1、?ii?iiiiv?11.1?. . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2?Peano?. . . . . . . . . . . . . . . . . . . .41.3?. . . . . . . . . . . . . . . . . . . . . . .211.3.1?. . . . . . . . . . . . . . . . . . . . . . . .211.3.2?. . . . . . . . . . . . . . . . . . . . . . .241.4?Dedekind?. . .
2、 . . . . . . . . . . . . . . .271.5?. . . . . . . . . . . . . . . . . . . . . . .39?492.1?. . . . . . . . . . . . . . . . . . . . . . .492.2ZFC?. . . . . . . . . . . . . . . . . . . . . . . . . .512.3?. . . . . . . . . . . . . . . . . . . . . . . . . . . .572.4?. . . . . . . . . . . . . . . . . . .
3、. . . . . .622.5?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .672.6?Zorn?. . . . . . . . . . . . . . . . . . . . . .732.7?. . . . . . . . . . . . . . . . . . . . . . . .762.8?. . . . . . . . . . . . . . . . . . . . . . . . . . . .81vvi?Peano?Dedekind?1.1?1,?,?a,b,c,?1,2,3,.,?2, , e
4、,?sinx, cosx,.,?I?2,?,?x,y,z,X,Y,Z,?3,?.?, ?“”,?“”,?x + y z?x,y,z?x,y?R?xRy,x,y,z?R?R(x,y,z), x1,xn?n?R?R(x1,xn).4,?.?n?f?f(x1,xn), n?Dom(f)?Ran(f).?5,?: “”(?“?”), “”(?“”), “”(?“?”),“”(?“?”)?“”(?“?”).12?6,?: “”(?“?”)?“”(?“?”).7,?: “( , )”,“ , ”, “ , ”.?1.1.1?“=”, “=”?(i) x = x;(ii)?x = y,?y = x;(iii
5、)?x = y?y = z,?x = z;(iv)?x?P(x)?x = y,?P(y)?1.1.2?f(x1,xn)?n?c1,cn?n?(c1,cn) Dom(f),?f(c1,cn)?f(x,y) = x + y?e +2?1.1.1?(term)?(i)?;(ii)?;(iii)?f(x1,xn)?n?t1,tn?n?(t1,tn) Dom(f),?f(t1,tn)?.?1.1.2?(formula)?(i)?t1, t2?t1= t2?(?atomic)?(ii)?R(x1,xn)?n?t1,tn?n?R(t1,tn)?(?atomic )?(iii)?(iv)? , , , ?(v
6、)?x ?x ?x?x?,?x?x ?x ?.?.?( x + 1 y ) ( z = y ) ( z y?z = y?“”?z y|zz = y| z |z z y x + 1 = z?x?y, z?x?1.1.3?,?.?x = limnxn?x = limnxn ? 0 N n (n N |xn x| 0)( N N)( n N) |xn x| y;(3)?v 6= 0?y = x + v,?x y.?. (i)?P(x,y)?x = y u ( u 6= 0 ) ( x = y + u ) v ( v 6= 0 ) ( y = x + v ).?y,?y = 0?0 = y,?y 6=
7、 0?y = 0 + y,?x = 0?(1)?(3)?P(0,y)?P(x,y)?x = y?(1.2.5)?(x) = (y) = y + 1;?u 6= 0?x = y + u,?(x) = (y + u) = y + (u),?P3?(u) 6= 0.?(x)?v 6= 0?y = x + v,?1.2.2?z?v = (z).?(1.2.5)?y = x + v = x + (z) = x + (z + 1) = x + (1 + z) = (x + 1) + z = (x) + z,?z = 0?(x)?z 6= 0?(x)?P5?x?P(x,y)?(ii)?1.2.4?x+0 =
