第一及第二数学归纳原理的推广

Generalizations Respectively of the First and the Second Principles of Mathematical Induction

将第一及第二数学归纳原理由自然数集N推广到全序整环Z的子集∑={ay+6∈Z|y遍历N中诸数,而a,b为∈Z的某二数,且a≠0},得到定理Ⅰ(第一数学归纳原理的推广):设S(■∑={a×1+b,a×2+b,…})具有性质1)a×1+b∈S;2)s∈S■s+a∈S,则S=∑及定理Ⅱ(第二数学归纳原理的推广):设S(■∑={a×1 +b,a×2+b,…})具有性质1)a×1+b∈S,2)■2≤k∈N,a×1+b,a×2+b,…,a×(k-1)+b∈S■a×k+b∈S,则S=∑.

Generalising the first and the second principres of mathematical induction from N to ∑={ay＋b∈Z/y running through N ,whilst a and b being two arbitrary numbers in Z and a≠0},the writer obtains Theorem I ： Suppose S（∈∑={a×1＋b,a×2＋b,…}）possesses the properties：1）a×1＋b∈S;2）s∈S=〉s＋a∈S,then S=∑ and Theorem Ⅱ ：Suppose S（∈∑={a×1＋b,a×2＋b…}）possesses the properties：1）a×1＋b∈S,2）∨2≤k∈N,a×1＋b,a×2＋b,…,a×（k-1）＋b∈S=〉a×k＋b∈S,则S=∑.