[2] **viXra:0901.0003 [pdf]**
*submitted on 14 Jan 2009*

**Authors:** Fu Yuhua, Fu Anjie

**Comments:** recovered from sciprint.org

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

[1] **viXra:0901.0002 [pdf]**
*submitted on 3 Jan 2009*

**Authors:** Tong Xin Ping

**Comments:** recovered from sciprint.org

N = p_{i} + (N-p_{i}) = p+ (N-p). If p is congruent to N modulo p_{i}, Then (N-p) is a composite integer,
When i = 1, 2,..., r, if p and N are incongruent modulo p_{i}, 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