8、 x = y+u = (x+v)+u = x+(v+u),?u?v?0.?1.2.4?0 = u+v,?1.2.2?z?v = (z),?0 = u + v = u + (z) = (u + z),?P3?1.2?Peano?9?(i)?(ii)?“?”.x y x = y?x y,?“x?y”.?1.2.3?x y?u?y = x + u,(1.2.6)?1.2.4?u?x = y?u = 0.?1.2.6?(1)?: x x;(2)?: x y y x x = y;(3)?: x y y z x z.?.?“”?1.2.5?x y?y z,?1.2.3,?u?v?y = x + u?z =
9、 y + v,?z = y + v = (x + u) + v = x + (u + v),?1.2.3?x z.?1.2.3?(1)?x 6= 0, 0 x;(2)?x?x = (u).?z x,?z u;(3)?x y,?(x) y.?.(i)?x 6= 0,?x 6= 0, x = x + 0 = 0 + x,?“”?0 x.(ii)?x = (u)?z x.?z x?v 6= 0?x = z + v,?1.2.2?v = (w),?u + 1 = (u) = x = z + v = z + (w) = z + w + 1,10?1.2.1?u = z + w,?1.2.3?z u.(i
10、ii)?x y?u 6= 0?y = x+u,?1.2.2?v?u = (v),?1.2.2,?y = x + u = x + (v) = x + (v + 1) = (x + 1) + v = (x) + v,?1.2.3?(x) y.?1.2.7?x y,?z, x+z y+z.?x+y y+z,?x y.?.?x y,?1.2.3,?u?y = x + u,?y + z = (x + u) + z = (x + z) + u,?1.2.3?x + z y + z.?x + z y + z,?1.2.3,?u?y + z = (x + z) + u = (x + u) + z,?1.2.4
11、?y = x + u,?1.2.3?x y.?1.2.8?x y?z w,?x + z y + w.?.?x y?z w,?1.2.4,?u,v?y = x + u,w = z + v.?y + w = (x + u) + (z + w) = (x + z) + (u + v),?1.2.4, x + z y + w.?1.2.9?M?N?v M,? x x M v x.?v?M?.?. (i)?N = x|x N, y y M x y,?0 N.? x x N (x) N1.2?Peano?11?P5, N = N,?M?x N?(x) / N.(ii)?y M, x6= y,?x N?x
12、y,?1.2.3?(x) y,? y y M (x) y,?(x) N,?(i)?y M?x= y,?x M.(iii)?x N?y M, x y,?x?M?x?x x, x x,?“”?x= x,?1.2.10?P(x,y)?hP(0,y) ? x z z x P(z,y) P(x,y)?i x P(x,y).?P(0,y)? z z x P(z,y)?P(x,y)? x P(x,y)?.?Q(x,y)? z z x P(z,x).(i)?1.2.3?(1), z 0?z = 0,?P(0,y)? z z 0 P(z,x)?Q(0,y)?(ii)?Q(x,y)? z z x P(z,x).?
13、1.2.3?(2),?z (x),?z x,?P(z,y)? z z (x) P(z,x)? x z z x P(z,y) P(x,y)?P(x),y)? z z (x) P(z,x)?Q(x),y)?(i), (ii)?P5, x Q(x,y)? x P(x,y)?.?M = x|x N, P(x,y).?M?1.2.9,M?x.?M?0 x?z x,?P(z,y)? z z x P(z,y),?P(x,y)?x M?M? x P(x,y)?12?1.2.4 (ii)?1.2.10?.(ii)?M?m?n?x + m = n,(1.2.7)?(1.2.7)?m n.?m = n,?x = 0
14、,?1.2.4, 0?(1.2.7)?m n,?n = m + u,?x = u,?1.2.4, u?(1.2.7)?(1.2.7)?n?m?,?m n.?m?n?“”?.?m,n, (m + n) m = n, m m = 0.?1.2.11?(1)?x m + n,?x (m + n) = (x m) n;(2)?m n?x m n,?x (m n) = (x + m) n;(3)?x y?y n, x n y n.?1.2.1?1.2.11.?1.2.12?N?x y(?),?x 0=0,(1.2.8)x (y)=x y + x.(1.2.9)?.?x N,?hx(y) = x y,?(
15、1.2.8)?(1.2.9)?hx(0)=0,(1.2.10)hx(y)=hx(y) + x,(1.2.11)?x N,?N?hx(y)?(1.2.10)?(1.2.11)?1.2?Peano?13(i)?x = 0?h0(y) = 0.?h0(y)?(1.2.10)?(1.2.11).?g0(y)?(1.2.10)?(1.2.11),?g0(0) = 0, g0(y) = g0(y) + 0?h0(0) = g0(0).?y?g0(y) = h0(y),?g0(y) = g0(y) + 0 = h0(y) + 0 = h0(y),?P5?y, g0(y) = h0(y),?h0()?(1.2
