## Tuesday, June 13, 2006

### Purpose

Today, I start a new blog track which will cover false proofs. This fits in with my Fermat's Last Theorem Blog. The idea is to cover false proofs for Fermat's Last Theorem or for other famous math problems.

It is my goal with this blog to stick to the mathematics and to the ideas. For this reason, I am seeking to analyze "incoherent" ideas from the perspectives of mathematics. What mistakes are made? What claims are made which are not backed up by evidence or argument? Are the proofs presented valid? If they are not valid, why not? Arben said...

Your wrote: "For any real number x, there is always a smaller number that also exists say x/2".

Sorry Larry, this is a bad choice of an example. F.e. for x=-1, -1/2 > -1. However, x-1 works, so no biggie.
PS: Thanks for the hard work on the blog.

10:52 AM Larry Freeman said...

Good point. My example is bad.

I should have said x - 1 is less than x.

Not only is -1/2 greater than -1 (as you point out), but also 0/2 = 0.

-Larry

2:26 PM Edgar said...

Two Fatal Defects in Andrew Wiles’ Proof of FLT

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 that proving that it is false.

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. In general, any vacuous concept yields a contradiction.

References

 Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
 Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
 Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
 Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
 Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
. Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
 Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
 Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
 Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier
Science, Ltd.), 2009, Paris.
 Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
 Kline, M., Mathematics: The Loss of Certainty, Cambride University Press, 1985.

E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University

3:45 AM E. E. Escultura said...

SOME IMPORTANT INFORMATION ABOUT THE NEW REAL NUMBER SYSTEM

We first note the sources of ambiguity in a mathematical space so that we can avoid or contain them; they are contained if their ambiguity is approximated by certainty, e.g., a nonterminating decimal which is ambiguous is approximated by its initial segment at the nth decimal place at margin of error 10^-n. We consider an ambiguous concept well-defined when its ambiguity is contained. The sources of ambiguity are: infinite set, large or small number (depending on context), self-reference, e.g., the barber paradox, vacuous concept, e.g., i = the root of the equation x^2 + 1 = 0, among the real numbers, ill-defined concept and statement involving ambiguous concept.

1) The new real number system is built on the the elements 0 and 1 defined by the addition and multiplication tables (these are the three axioms).

2) The basic digits 0, 1, …, 9 are built first, then the integers and the terminating decimals. They are the well defined decimals.

12:22 AM E. E. Escultura said...

3) Then the inverse operation to multiplication called division; the result of dividing a decimal by another if it exists is called quotient provided the divisor is not zero. Only when the integral part of the devisor is not prime other than 2 or 5 is the quotient well defined. For example, 2/7 is ill defined because the quotient is not a terminating decimal (we interpret a fraction as division).

4) Since a decimal is determined or well-defined by its digits, nonterminating decimals are ambiguous or ill-defined. Consequently, the notion irrational is ill-defined since we cannot cheeckd all its digits and verify if the digits of a nonterminaing decimal are periodic or nonperiodic.

5) Consider the sequence of decimals,

(d)^na_1a_2…a_k, n = 1, 2, …, (1)
where d is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (1) d-sequence and its nth term nth d-term. For fixed combination of d and the a_j’s, j = 1, …, k, in (1) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (1) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.

6) As convention when d* appear in any equation or expression, it means that either is unaltered if d* is replaced by any dark number, i.e., element of d*. The dark number d* satisfies the following:

N.99… - (N – 1)… = d*, N = 1, 2, …; if x is any nonterminating decimal different from zero, xd* = d*x = d*; (d*)^N = d* (1)

12:23 AM E. E. Escultura said...

7) Nonterminating decimals. Now we define a nonterminating decimal for the first time without contradiction and with contained ambiguity, i.e., approximable by certainty. We build them on what we know: the terminating decimals, our point of reference for all their extensions.
A sequence of terminating decimals of the form,

N.a_1, N.a_1a_2, …, N.a_1ª_2…a_n, … (2)

where N is integer and the a_ns are basic integers, is called standard generating or g-sequence. Its nth g-term, N.a_1a_2…a_n, defines and approximates its g-limit, the nonterminating decimal,

N.a_1a_2…a_n,…, (3)

at margin of error 10n. The g-limit of (2) is nonterminating decimal (3) provided the nth digits are not all 0 beyond a certain value of n; otherwise, it is terminating. As in standard analysis where a sequence converges, i.e., tends to a specific number, in the standard norm, a standard g-sequence, converges to its g-limit in the g-norm where the g-norm of a decimal is itself.

8) Decimal integers. A nonterminating decimal of the form N.99… , N = 0, 1, …, is call decimal integer because the set of such decimals for all N is isomorphic to the integers, i.e., the integral parts of the decimals, under the mapping d* -> 0, N -> (N – 1).99…, N – 1, 2, … From the kernel of this isomorphism it follows that (0.99…)^N = 0.99… and ((0.99…)10)^N = (0.99…)10^N.

