Απο Την Διαισθηση Του Συνεχους Στην Θεμελιωση Των...
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pipi - pi pi
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pipi: pi
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1 . 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 . . . . . . . . . . . . . . 121.1 . . . . . . . . 13
1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 pi . . . . . . . . . . . 201.2 . . . . . . . . . . . . . . . . . . . . 26
2 pi . 392.1 . . . . . . . . . . . . . . . . 392.2 pi . . . . . . . . . . . . . . . . . . . . . 432.3 pi . . . . . 48
3 pi . 553.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 . . . . . . . . . . . . . . . . . . . . . . . . 613.3 pi Q+0 . . . . . . . . . . 61
3.3 pi 0 pi - . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 pi 0. . . . . . . . . . . . . 633.3 (
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vi
3.4 G . . . . . . . . . . . . . . . . . . 1123.4 G . . . . . . . . . . . . . . . . . . . 115
3.5 R pi . . . . . . . . . . . . . 1193.5 pi . . . . . . . . . . . . . . . . . . . . . . . . 1213.5 R. . . . . . . . . . . . . . . . . . . 1213.5 R. . . . . . . . . . . . . . . . . . 1243.5 R. . . . . . . . . . . . . . . . . . 1293.5 pi R. . . . . . . . . . . . . . 1333.5 R. . . . . . . . . . . . . . . . . . . 136
Dedekind. 139
R . 147
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1.1 pi . . . . . . . . . . . . . . . . . 51.2 . . . . . . . . . . . . . . . . . . . . . . . 71.3 . . . . . . . . . . . . . . . . . . . 91.4 Bolzano . . . . . . . . . . . . . . . . . . . . . . . . 111.5 . . . . . . . . . . . . . . . . . . . 281.6 pi . . . . . . . . . . . . . . . . . . . . . 371.7 . . . . . . . . . . . . . . . . . . . . . . . 38
2.1 pi . . . . . . . . . . . . . . . . . . . . . 432.2 pi - . . . . . . . . . . . . . . . . . 44
3.1 pi . . . . . . . . . . . . . . 583.2 . . . . . . . . 59
.1 . . . . . . . . . . . . . . . . . . . . . . . . . 139.2
2 . . . . . . . . . . . . . . . . . . . . . . . . 141
.3 x, x . . . . . . . . . . . . . . . . . . . . . . . . 143.4 x . . . . . . . . . . . . . . . . . . . . . . . . . 144.5 x+ x . . . . . . . . . . . . . . . . . . . . . . . . . 145
vii
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ix
pipi pi - . pi pi - , pi pi pi pi pi pi pi .
pi pipi pi-pi, . pipi pi - , pi pi pi pi . pi , pi - pi pi pi. pi pi, pi pi .
pi : . pi pi pi, pi pi pi.
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pi pi - pi . pi -pi, pi ( - ) pi pi pi pi . , pi , . pi pi.
pi pi pi pi , pi pi pi - pi pi - , pi .
, pi . pi - pi . pi, pi . , pi . - , pi pi . pi pi pi , pi .
, ( pi) pi pi, , ( ) pi pi pi pi pi pi. pi, pi pi Fischbein.
, pi pi- pi pi. pi pi
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: pi pi, , , , pi- pi pi pi ( pi) pi pi . , pi pi pi , pi pi pi (reification) Sfard.
pi, , - Dedekind pi- Dedekind. pi pi pi pi pi pi pi pi , -, pi pi .
, pipi pi -pi . , pipi pi pi , 1 2, pi pi pi , pi - . pi, pi pi , pi pi pi.
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pi pi . pi pi 19o - 20o . pi pi ;
1.1 .
, pi (mathemati-cal reasoning) pi pi, pipi pi pi - (Bruner 1960 pp. 66 77, Fischbein 1987 pp. 11,Longo 1998, Poincare 1997, 1999 : 323).
pipi pi pi , - pi pi pi , - . pi pi pi , . pi, pipi .
pi pi pipi pi pi , .
(Vinner 2002):
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1. .2. pipi pi pi .3. pi
pi.
( insight).pi pi (immediate), pi
pi pi pi pipi, , , - pi pi . ; pi pi . , pi , pi pi- pi , pi , pi, pi pi .
Hilbert, 19o , - , pipi- pi pi (Hilbert 1995). , . , pi pi Hilbert, pi pi pi pi pi.
pi pi, - . , , . , , pi-pi pi pi (compression process) pi pi pi (structural schema) (Fischbein 1999, pp. 49).
pi - pi pi , pi - pi pi pi pi pi, pi pi pi pi pi pi .
Bruner pi Webster : - (Bruner 1960, : 70).
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1.1. . 3
, pi pipi, . pi pipi Fischbein, .