16、.10)?(1.2.11)?(ii)?hx()?(1.2.10)?(1.2.11)?h(x)(y) = hx(y) + y,?(1.2.10), h(x)(0) = hx(0) + 0 = 0,?h(x)(0)?(1.2.10).?h(x)(y)?hx(y)?(1.2.11),h(x)(y) = hx(y) + (y) = hx(y) + x + (y)=hx(y) + x + (y + 1) = hx(y) + y + (x + 1) = h(x)(y) + (x),?h(x)(y)?(1.2.11).?g(x)(y)?(1.2.10)?(1.2.11)?g(x)(0) = 0, g(x)(
17、y) = g(x)(y) + (x)?g(x)(0) = 0 = h(x)(0).?g(x)(y) = h(x)(y),?g(x)(y) = g(x)(y) + (x) = h(x)(y) + (x) = h(x)(y),?P5?y, g(x)(y) = h(x)(y),?h(x)(y)?(1.2.10)?(1.2.11)?(i)?(ii),?P5?x, hx(y)?(1.2.10)?(1.2.11)?x + y?(1.2.8)?(1.2.9)?x y?x?y?,?xy.14?1.2.13 N?,? x y z x (y + z) = x y + x z.?. (i)?P(z)? x y x
18、y + x z = x (y + z).?(1.2.8)?x,y,x y + x 0 = x y = x (y + 0),?P(0)?(ii)?P(z)? x y x y + x z = x (y + z).?x,y,x y + x (z) = x y + (x z + x) = (x y + x) + x z = x (y) + x z,?P(z)?x y + x (z) = x (y) + x z = x (y) + z)=x (y + 1) + z = x y + (z + 1) = x (y + (z),? x y x y + x (z) = x (y + (z),?P(z)?(i)?
19、(ii),?P5? zP(z)?1.2.14 N?,? x y z (x y) z = x (y z).?.?x,y?P(z,x,y)?(x y) z = x (y z).?(1.2.8),(x y) 0 = 0, x (y 0) = x 0 = 0,?P(0,x,y)?P(z,x,y)?(x y) z = x (y z).?(x y) (z) = (x y) z + x y = x (y z) + x y,?(1.2.9),x (y z) + x y = x (y z + y) = x (x (z),?(x y) (z) = x (x (z), P(z),x,y)?P5?1.2?Peano?
20、15?1.2.2 N?,?y z?x (y z) = x y x z.?1.2.15 N?,? x y x y = y x.?.?y?P(x,y)?x y = y x.?1.2.8?0y = h0(y) = 0,?(1.2.8)?yx = 0,?P(0,y)?P(x,y)?x y = y x.?1.2.8?(x) y = h(x)(y) = hx(y) + y = x y + y = y x + y,?(1.2.8)?(1.2.9), y 1 = y (0) = y 0 + y = y,?y x + y = y x + y 1 = y (x + 1) = y (x),?(x) y = y (x
21、),?P(x),y)?P5, x P(x,y)?y?1.2.4?x y = 0,?x = 0?y = 0.?.?x 6= 0?y 6= 0,?1.2.2,?u?v?x = (u) = u + 1,y = (v) = v + 1,?0 = x y = (u + 1) (v + 1) = u v + u 1 + 1 v + 1 = 1 + (u v + u + v),?1.2.3?1 0,?1 = (0) = 0 + 1?1.2.3?1 0,?“”?1 = 0,?(0) = 0,?P3?x = 0y = 0.?1.2.16?x z = y z?z 6= 0,?x = y.16?.?x 6= y,?
22、x y.?x y?x = y.?1.2.17?x y, z w,?xz yw.?.?1.2.3,?u,v?y = x+u, w = y +v,?yw = (x+u)(z +v) = xz +xv +uz +uv = xz +(xv +uz +uv),?1.2.3, xz yw.?1.2.18?x 6= 0?y,?m?r?r y, y = m x + r.m?y?x?, r?y?x?.?.?x 6= 0?P(y,x)?! m ! r y = m x + r.(i)?y = 0?0 = 0 x + 0,?m = 0, r = 0.?0 = y = m x + r,?1.2.3, 0 = y r,?
