Applying the new primer on prime numbers
WSEAS Transactions on Mathematics
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By extending the operations +,× on natural numbers to the operations on finite sets of natural numbers, we founded a new formal system of a second order arithmetic 〈P(N), N,+,×,0,1, ∈〉. We designed a recursive sieve method on residue classes and obtained recursive formulas of a set sequence and its subset sequence of Sophie Germain primes, both the set sequences converge to the set of all Sophie Germain primes. Considering the numbers of elements of this two set sequences, one is strictly monotonically increasing and the other is monotonically increasing, the order topological limits of two cardinal sequences exist and these two limits are equal, we concluded that the counting function of Sophie Germain primes approaches infinity. The cardinal function is sequentially continuous with respect to the order topology, we proved that the cardinality of the set of all Sophie Germain primes is χ0 using modular arithmetical and analytic techniques on the set sequences. Further we extended this result to attack on Twin primes, Cunningham chains and so on.