Fischbein pi pipi , pi pi (self-evidence), pi pi(extrpolativeness), (coerciveness), (globality) (Fischbein 1987, pp. 14). , pi pipi pi pi, pi ,pi pi pi (Fischbein1987, pp. 26).
pipi pi , pi pi pi pi pi ,pi pi pi.
pi pi. pi pi, pi pi pi pi pi, pi pi . pi pi , , pi pi pi pi pi pi pi ( 1999, : 146).
pi pi pi pi pi pi , pi . pi - pi .
pi , . - pi ( , pi )pi , , pi pi - pi (Kronfellner 2003, : 249).
pipi , pi -
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4 1. .
pi pi pi (processes) pi pi , pi pi (Tall & Vinner 1981, pp. 152). pipi pi pi pi pi - , pi pi pi . pi pi pi .
pi pi, - . pi pi pi , pi pi pipi (mental images) (Tall & Vinner 1981, pp. 151).
pi (2005, : 257) pi (enactive) pi - procept, pi pi process concept, Gray & Tall. pipi pi pi pi (Gray& Tall 2001).
proceptual pi pi - (Gray & Tall 2001). procepts, -pi (Tall, Thomas, Davis, Gray & Simpson2000) pi pi (Tall 1995, pp. 7). pi pi (transformation) . , pi - pi (Tall, Thomas, Davis, Gray & Simpson 2000, pp. 225).
procepts (concept image) (Tall & Vinner 1981, Tall 1995, Gray & Tall 2001). pi , pi pi (mentalpictures), (processes) . pi , -pi , pi - pi pi pi pi pi pi (formal concept definion) (Tall & Vinner1981).
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1.1. . 5
1.1: pi (pi Tall 1995, pi 2005 : 265).
pi , pi pi pi - pi . -, pi pi pipi .
pi pi - pi pi pi pi . . . pi ( & 2004, : 9293).
pi pi pi - pipi, pi pi- Kant (Aquila 2004, pp. 250). pi, pi pi - , pi (representation). . . (Aquila 2004, pp. 250).
pi pi pi pi
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pi pi pi., pi pi , pi pi pi pipi pi, pipi.
pi- - pi pi pi pi pipi . pi pi , .
pi pi pi . (250 pi..),pi , - pi pipi pi (, , [90b3 7]). pi pi (450 ..) (, pi , [85,1-96, 15]). (300 pi..) -, pi pi pi pi . pi pi . pi- pi pi pi.
pi- pi pi pi pi pi . pipi pi pipi , pi pi pi pi pi , pi , pi (Tall 1995, pp. 7).
pi , pi , pi - . pi , pi pi , pi -pi (Tall & Thomas & Davis & Gray& Simpson 2000, pp. 238). pi, pi , pi - pi
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1.1. . 7
1.2: - (pi Tall 1995, pi 2005 : 265).
, (Gray, Pinto, Pitta & Tall 1999, pp. 125 Tall & Thomas & Davis & Gray &Simpson 2000, pp. 238).
- pi pi- pipi:
pi , pi, pi, (mental ob-ject), pi pi pi (Harel & Tall 1991, pp. 39).
pi pi: . . . , pi pi pi pi pi (encapsulation) (mental obje-ct) pi (Tall& Thomas & Davis & Gray & Simpson 2000, pp. 233).
pi , pi pi pi - . pi pi pi ,
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8 1. .
pi , pi. pi- pi pi pi, pi, pi.
pi , pi - pi pi . pi pi (pi-).
pi pi pi , . pi pipi (PET ) ( 2001, : 70 72).pi pi pi pi pi . (solution processes) pi , pi pi pi (Gray & Tall 2001, pp. 6).
pi pi pi pi-, pipi, pi . pi , pi pi pipi .
, (evolution of brain) pi pi pi pi pi pi pipi pi pi (Zervos 1972, pp. 403). pi :
pi pi pi pi pipi pi . , pi pi, pi pi pi , ( , , ) pi pi . , pi pi pipi, pi, , pi (Zervos 1972, pp. 402).
pi pi pi Poincare,pi pi pi pi pipi, ,
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1.1. . 9
1.3: pi (pi Tall1995 pp. 172, pi 2005 : 270).
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10 1. .
, (Poincare 1997, : 197). pi -
pi pi pi pi pi, pi pi pi. pi pi, pi pi pi pi pi pi pi pipi.
[] pi pi, pi, pi . pi, pi pi pi pi pi , pi pi, pi pi pipi (Dreyfus 2003, pp. 126).
pi , pi, . 20o pi, - pi pi pi (Dreyfus 2003, pp. 124). pi (- 1973, : 69). , pi pi pi pi pi pi - pi pi pi pipi pi pi , pi, pipi pi pi , pi pi pi pi (pi 1985, : 278).
, pi ; pi, pipi -
pi
1o : 19o Jordan pi, pi - (pi 2005, : 236). pi piJordan pi pi pi. pi , pi pi pi Jordan, pi- pi pi. pi pi pi ,
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1.1. . 11
pi .
2o : Bolzano1: f [, ] f() f() < 0 f x0 (, )( 1996, : 57 58).
pi pi , pi pi pipi pi . pi - , pi pi pi pi - pi pi pi pi.
1.4: pi pi Bol-zano ( 1996, : 58).
pi pi pi, pi , pi , pi pi, pi . pipi. pipi, pipi , pi .
pi pi - , pi pi .
1 Bolzano-Weierstrass.
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12 1. .
pi pi . pi. pi -pi pi 20o , pi pi pi .
Poincare 1905 "La valeur de la science" :... pi pi pi pi, pipi2 pipi pi, (Poincare 1997, : 34).
pi Brunner 1960 "The process of education" : pipi pi pi (Bruner 1960, :69).