23、r 0?”?r = 0,?0 = y = m r?1.2.4?m = 0.?P(0,x)?(ii)?P(y,x)?m,r?y = m x + r,?(y) = y + 1 = m x + r + 1,?r+1 = x?m= m+1, r= 0,?(y) = (m+1)x = mx+r;?r+1 x?m= m, r= r+1,?(y) = mx+r+1 = mx+r.?(y) = mx+r, r x.?r 1,?1.2.2, r= u+1,?y + 1 = (y) = m x + r= m x + u + 1,1.2?Peano?17?1.2.1?y = m x + u.?P(x,y)?m= m
24、, r = u,r= u + 1 = r + 1,?m?r?r 1,?r= 0.?m 6= 0,?(y) = 0.?1.2.2,?m= (u), x = (v),?y + 1 = (y) = m x = (u + 1) x = u x + x = u x + v + 1,?1.2.1?y = u x + v.?P(x,y)?m = u, r = v,?m= (u) = m + 1, x = (v) = v + 1 = r + 1,?m?r?P(y),x)?(iii)?(i), (ii)?P5,?y?x 6= 0, P(y,x)?m?r?,?.?a2a1?a2 10 + a1?a3a2a1,?a
25、3 100 + a2 10 + a1?35?1+1 = 2, 1+2 = 3, 1+3 = 4, 2+2 = 4, 2+3 = 5, 3+3 = 6,.?a2a1+ b2b1,?(a2+ b2) 10 + (a1+ b1).?(1)?a1+ b1 10, a2+ b2 10,?a2a1+ b2b1= (a2+ b2) 10 + (a1+ b1) = (a2+ b2)(a1+ b1);(2)?a1+ b1 10, a1+ b1= 10 + c1, a2+ b2+ 1 10,?a2a1+ b2b1= (a2+ b2) 10 + (a1+ b1) = (a2+ b2+ 1)c1;(3)?a1+ b1
26、 10, a2+ b2 10, a2+ b2+ 1 = 10 + c2,?a2a1+ b2b1= (a2+ b2) 10 + (a1+ b1) = 1c2(a1+ b1);18?(4)?a1+b1 10, a1+b1= 10+c1, a2+b2+1 10, a2+b2+1 = 10+c2,?a2a1+ b2b1= (a2+ b2) 10 + (a1+ b1) = 1c2c1;?a2a1+b2b1(a2+ b2)(a1+ b1)a2a1+b2b1(a2+ b2+ 1)c1a2a1+b2b11c2(a1+ b1)a2a1+b2b11c2c1?35?1 1 = 1, 1 2 = 2, 1 3 = 3
27、, 2 2 = 4, 2 3 = 6, 3 3 = 9,.?a2a1?b2b1?(a2a1) (b2b1)=(a2a1) b2 10 + (a2a1) b1,(a2a1) b2 10=a2 b2 100 + a1 b2 10,(a2a1) b1=a2 b1 10 + a1 b2,?1.2.1?a b?b?a?0,?a?b,?a|b,?a?b?, b?a?.?a?b,?a b.1.2?Peano?19?b?1?b.?1.2.2?a,b?c?c|a?c|b?c?a?b?.?d?a?b?a?b?c,?c|d,?d?a?b?,?(a,b).?1.2.5?x = m y + r,?r y,?(x,y)
28、= (y,r).?.?d|x, d|y,?d|r,?d|(y,r),?(x,y) (y,r).?d|y, d|r,?d|x,?d|(x,y),?(y,r) (x,y),?(x,y) = (y,r).?123?25?123=4 25 + 23,25=1 23 + 2,23=11 2 + 1,2=2 1,?123?25?25?23?25?23?23?22?23?2?2?1?1,?1=23 11 2 = 23 11 (25 23) = 12 23 11 25=12 (123 4 25) 11 25 = 12 123 48 25 11 25=12 123 59 25.?1 = 12 123 59 25
29、.?a,b,?a=m1 b + r1,b=m2 r1+ r2,r1=m3 r2+ r4,.,rk=mk+2rk1,20?rk1?a?b?.?m,n?m a n b = (a,b)m b n a = (a,b).?(a,b) = 1?a?b?.?1.2.19?x,y?0?d?x?y?m,n,?m x n y = d?m y n x = d.?.?M = c|c N, c 1, m n ( mx ny = c ) ( my n x = c ),?x,y M.?1.2.10, M?e.?z M,?z?e,?r pm, r1,rm?x = pr11prmm.?.?x = 2?z x?x?x?x?a x
30、,b pm,?1.2.19?p1= p1,?pr111prmm= p1r11pmrm,?x,?p1= p1, r1= r1, ,?x?x?1.3?211.3?1.3.1?x?x + m = n(1.3.1)?m n?N?“?”,?Z,?(1)?Z?N?Z?(2)?(1.3.1)?m,n Z,?(1.3.1)?Z?(3)?Z?N?Z?Z?+,?.?(1.3.1)?x + m = 0, m N,m 0.(1.3.2)?(1.3.2)?m,?Z?m,|m N,m 0.?n?y = (m)+n?y +m = (x+n)+m = (x+m)+n = 0+n = n.?m n?y + (m n) = 0.
