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0901 Submissions

[3] viXra:0901.0003 [pdf] submitted on 14 Jan 2009

A Revision to Gödel's Incompleteness Theorem by Neutrosophy

Authors: Fu Yuhua, Fu Anjie
Comments: recovered from

According to Smarandache's neutrosophy, the Gödel's incompleteness theorem contains the truth, the falsehood, and the indeterminacy of a statement under consideration. It is shown in this paper that the proof of Gödel's incompleteness theorem is faulty, because all possible situations are not considered (such as the situation where from some axioms wrong results can be deducted, for example, from the axiom of choice the paradox of the doubling ball theorem can be deducted; and many kinds of indeterminate situations, for example, a proposition can be proved in 9999 cases, and only in 1 case it can be neither proved, nor disproved). With all possible situations being considered with Smarandache's neutrosophy, the Gödel's Incompleteness theorem is revised into the incompleteness axiom: Any proposition in any formal mathematical axiom system will represent, respectively, the truth (T), the falsehood (F), and the indeterminacy (I) of the statement under consideration, where T, I, F are standard or non-standard real subsets of ]-0, 1+[ . With all possible situations being considered, any possible paradox is no longer a paradox. Finally several famous paradoxes in history, as well as the so-called unified theory, ultimate theory and so on are discussed.
Category: Number Theory

[2] viXra:0901.0002 [pdf] submitted on 3 Jan 2009

The Chinese Remainder Theorem . Goldbach's Conjecture (A) . Hardy-Littewood's Conjecture (A)

Authors: Tong Xin Ping
Comments: recovered from

N = pi + (N-pi) = p+ (N-p). If p is congruent to N modulo pi, Then (N-p) is a composite integer, When i = 1, 2,..., r, if p and N are incongruent modulo pi, Then p and (N-p) are solutions of Goldbach's Conjecture (A); By Chinese Remainder Theorem we can calculate the primes and solutions of Goldbach's Conjecture (A) with different system of congruence; The (N-p) must have solution of Goldbach's Conjecture (A), The number of solutions of Goldbach's Conjecture (A) is increasing as N → ∞, and finding unknown particulars for Hardy-Littewood's Conjecture (A).
Category: Number Theory

[1] viXra:0901.0001 [pdf] submitted on 3 Jan 2009

On Dark Energy, Weyl Geometry and Brans-Dicke-Jordan Scalar Field

Authors: Carlos Castro
Comments: recovered from

We review firstly why Weyl's Geometry, within the context of Friedman-Lemaitre-Robertson-Walker cosmological models, can account for both the origins and the value of the observed vacuum energy density (dark energy). The source of dark energy is just the dilaton-like Jordan-Brans-Dicke scalar field that is required to implement Weyl invariance of the most simple of all possible actions. A nonvanishing value of the vacuum energy density of the order of 10-123M4Planck is derived in agreement with the experimental observations. Next, a Jordan-Brans-Dicke gravity model within the context of ordinary Riemannian geometry, yields also the observed vacuum energy density (cosmological constant) to very high precision. One finds that the temporal flow of the scalar field φ(t) in ordinary Riemannian geometry, from t = 0 to t = to, has the same numerical effects (as far as the vacuum energy density is concerned) as if there were Weyl scalings from the field configuration φ(t), to the constant field configuration φo, in Weyl geometry. Hence, Weyl scalings in Weyl geometry can recapture the flow of time which is consistent with Segal's Conformal Cosmology, in such a fashion that an expanding universe may be visualized as Weyl scalings of a static universe. The main novel result of this work is that one is able to reproduce the observed vacuum energy density to such a degree of precision 10-123M4Planck, while still having a Big-Bang singularity at t = 0 when the vacuum energy density blows up. This temporal flow of the vacuum energy density, from very high values in the past, to very small values today, is not a numerical coincidence but is the signal of an underlying Weyl geometry (conformal invariance) operating in cosmology, combined with the dynamics of a Brans-Dicke-Jordan scalar field.
Category: Relativity and Cosmology