1.1 .
, pi . pi pi - (Fischbein 1987, 1999) . pi .
pi , pi. .
Fischbein (1987, pp. 13) pi pi:
pi ( immediate cognition). - pi pi pi . pi . pi. .
. pi pi pi . pi, pi .
pi pi , (sensorial cognition), (Fischbein, 1999 pp.18). -pi pi , pi pi pi pi pi pi pipi .
2 pipi Poincare (Poincare 1997, : 204).
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1.1. . 13
pi pi pipi pi pi pi . , pi pi pi
pi (Fischbein 1987, Longo 1998, Vinner 2002), pi pi pi pi pi, pi pi - . pi , pi pi pipi pi . , pi - pi pi pi pi . pi pi- pi pi pipi pipi pi , pipi . pi pi pi pi pi pi.
pi pi pi pi- pi pi pi (Fischbein1987, pp. 14).
( , , pi, pi, )...(Fischbein 1999, pp. 18).
pi pi pi Fischbein (1987, pp. 14):
pi pi pi , pi pi pi . pi- pi . pi pi pi , -. pi pi .
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, pi pi pi pi pi pi pi. - pi pi pi pi . , . pi pi pi pi pi-, pi, pi pi pi pi.
pi pi pi pi, pi, -pi pi pi pi
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pi . - , pi, pi pi.pi pipi pi .
pi Bruner : pi ,
pi -. , , , pi , pi. , pi pi pi pi pi . , pi pi pi, , pi pi pi (Bruner 1960, :70).
pi pi :
pi, , . pi pi. pi pi pi pi . pi pi pi pi , . pi pi pi, pi .
, pi pi, . , , pi pi -, pi , pi. , , . pi- pi pi pi pi pi pi . - , pi pi pipi , , pi- , pi pi pi (Bruner 1960, : 68 69).
, , Fischbein :
pi pi pi pipi -
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1.2. . 15
(Fischbein 1999, pp.18):
pi pi pi. 3.
pi pi , pi. (logically-based cognitions).
pi , pi pi . pi pi .
. . . pi pi pi pi pipi pi (Poincare1997, : 37).
pi pi pi - . pi , pi pi pi , (Hadamard 1995, : 113 114).
pi pi - 4 (Bruner, 1960, : 75).
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pi - (, pi , [8, 1 6] [8, 22 25]), pi . (450 pi..), , - .
pi pi pi . pi pi .
3 .4 Bruner (Bruner 1960, pp.
66)
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pi . . . , pi pipi pi pi pi (pi 1985, : 67).
pi Whitehead :
[] pi pi . pi pi- pi. pi pi - pi: pi pi , pi . . . pi pi pi- pi pi pi . pi pipi pi, pi ( 1973, : 426).
pi pipi Whitehead pi pi . pi - -pi (pi1985, : 67), Whitehead -pi ( 1973, : 414). pi pi .
pi pi , , pi. pipi pi, pi, .
pi , pi , pi . pi pi pipi pi ( 1973, : 56).
pi pi pi. - pi pipi pi, pi. pi pi , pipi . , pi pi pi.
pi pi pi pi -
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1.2. . 17
pi pi pi pi . . . , - - . pi -. , , pi pi , pi pi pi pi - pi pi pi pi (pi, : 38).
pi pi pi, pi - pi pi pi pi pi pi , .
pi pi -, , pi - pi pi , (Longo, 1998, pp.10).
, , Fischbein. (lay intuitions) pi pi pi (Fischbein 1987, pp. 60) (expert intuitions) pi pi pi, pi pi pi . pi (conjectural intuitions).
pi pi pi - . pipi, - . .
pi pi . . pi pi- pi pi pi pipi pi-. pi pipi pi , pi pi . pi-
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pi R . pi pi ;
20o Poincare : , pi - pi pi (Poincare 1997, :103).
20o Rucker : R . - pi pipi pi. pi ( ), - R. , pi pi . . . pi, pi pi - pi , (Rucker 2004, pp. 265).
pi pi pi - pi pi . pi pi ( ) pi- pi.
pipi pi pi pi . pi pi pi . pi , pi pi , pi , .
Longo - pi pipi: , .
pipi ,
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1.2. . 19
pi pipi. pi- - (conceptualconstructions) (Longo 1998, pp. 21).
, pi-pi pi pi pi pi- . ,pi , pi pi pi pi pi -, pi pi pi.
pi pipi pi pi- , pi , pi, pi pi , pi pi pi . pi pi pipi. pi pipi pi - (original intuitions), pi pi pi pi (mathematical praxis). pi pipi, pi pi ( -pi pi pi , , pi ) pi . (Longo 1998, pp. 20).
, pi pi , pi pi pipi. , pi pi pipi pi pi. - pi pi pi (2003, : 9).
pi pi pi , . pi pi, , pi pi. pi pi- pi pi , pi pi- . pi , pipi pi, pi .
pi pi , pi
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, pi pi - pi pi , pi pi . , pi, pi . pi (2003, : 11) -.
pi pi pi pi pi pi . pi pi ( 2003, : 10).
1.2 pi .
pi pi , pi pipi. pi pi pi pi pi pi pi pi pi pi .
pi , pi pi. pi . pi pi - pi . pipi pi , pi pipi , , pi pi . pi pi - pipi pi . pi - pi pi.
pi pi pi -. pipi , , .