31、?m n,?y = (n m).?(m) + n =(m n),?m n,n m,?m 0.?n?x = m?x + m = 0,?n x + n m = n (x + m) = 0,?n (m)?x + nm = 0?n (m) = m.?x = m?y = n?x + m = 0?y + n = 0,?0 = (x + m) (y + n) = x y + m y + n x + m n=x y + n (m) + m (n) + m n=x y + (n m),?m n,?m n = x y,?(m) (n) = m n.?Z?a,b,c Z,?a + c = b,?c?a?b?c =
32、a b.?m n=(n m),?n m,m n,?n m.m (n)=m + n,(m) n=(m + n),(m) (n)=(n m,?n m,(m n),?n 1SN?Z,?. Z?, Z?,?.?m?m 1,?m?+m.?1.3.1?Z?(1) (x + y) + z = x + (y + z), (x y) z = x (y,z), x,y,z Z;(2) x + y = y + x, x y = y x, x,y Z;(3) x (y + z) = x y + x z, x,y,z Z;(4) x + 0 = x, x 1 = x;(5) x + (1) x = 0, x Z;(6)
33、 x y = x + (1) y, x,y Z;(7)?x z = y z, z 6= 0,?x = y, x,y,z Z;(8)?x + a = b?a,b Z?Z?1.3.1?1.3.1.?1.3.1?y x N,?x?y,?x y.?x y?x 6= y,?x?y,?x x.?1.3.2?(1)?x y, y x,?x = y;(2)?x y, y z,?x z;(3)?x,y Z,?x y,?y x;(4)?x y, z w,?x + z y + w;(5)?x + z y + z,?x y;(6)?x y, z 0,?x z y z;(7)?x z 0,?x 0,?qp?;?p 0,
34、?qp?,|p|q?qp?.?1.3.4?Q = qp|p,q Z, p 1, (p,q) = 1, Q?(1) (x + y) + z = x + (y + z), (x y) z = x (y,z), x,y,z Q;(2) x + y = y + x, x y = y x, x,y Q;(3) x (y + z) = x y + x z, x,y,z Q;(4) x + 0 = x, x 1 = x;(5) x + (1) x = 0, x Q;(6) x y = x + (1) y, x,y Q;(7)?x z = y z, z 6= 0,?x = y;(8)?a,b Q, a 6=
35、0,?a x = b?Q?1.3.3?1.3.4.?1.3.4?y x =qp Q, p 0,?x?y,?x y.?x y?x 6= y,?x?y,?x x.?1.3.5?(1)?x y, y x,?x = y;(2)?x y, y z,?x z;(3)?x,y Z,?x y,?y x;(4)?x y, z w,?x + z y + w;(5)?x + z y + z,?x y.(6)?x y, z 0,?x z y z;(7)?x z 0,?x 0,?n N?n x y.1.4?Dedekind?27?1.3.4?1.3.2.?a x = b, a 6= 0?a =qp, b =nm,?np
36、mq,?b/a,?b?a?.?1.3.6?(1)?x 6= 0, x/x = 1?y/x = y (1/x), y Q;(2)?0 x y?0 1/y 0,?n N?y/n 0,O?P?OP?|x|;(2)?x 0,O?P?OP?|x|;(3)?x = 0,?x?O.?(x),?28?.?,?x?x.?P,?OP?(?),?OP?M = n|n N,On?OP?,?M?M?a1.?P?a1 1?a1?a11?b1, a1?c1.?b1c1?b1c1?P?P?b2c2?b2?c2?12.-xqqO1qqb1c1c21qb3b2qc3rP?bn, cn, n = 1,2,?(1) bncn?12n
37、,?bn?cn?(2) bn bn+1, cn cn+1;(3) P?bn?cn?bn,n = 1,2,?P,?cn,n = 1,2,?P.?OP?bn, cn, n = 1,2,.?“?”,?“?”.?“?”?Q?“?”,?R,?R?“”?(1)?1.3.4,?1.3.5?1.3.6?R?(2)?R?(x),?(i) x 0, r Q.?(?O)?P?A?B?OP?0,?OP?A?B?1.4.1?A?B?Q+?(1) A B = Q+, A B = ;(2) x A, y B,?x y;(3) A? x A, y A,?x 0,?x A,y B?y x r.?.?x A,?1.3.6?(3)
