pairs
of real numbers. Again, nothing ill-defined about that.
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Yes, this has been done by Hamilton. But he relied on the real numbers which are ill-defined.

What is really needed here is fix the real number system first which I did in the paper I cited elsewhere on this thread.

Moreover, you seem to be suggesting that since a polynomial equation
has no real roots implies that C is ill-defined. Maybe you can explain
why I'm wrong but that's like saying that since 3x+1=-1 has no roots
among the integers that Q is ill-defined, and like saying that since
x^2 = 2 has no rational roots that the set of irrationals are ill-
defined. There is a progression of number sets that allow for more
polynomial equations to be "solved" (ie, roots found), starting with
N, then Z, Q, R, and finally C. That C is algebraically closed
implies that polynomials will not yield further number sets.

Certainly, the root of a polynomial that has no root is ill-defined, in fact, a contradiction.

What you are suggesting is an extension of the reals that will yield roots to such polynomials. But an extension of any mathematical space belongs to its complement and is not covered by its axioms assuming that they are consisent. Therefore, you need a new set of consistent axioms for them.

Congratulations. You have the most advanced commments so far in contrast to the name callers whose posts come from the flat of their foot because the top is quite empty.

Cheers.

Cheers.

Cheers.

Between that and the Dedekind cut construction of the real numbers,
what is inconsistent?
http://en.wikipedia.org/wiki/Dedekind_cut#The_cut_construction_of_the...

I vaguely recall something about a way to define real numbers as
equivalence classes of sequences of rational numbers where the
equivalence relation between two rational sequences is that two
sequences are considered equivalent if the terms in the sequence get
arbitrarily close to one another (ie, the tails of the two sequences
differ by an arbitrarily small rational number). For example, the
square root of 2 is defined to be the equivalence class of the
sequence {1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...}. The addition
and multiplication of two real numbers (ie, two equivalence classes of
rational sequences) is done by taking the equivalence class of the
sequence whose nth term is the sum or product of the nth term of
representative sequence1 with nth term of representative sequence2.
(The sum and product need to be shown independent of which
representative sequence you use, of course.) From those definitions,
you can define subtraction and division.

Between that and the Dedekind cut construction of the real numbers,
what is inconsistent?
http://en.wikipedia.org/wiki/Dedekind_cut#The_cut_construction_of_the...
----

The Dedekind cut or its equivalent, the completeness axiom of the field axioms of the real numbers do not apply apply to infinite set such as the digits of a nonterminating decimal because ot the latter's ambiguity since not all its digits are known. Any statement about ambiguous set or concept is ambiguous and is not admissible as an axiom for it erodes the validity of any theorme. The other source of inconsistency of the real numbers is the trichotomy axiom to which Brouwer and myself has constructed counterexamples.

That would be an improvement. However, there must be a reason people do not use this Hamiltonian scheme and resort to the standard notation. Cumbersome, perhaps? At any rate there is simple remedy by looking a i as on operator on the Euclidean plane vectors and e^itheta. This is found in the appendix to my paper, The generalized integral as dual to Schwartz distribution, in press, Nonlinear Studies.

Since a mathematical systems is completely well defined solely by its axioms reasoning within it, i.e., the rules of inference must follow from the axioms. In fact, in practice mathematicians do not use formal logic.