, pi (, , [219b, 1 2])
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1.2. . 21
pi - (pi 1985, : 73). -pi pi pi pi , pi - pi pi , pi , .
pi - pi, , (pi 1985, : 71). pi, pi . pi pi . pi, a priori, pi pi pi. pi, , pi pipi pi pi pi .
Weyl , . : [] pi -,. . . , pi pi , pi (pi, , pi pi ) (Weyl1987, pp. 92).
, pi , pi pi pi, pi . pi, , pi pi pi pi . pi pi . pi , .
Poincare pi pi pi - .
[] pi pi pi- pi pi , , pi , pi , pi , pi pi pi, pi (Poincare 1997, : 41).
,
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22 1. .
pi pi (Poincare 1997, : 42).
Weyl pi (phenomenaltime)- pi pi (objective time)- pi - pi, pi (Weyl1987, pp. 88). , (Weyl 1987, pp. 90).
pi Kant pi - . . . pipi (sensible intuition)(Aquila 2004, pp. 250).
Whitehead , pipi (1) - (sense-time) - (sense-space), (2) - ( thought-time of perception) -( thought-space of perception).
- - pi (actually)pi - ( time-relations) - (space-relations) (sense-objects). - - , , pi pi- , pi pi pi, - - pi pi ( fragmentary).
- - pi - ( thought-objects of perception). - - pi . - - ( ). (Whitehead 1917, pp. 162)
Whitehead pi pi . ( ) pi pi . ( ) .
- Fischbein- , pi. Whitehead pi Fischbein,
-
1.2. . 23
Whitehead pi Fischbein. pi pi.
Whitehead pi ( pi ) pi (events) ( 1973, : 406 407). pi ( 1973, : 408).
pi pi pi . pi . pi pi pi pi , .
pi pi pi pi pipi:[] pi pi -
( Longo 1998, pp. 3). pi , pi
pi pi pi pi : , , . . . (Longo 1998, pp.4).
pi . - pi pi (pi, - : 38: Poincare 1997, : 55). - pi- pi pi pi (Weyl 1987, pp. 92). , - ( 1973, : 415).
(conceptionof space) (conception of time)- (Zervos 1972, pp. 412).
Poincare pi pi pi (Poincare 1997, : 90). pi pi , pi . pi pi pi-pi pi .
pi pi pi , pi , , pi pi
-
24 1. .
pi . .
pi pi pi pipi , . , pi a priori , pi, pi .
Rucker , pi , pi pi pi pi. :
[] pi pi , pi pi . . . [] , pi pi . pi, pi pi (Rucker2004, : 11).
pi Rucker 5 [] pi - . pi, pi pi pi . , pi pi pi pi (Rucker 2004, : 183).
pi pi, pi pi pi, pi- pi pi Weyl pi (Weyl 1987, pp. 88) pi .
, pi- pi .
pi pi pi pi . pi , , , , , . pi Longo (Longo 1998, pp. 6).
pi pipi pi pi pi pi pi. pi pi pi-
5 pi Kurt Godel pi 1949 - "A remark on the relationship between relativity theory and idealistic Philosophy".
-
1.2. . 25
, pi pi pi . pi pi (Zeki S. 2002, :87). pi -pi pipi pi. - pipi pi pi. pi pi pi pi , [] , pi pi pi- , pi (Lakoff & Nunez 2000, pp. 21). pi, pi pi pi pi (Lakoff and Nunez 2000, pp. 22)
pi, , , pi, pi, pi . - pi.. pi. pi pipi, pi pi ( 1973, :435).
pi pi , pipi pi pi - pi , pi pi pi , pipi. pi- , pi , pi pipi. pi , pi pi - , pi pi .
pi, pi pi pi , pi, , - pi pi . pi pi, (pipi) - pi pi pi, ,pi pi
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26 1. .
. pi pi pi
pi pi , pi pi pi - . pi pi pi pi pi .
pi pi pi- pi , pi pi pi pi minima pi . pipi pi .
pi pi pipi: pi, ,
pi pi minima, pi pi pi (,11 , : 329).
1.2 .
. : [] ( 1973, :73), (Longo 1998, pp. 24).
, pi pi pi pi, pi , pi pi pi-, pi pi pi . pi 6, , pipi .
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6 [299a25 30], [296a16 17] [299a30 31] pi .
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1.2. . 27
, , pi-pi .
(, , [299a25 30])
: pi (De caelo II. 13, 296 a 17) ( ibid. III. I, 299 a 30). , . . . pi pipi pi pi pipi pi (Zervos 1972, pp. 419).
.
(, , [296a16 17])
- , .
(, , [299a30 31])
pi pi pi , pi- pi 19o , Cantor pipi J1, J2 J1 J2. pi 1.5, pi J1, J2 pi pi pi pi pi pi pi (Kamke 1963, :24). , , , , pi 7 [0, 1], (Kamke 1963, : 25).
pi pi pi 8. pi- pi pipi ( ), [0, 1]. - pi . n - pi pi [0, 1](Kamke 1963, : 66) , pipi n -
7 [0, 1], [0, 1), (0, 1] (0, 1) , [0, 1] [0, 1) (0, 1] (0, 1) (Kamke 1963, : 23).