12:25 AM E. E. Escultura said...

9) We note these important results.

a) Theorem. The d-limits of the indefinitely receding (to the right) nth d-terms of d* is a continuum that coincides with the g-limits of the tail digits of the nonterminating decimals traced by those nth d-terms as the a_ks vary along the basic digits.

b) Theorem. The g-closure (closure in the g-norm) of the decimals is a continuum R*; this the new real number system; it is a continuum, countably infinite, non-archimedian and nonhausdorff but its subspace of decimals is also countably infinite, discrete, Archimedean and hausdorff/

c) Theorem. In the lexicographic ordering R* consists of adjacent predecessor-successor pairs each joined by the continuum d*.

d) Theorem. The rationals and irrationals are separated, i.e., they are not dense in their union (this is the first indication of discreteness of the decimals).

e) Theorem. The largest and smallest elements of the open interval (0,1) are 0.99… and 1 – 0.99… = d*, respectively.

f) Theorem. An even number greater than 2 is the sum of two prime numbers (this used to be called Goldbach’s conjecture; now it has a proof in R*).

g) The counterexamples to FLT. The exact solutions of Fermat’s equation, which are the counterexamples to FLT, are given by the triples (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (4)

for n = NT > 2. Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false. One counterexample is, of course, sufficient to disprove a conjecture.

Remark. Nondenmerable set does not exist. Cantor’s diagonal method generated only countable set. Any set which is a union of countable set or whose elements can be labeled by integers is countable. Only countable set has cardinality, a continuum has none.

References

 Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University
Press, Cambridge, 52 - 61.
 Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
 Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s
last theorem, Nonlinear Studies 5(2), 227 – 2254.
 Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and
Computations, 130(1), 145 – 169.
 Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and
Computation, 138(1), 127 – 149.
 Escultura, E. E., The new real number system and discrete computation and calculus, Neural,
Parallel and Scientific Computations 17 (2009), 59 – 84.
 Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied
Mathematics Letters 21(10), 2007, 1074-1081.
 Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A: Theory, Methods and Applications; online at Science Direct website
 Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear
Studies.
 Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C:
Hybrid Systems, 3(2) May 2009, 101-107.
 Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
 Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
 Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.

E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
and Departments of Mathematics and Physics
GVP College of Engineering, JNT University

12:26 AM Keven said...

Sir,

I know this post of yours is rather old, but if you still read it, here is a nice little false proof:

We have to know two things from complex analysis, first if f and g are entire functions then f composed of g is also entire. Second is Picards little theorem, which states that a non-constant entire function defined on C takes on all values of C except at most one. So here it is:

let g(z) = exp(z), exp(z) is an entire function and thus it takes on every complex value except for at most one, in this case g(z) != 0. Now define f(z) = exp(g(z)). So f(z) is g composed with itself, and since g is entire so if f. As f is an entire function defined on the complex plane, it can omit at most one value. Now since g(z) != 0, we have f(z) != exp(0) = 1.

So we have proven that exp(z) != 0 and exp(z) != 1, thus since it is entire and omits more than one value, exp(z) must be a constant function.

If anyone wanted they could extend it and show exp(z) omits a countably infinite set of complex numbers. Define f_0(x) = exp(x), and f_(n+1)(z) = f_0 composed with f_(n-1), or f_n(z) = f_0( f_(n-1)(z)). Now f_0 != 0, f_1 != 1, f_2 != e, f_3 != e^e, f_3 = e^(e^e), and so on. This doesn't change anything, but it might be more convincing to those who prefer "convincing arguments" to "a proof".

Hope you like it.

1:29 PM j2kun said...

I would be very interested to see an explanation of the falsity in the recent false proof that P != NP...

7:17 AM E. E. Escultura said...

Reply to the preceding commeent on Esculra's disproof of FLT:

I have responded to this comment many times on many blogs including this one. Here is the chain of reasoning in the disproof:

1) The real number system that includes the integers is inconsistent because one of its axioms, the trichotomy axiom is false, counteramples to it having been counsted by me (Escultura, E. E. The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 8) and L. E. J Brouwer (Benacerraf, P. and Putnam, H. Philosophy of Mathematics, Cambridge University Press: Cambridge, 1985).

2) Therefore, the originsl FLT being formulated in it is nonsense.

3) Consequently, I had to fix the real number system first by reconstructing it into the new real number system using only three consistent axioms. This is done in the same reference in 1).

4). Then, to make sense of FLT, I reformulated it in the new real number system where I found countably infinite counterexamples to it (see the same reference above) although only one counterexaple is needed to disprove it.

E. E. Escultura

3:57 AM E. E. Escultura said...

Sorry, my reply (last post on this thread)got posted on the wrong thread. It should apply to Larry's comment that I misunderstood FLT. I didn't and the post explains how I disproved FLT. E. E. Escultura

4:04 AM E. E. Escultura said... jamestmsn said...