38、,?n N?x/n y,?kr B,?M?1.2.9?M?k0.?M?x = (k0 1) r A, y = k0 r B,?y x = r r.?R+?+?.?1.4.2? = (A1,B1), = (A2,B2)?Dedekind?A = x + y|x A1, y A2,B = y|y Q+, y / A,? = (A,B)?Dedekind? = x + y.?,? = + ?.?A?B?Q+?z Q+,?x A1,y A2?z x + y,?1.3.6,x/(x + y) z = x 1/(x + y) z = z 1/(x + y) x = z z/z x x,y/(x + y)
39、z = y 1/(x + y) z = z 1/(x + y) y = z z/z y w.?A1?A2?A? = (A,B)?Dedekind?,? = + ?1.4?Dedekind?31?1.4.3? = (A1,B1), = (A2,B2)?Dedekind?A = x y|x A1, y B2,B = y|y Q+, y / A,? = (A,B)?Dedekind? = .?,? = + ?1.4.4 R+?.? = (A1,B1), = (A2,B2), = (C,D)?Dedekind?( + ) ?A = (x + y) z|x A1, y A2, z C, + ?A = x
40、 z + y w|x A1, y A2, z,w C,?A A.?x z + y w A,?z w,?x z + y w x z + y z = (x + y) z A?A?Dedekind?x z + y w A,?x z + y w?A A,?( + ) = + .?r Q+,?Er= x|x Q+, x r, Fr= s|s Q+, x r,?dr= (Er,Fr)?dr+ ds= dr+s, dr ds= drs,?dr|r Q+?Q+?dr?r.?1.4.5? R+, 1 = .?.? = (A,B), 1 = (C,D), E = x|x Q+, r 1.?x A, y E,?1.
41、3.5, x y y.?u = y/z,?u = y/z = y (1/z) z 1/z = 1, u E?z u = z (y/z) = z y (1/z) = y z (1/z) = y 1 = y,?A C,?A = C,? 1 = .?32?1.4.6? R+,? R+? = 1. ?1.?.? = (A,B),?C = 1/x|x B, x?B?,D = r|r Q+, r / C.?1/x C, x B,?x?y B?y x,?1.3.6?(2), 0 1/x /y,?1/y C.?z Q+, z 1/x, x B,?1.3.6, 1/x 1/z Q,?1/z B,?z = 1/(
42、1/z) C.?(C,D)?Dedekind?.Dedekind? ?(E,F).?y A, 1/x C,?x B?1.3.6,y 1/x x (1/x) = 1,?u x (1 u) x,?y 1/x x (1 u) x 1/x = 1 (1 u) = u,?E = x|x Q+, r 1,? = 1.? R+? = 1,? = 1 = ( ) = ( ) = ( ) = 1 = ,?R+? = (A1,B1), = (A2,B2), A1 A2. ? 6= ? . ? , .?1.4.7?, R+, ?.? = (A1,B1), = (A2,B2), x x A1 x A2? x x A1
43、 x / A21.4?Dedekind?33?A1 A2,? ,? = .?x A1?x / A2,?A2 B2= Q+?x B2,?(A2,B2)?Dedekind? y A2,?y .?1.4.8?(1) ;(2)? ? ,? = ;(3)? , ,? ;(4)?, R+,?n?n .?. (1)-(3)?(4).? = (A,B), = (C,D).?r1 A,r2 D,?1.3.6?(3),?n N?n r1 r2,?n r1/ C.?1.4.7?(n + 1) ?(n + 1) ,?(n + 1) ,?n r1 .?1.4.9?, R+, + .?.? = (A,B), = (C,D