8 pi pi.
-
28 1. .
1.5: J (Kamke 1963, : 24).
( n - ) pi [0, 1].
pipi pi ( 1973, : 72). pi, pi pi pi pi. pipi .
pi pi- pi - , pi pi , pi (, pi , [95, 22]). []pi pi - : pi pipi pi pi qua , pi pi (Heath 1956, vol. 1, pp. 155).
pi : pi (,, [24e4 5]).
pi , pi . pi . . pi ( 1972, : 419).
(300 pi..) , . pi pi pi
-
1.2. . 29
pi :
, ( 1975 , : 38).
pi , [85, 1-96, 15], pi 9. -, pi , pi, pi pi pi . pi pi pi pi .
Hilbert pi pi pi pi , (Hilbert 1995, : 131).
Weyl - pi 10 ( immediately given) . pi pi pi - , pi (judgment schemes). pi pi , pi pi pi - pi pi - , pi pi (Weyl 1987, pp. 8).
pi pi , pi , pi pi pi, pi. pi ; pi pipi pi , -pi pi pi pipi , pi pi pi . pi pi pi -pi pi pi.
pi , a posteriori, .
(conceptualconstruction), pi .
9 [88, 5 13], [88, 17 24]10 pi .
-
30 1. .
, , pi pi , . pi - pi - - pi pi pi pi Weyl, Wittgenstein Thom, (Longo 1998,pp. 5).
pi pi pi, Cantor-Dedekind pi , pi pi pi pi pi (Longo 1998, pp. 7).
, , :
pi , pi , pi pi pi . pi pi pi pipi pi ( pi pi pi ), pi pi pi pi (Zerbos1972, pp. 431).
pi pi pi - pi pi . pi pi- pi pi pi . [510b6 7] pi11, pi pi - , pi ( 1939, :469). , pipi pi pi pi, pi pi pi pi pi -pi pi pi (, ,[1006a5 15]).
pi , pi . pi pi ( Dedekind) pi 12.
11 pi pi [511b7].12 Cauchy.
-
1.2. . 31
pi pi pi pi pi- pipi:
pi - (Weyl 1987, pp. 90).
, pi ( temporal continuum), . . . , pi pi (spatial continuum) (Longo 1998, pp. 5).
pi Poincare pi, , - : [] , (Poincare 1997, : 66).
pi ( 1972, : 419), pi , . pi pi pi pi pi- . , pi , pi pi pi - pi pi .
13 (continuum) pi pi - , pi . pi pi pi , pi - 14 . 19o pi . pi , - , , . pi 15 (Kamke1963, : 97). .
13 pi pi - .
14 pi pi (Kamke 1963, : 97).
15 pi M = A + Z M A Z , A Z pi . pi, pi (0, 1), (1, 2) , (0, 1) + (1, 2) (Kamke1963, : 96).
-
32 1. .
Dedekind pi (Dedekind 1963, pp. 9). . , (extensive magnitudes). Dede-kind pipi pi pi (Dedekind 1963, pp. 10), pi , . pi , pi - , pi pi pi . pi pi , Dedekind.
pi (R) pi (Q) Dedekind. pi (R) pi (Q) Cauchy Cantor. 1872 (Rudin 2000, : 30 31).
pi ( pi ) . , pi pi-. pi , pi (pi,pipi, pi 0, 1, pi ).
pi , pi . - pi pi, , , pi, pi pi16 pi- pi . pi pi pi pi R, Q,. pi ( ) pi pi pi R, pi Q,. -, pi pi pi pi (supremum), R, Q,. , pi -
16 pi ( 2003, : 62).
-
1.2. . 33
pi pi pi pi . pi pi pi.
pi pi Dedekind - , pi pi - (pi ) (pipi ).
Dedekind pi - , , pi- .
- . pi pi , , pi .
pi pi - pi . ( : 48). - pi pi pi pi ( ) pi-. pi . pi pi pi - , (pi 1985, : 211).
pi . - pi, pi, pi - , pi pi pi pi . pi pi , ,pi , pi , - ( pi ).
, pi , , pi, -pi. pi pipi Cantor Dedekind.
-
34 1. .
: Cantor De-dekind pi, pi , pi( 1973, : 65).
pi . , pi Cantor "Grundlangen einer allegemei-nen Mannigfaltigkeitslehre" 1883: pi pi pi , pi pi, , pi , pi pi pi pi pi, , (Zervos 1972, pp. 437).
Dedekind "Essays on the theory of numbers" pi , pi (system). pi pi:
pi pi, a, b, c, . . . pi pi pi pi, pi (associated) S - pi a, b, c, . . . S, pi pi S , S pi17 (consists) pi (Dedekind 1963, pp. 45).
pi - pi , pipi pi. pi- . [] pi - ( 1904) pi (Kuratowski1972, pp. 79).
pi pipi pi pi Zermelo (18711953) pi pi Fra-enkel (1891 1967), Zermelo-Fraenkel (- ZF) Zermelo-Fraenkel pi18(axiom of Choice) ( ZFC).
17 pi , pi Dedekind pi .
18 pi : - - - - (Halmos 1960, pp. 59).
-
1.2. . 35
pipi pi : Zermelo, pipi
pi [ ], , pi pi Hilbert (18621943), pi pi ( 1973, : 65).
, pipi pi pi -, , , . ZFC pi pi - pi (Rucker 2004, : 277), pi pi pi. - , -. , pi pi . pi pi pi - , pi pi pi .
Cantor pi pi pi , -pi (Mankiewicz 2002, : 152).
pi , pi pi - pi pi , pi . , pi - , pipi pipi pi pi , , , - pi pi.
pi - , . pi, , , - , . , pi pi ( 1973, : 58).
pi pi pi - ( 1973, : 47).
pi 1887 pi Dedekind -
-
36 1. .
, pi ( 1973, : 48). Dedekind pi pi.
pi pi pi pi pi , pi pi pi . pi pi Cantor , pipipi pi19. pi pi pi - -20 , pi 20 = 1, pi pi (HC) pi . pi pi pi pi pipi , - ( : 63). pi pi ( ) pi ( ).
pi pi pi 20, 0 pi ( 1.6). pi pi pi- pi . pi pi - pi. pi pi- , pi . pi pi 20 ( pi- ), 0 ( ) (Rucker 2004, :263 265).
pi, , pi Cantor . Cantor pi pi I = [0, 1] (pi1994, : 42).
pi pi , pi , pi pi ,
19pipipi pi pi pi pi-pi pi (Kamke 1963, : 30).
20 pi . , a - a. , 0 , pi (Rucker2004, : 82).
-
1.2. . 37
1.6: pi (Rucker 2004, : 264).
21( 1.7). C t
t = t1/3 + t2/9 + + tn/3n + ,pi tn pi pi 0 2.
(t) =1
2(t1/2 + t2/4 + + tn/2n + ) .
21 step function staircasefuction.
-
38 1. .
- pi Cantor. pi f ., t C , f(t) = (t) (Kuratowski 1972, pp. 210 214).pi Cantor pi- (pi 1994, : 42).
1.7: (Engelking & Sieklucki1992, pp. 200).
, pi pi, pi- pi pi. pi pi pi, pi pipi pi .
pi , pipi pi , pi , pi pi.
-
2
pi .
pi pi pi pi , , pipi pi -. , pi , - pi - pi pi .
2.1 .
pi pi ( 1982, : 43). pi pi , pi , . , pi, pi pi - pi pi pi ( 1982, :51).
pi pi pi pipi pi pi pi pi , -pi pi ( , 1975, : 9).
, pi- , , , pi
39
-
40 2. .
( , 1975, : 9 10; 1982, : 52). pi , , pi, ( , 1975, : 10; 1982, : 15). pi pi pi pi - , pi pipi . pi , pi pi pi- pi ( , , : 10).
, pi pi pi pi , pi ( 1982, :41). pi pi pi-pi pi ( 1982, : 41). , - pi, pi ( 1982, : 42).
, pi pi, pi, - , , , pi, , ( , , : 11 12). , , pi pi , pi pi pi 1, pi pi . pi pi, pi pi, pi pipi pi pi pi pi- pi (pi 1985, : 59).
pi pi . pi pi : pi- ... , pi pi ( ) pi . pipi pi pi. , -
1 : pi, . pi - pi pi pi pi (, 11 , :334).
-
2.1. . 41
pi ( 1973, : 52).
pi pi , pi pipi , , pi , pi pi ( 1982, : 55).
pi pi pi pi pi , pi pi , pi pi -.
pi ( 6o ..), , pipi2 pipi pi pi pi - (pi, ,[9, 140, 29]), pi . - pi pi - pi ( 1973, : 49). pipi pi , pi pi pi pipi , , pi pi pi pi - pipi : pi pipi , pi pi, pi , - pi . pipi, pi pi ,pi , pi. pi pi pi ( 1973, : 49 50).
pi pi pi - pi pi pi pi pi . pi
2pi, , [9, 139, 11 15], [9, 140, 29 33], [9, 141, 1 8]
-
42 2. .
pi pi ( 1973, : 49).
pi - pi pi pi pi pi pi . -pi pi, pipi pi pi . , - . - pi pi pi . pi pi:
, pi pi , . , pi pi ( 1973, : 57).
pi pi pi - pi - . . - pi pi - .
pi, pi - . pi pi pi pi . pi pi 19o pi Dedekind Cantor.
pi pipi E. W. Beth: pi pi pi- . , pi pi (Zervos 1972,pp. 430).
-
2.2. . 43
2.2 pi .
pi pi pi, pi pi, , , pi, . pi , . pipi , pi pi pi .
pi pi pi pi pi , [239b5-240b7]. pi - pi pi pi(, , [239b5]), pi , pi pi, pi. pi .
pi : pi pi AB. pipipi AA1, pi A1 , AA1 = AB/2. A1A2 = AB/4, A2A3 = AB/8 (pi 1985, : 59). pipi pi pi .
2.1: pi (pi, : 59).
pi : . pi pi pi , A X, A pi pi pi X. pi pi , A1, pi pi X ,
-
44 2. .
X1, pipi A2, pi X1, pi A2, (pi 1985, : 5960; Fischbein 2001, pp. 317).