44、).?y C,?1.4.1?x1 A,x2 B?x2 x1 x2,?x1+ y / A.?1.4.7? + ? + ,? + ,?x1+ y,? + .?1.4.10?, R+, ,? R+?+ = .?.? = (A,B), = (C,D).? x.?E?Q+?z Q+?z x,?x + z x,?C?Dedekind?z C, z y,?z / A?y = (yx)+x, z = (zx)+x34?y x z,?z + (y x) y, z E,?1.4.1,?u A, v B,?v u u, v = u + (v u) v,?z v E, (z v) + u G,?1.3.4,(z v)
45、 + u = z + (1) v + u = z + (1) v + (1) u=z + (1) (v u) z + (1) (z y) = z + (1) z + y = y,?y G,?G = C,? + = .?1.4.11? ,? + + .?.? + + .? = (A,B), = (C,D), = (E,F).?x1,x2 C, x1,x2/ A, x1 x2,?1.4.1,?y1 E,y2 F,?y2 y1 x2 x1.? + = + ,? + = (G,H),? + ,?x2 C?y1 E?x2+ y1 G.? + = (G,H),?x A, z E,?x2+ y1= x +
46、z,?x2 x = x2+ y1 (x + y1) = x + z (x + y1) = z y1,?x A?x1/ A?x x1,?z E?y2 F?z y2,?x2 x1 x2 x = z y1 y2 y1,?y2 y1 x2 x1? + ,?( ) = ( + ).1.4?Dedekind?35?.? ?1.4.11? = ( ) + + ,? ( + )?1.4.4,( ) + ( + ) = ( ) + + = ( ) + = ,?( ) = ( + )?1.4.4?1.4.13?, R+, , ( ) = .?R?R?R= x|x R+,R = R+SRS0. R?, R+?, R
47、?, R?.?:x + y=x + y,?, R+,x ,?x, R+, y = ?x ,( x),?x, R+, y = ?x y,y ,?,y R+, x = ? y,0,?x = y, y R+?x = y,x y=x y,?x,y R+,x ,?x, R+, y = ,0,?x = 0y = 0, y,?,y R+, x = , ,?, R+, x = , y = .?R?R+?x R, x + 0 = x, x 0 = 0, x 1 = 1, (1) + 1 = 0.?x R, x 6= 0,?x1=(x1,?x R+,1,? R+, x = .36?x R, x 6= 0, x x
48、1= 1.?1.4.4,?1.4.12?1.4.13?R?1.4.1 R?R?x y = x + (1) y.?(x y) + y = x + (1) y + y = x + (1) + 1 y = x + 0 y = x,?x y?X?X + y = x?X + y = x,?xy = (X +y)y = X +y +(1)y = X +1+(1)y = X +0y = X,?x y?X + y = x?1.4.0?1.4.1,?x,y R+?y x?x y?0?X +y = y?y y?X +x = x?y y = 0.?x y = 0,?y = 0+y = (xy)+y = x+(1)y
49、+1y = x+(1)+1y = x+0y = x,?x = y?x y = 0.?1.4.2?y x R+,?x?y,?x y.?x x.?1.4.1 (1)?x,y R+?x y?z R+?y x = z,?y = x + z,?1.4.9?1.4.10?x y?R+?x y.(2)?x = 0,?0 y?y 0 = y R+.?0 y y R+,x 0.(3)?x = y?y x = 0,?x y y x 0.1.4?Dedekind?37?1.4.2?1.4.1,?1.4.1?1.4.8?(4),?R?(1)?x y, y x,?x = y;(2)?x y, y z,?x z;(3)?
50、x,y R,?x y,?y x;(4)?x y,?x + z y + z;(5)?0 x, 0 y,?0 x y;(6)?x,y 0,?n N?n x y.?1.4.2?x y(?x y), z y z).?1.4.3? R?K R,? ,? K(? K),?(?),?K?(?).?(?)?(?),?sup(?inf ).?.?1.4.2? R? = | .?sup = inf, inf = sup.?1.4.3 (?)? R?.?+= | , 0,?(a) +6= ;(b) += ?( r Q)r 0 ( ) r;(b) += ?( r Q)r 0 ( ) r.(i)?+?+? +,? =