2.2: pi - (pi, : 60).
pi : ( pi3) pi , ( pi) pi . , pi , (Vlastos, 11 , : 301; Fischbein 2001, pp. 317).
pi pi pi, pi pi - pi pi pi. pi pi.
pi pi pi pi , pi , pi. pi- pi (Derrida, : 187). pi pi , pi - , pi pi.
pi pi , pi pi , pi : , pi , - pi pi .
3 pi : [] pi ( pi ), pi pi , pi pi (Vlastos, 11 , : 303).
-
2.2. . 45
pi pipi : pi ( ) . , .. pi, , - pi -, ... , pi , ( ), pi pi - pi pi , pi ( ) (Vlastos, 11 , : 310).
, pi pi pipi, pi pi- pi pi pi, pi pipi .
pi, pi pi pi . pi : , pipi pi s t pi , pi pi pi , :
v = 0/0
pi pi
v = 0/t
pi pi pipi (, 11 , : 329).
pi , pi pi pi pi , pi, - pi (pi, : 36; 1973). , pi , , , , pi ( 1982, : 2021). pi pi pipi pi, pi -. pi pi . ,
-
46 2. .
pi pi , -. pi pi pi pi, pipi, pi .
pi pi pi ( ), pi pi pi pi . pi pi pi pi pipi .
[] pi , pi pi. pi , pi pi pipi pi pi pi (pi 1985, : 75).
pi pi - , pi - . pi pipi pi:
[]pi pi pi pi pi (pi- 1985, : 75).
pi pi pi , pi .
pi , pi pi pi pi (, 11 , : 333). pipi pi.
pi .
pi pi , pi pi - pi , pi . , - pi
-
2.2. . 47
( 1973, : 54). pi pi ,
pi , pi pi : pi pi pi - pi pi pipi pi pi pi - (, 11 , : 326).
pi pi pi . pi pi, pi pi (pi2005 2006, : 2).
, pi pi, pi , , (pi pi - , pi [0, 1]) : pipi ( pi ) pi (pi ). , pi - pipi pi, pi pi pi pi , .
pi, , pi pi -, pi pi pi pi (, 11, : 326 328).
pi pi pi pi pi (Tall2001; Tall & Tirosh 2001), pi pi, pi (Vamvakoussi & Vosniadou 2004, Vamvakoussi & Vosniadou (in prepara-tion)). pi, pi pi pi pi ( pi pi) . pi , pi pi .
pi pi pi pi pi -.
pi - -: , pi n N n > pi m N m > (pi 2004 2005).
pi pi ( )
-
48 2. .
pi 4 (pi 2004 2005)., pi ,
0 < >
(vvA
)k> > 0, k N
pi,
a0 > a1 > . . . > ak > . . . > 0
, (an)nN . (infimum). pi pipi, (infimum) (an)nN . ,
limn
an = 0.
, pi, pi pi pi -. pi pi
1
2+
1
4+
1
8+ .
,
n=1
rn = r + r2 + r3 + , r = 12.
-
52 2. .
,n=0
rn = 1 + r + r2 + r3 + , |r| < 1.
,
sn = 1 + r + r2 + r3 + + rn
rsn = r + r2 + r3 + + rn + rn+1
pi,
sn =1 rn+11 r , r 6= 1.
,n=0
rn = limn
1 rn+11 r =
1
1 r , |r| < 1.
pi pi pi pipi pi - :
n=1
(1
2
)n=
n=0
(1
2
)n 1 = 1
1 12
1 = 1,
pi pi (12+ 1
4+ 1
8+ ) pi 1, pi
. pi pi
pi , - pi pi pi pi . pi pi - pi pi pi. pi pi- pi pi pi pi .
Fischbein, [] pi , , , ,
-
2.3. . 53
pi ( ) pi- pi (Fischbein 2001, pp. 320).
pi - pi - Fischbein :
pi , pi , , , pi. pi . pi . pi pi pi pi pi, pi pi , pi (Fischbein, 2001,pp. 318).
pi pi, pi , pi- ( ) pi ( -) pi pi . pi pi pi pi pipi pi, pi , , . AB B , . pi pi .
pi , , pi (space model). pi pi, pi (Fischbein, 2001, pp. 317).
-
3
pi .
, pi pi. pi Q+0 . pi pi 0, , () pi (pi) pi pi 6= 9 pi pi () pi pi pi pi . pi pi pi pi pi pi pi .
pi 0 - pi ( ) Q+0 .
G pi pi pi pi pi pi pi 6= 9. , pi pi
-
56 3. .
. pipi pi . pi 3.4.14 G pi pi G pi pi pi G .
pi pi pi pi- + , , ( ) / ( > 0), pi, 0, pi pi . - pi pi pi , pi pi.
pi pi, pi-pi, G . pi pi pi -, , pi, pi . pi pipi- pi (pi ) pi pi pi pi .
pi , pi pi, G - pi . pi G pi pi - pi . pi , 0 pi , pi pi- pi (, , , pi).
G , pi 0, -. , . , pi - R pi , pi ( pi , 0 -). pi R, ( pi ) pi pi, , pipi
-
3.1. . 57
. pi pi , R pi ( pi - pi pi G pi).
pi pi pi pi pi - pi .
pi R pi pi Dedekind ( ) pi pi ( ). pi , pi pi - , pi .
3.1 .
pi pi - pi , pi pi. , - pi pi pi . pi pi pi , , pi pi pi pi . pi pi -, pi . pi 2. pi pi 1
3,5, . . . ,
17. pi
. - pi pi . - .
pi pi pi - 3.1. pi pi
1 [145c7 148e5].
-
58 3. .
3.1: pi (Sfard 1991, pp. 13).
-
3.1. . 59
3.2: ( Sfard1991 pp. 13, pi 2005 : 269).
-
60 3. .
, pi . pi- pi , pi pi . pi - , pi pi pi - pi pi pi - pi (Sfard 1991, pp. 13 14).
, pi Sfard pi : pipi [-], pi pipi (Lakoff & Nunez 2000, pp. 294).
pi pipi (reification) Sfard -pi , - ( pi pipi pi) pipi pipi pi pi pi -, pi pi (. 3.2). pi pi pi (Sfard 1991, pp. 21). - pi (pseudostructural conceptions)(Sfard & Linchevski 1994, pp. 220 221).
pi pi (Sfard 1991, pp.33, Sfard & Linchevski 1994, pp. 191), , pi pi , pi pi pi pi.
pi pi pi pi, -pi , pi pi- pi pi (operational) (structural). pi, - pi pi 2(counting) pi pipi pi (Sfard 1991, pp. 5). pi pi (Sfard 1991,pp. 5).
2 counting pi.
-
3.2. . 61
3.2 .
:
N = {1, 2, 3, . . .} (3.1)N0 = {0, 1, 2, 3, . . .} (3.2)Q+0 =
{ : =
m
n, pi m N0 n N
}(3.3)
(3.1) pi (N).pipi 0 pi 0 (N0), pi pi (3.2). pipi pi 0 m
n, m N0 n N,
pi 0 (Q+0 ) pi pi (3.3). = m
n, pipi pi
. pi, qm
qn, pmpn
pi p, q N. m
n pi pipi
mn. :
N N0 Q+0 .
pi , N, N0 Q+0 , (), pi (pi, , pi-pi, ) .
3.3 pi Q+0
pi pi -pi pi , . pi - pi .
[] pi (alternate representational formats), (OConnor 2001, pp. 146).
-
62 3. .
pi. pi .
0 pi pi 0 pi pi .
pi pi pi - 0 pi .
3.3 pi 0 pi .
pi pipi mn m N0
n N , =
m
n.
pi a, a N0, pi pi
an m < (a+ 1)n.
pi
m = an+ 1, a 0, 0 1 < n, (3.4)
n m a 0 pi a n ( an) 1 pipi . pipi pi 1 = 0 .
pi 10 (3.4) pi,
10m = 10an+ 101, 10a 0, 0 101 < 10n. (3.5)
n pi pi pipi 1, 101n. pi (3.5), pi 101 n 0 < 10,
-
3.3. Q+0 63
1. 2 pipi :
101 = 1n+ 2, 0 2 < n. (3.6) pi k, k, k+1, pi k N,
0 k < n, 10k = kn+ k+1, 0 k+1 < n. pi , 10k+1n
10k+1 = k+1n+ k+2, 0 k+2 < n. , pi pi,
10k = kn+ k+1, 0 k+1 < n, k N. (3.7) pi pi pi (k)kN - .
2
1 pi pipi
=m
n=mp
np, p N
a (k)kN , pi (3.4) (3.7) pi :
m = mp = anp+ 1p = an + 1, 0 1 = 1p < np = n,10k = 10kp = knp+ k+1p = kn
+ k+1, 0 k+1 < n, k N.
3.3 pi 0. 3.3.1 pi pi pi a - (k)kN, pi
a, 123 . . . a, 1 . . . k . . . .
:
= a, 1 . . . k . . . a, 1 . . . k . . . = .
-
64 3. .
a pi (k)kN pi.
3.3.2 a, 1 . . .
a 10 + 1101 + + 10
= a+110
+ + 10
(3.8)
pi - .
, pi , pi 0 pi pipi , pi3, pi pi pi.
3.3.3 0, 999 . . . = 0, 9 1. , 0, 9 = 1.
: r 0, 999 . . .. , r = 0, 999 . . .. pi r 10 pi 10r = 9, 999 . . . ( pi pi pi r pi ). r pi 10r pi 9r, pi pi , 9r pi . 9r 9 pi r, pi pi 1. :
10r = 9, 999 . . . r = 0, 999 . . .
9r = 9r = 1
pi 0, 9 = 1, pi pi ,pi pi pi, 9 pi pi pi pi pi .
3 pipi pi pi pi 0, pi pipi.
-
3.3. Q+0 65
3.3.4 pi, pi - 0, 9 pi pi pipi pi . :
= a, 1 . . . k9, (3.9)
k = , k N
= a, 1 . . . k9 = a, 1 . . . k,
k = + 1 , k N.
pi. = a, 1 . . . k. (n)nN
1 = a, 1 . . . k90 . . .
2 = a, 1 . . . k990 . . ....
m = a, 1 . . . k
m 9 . . . 9 0 . . .
...
limn
( n) = 0limn
limn
n = 0
limn
n = 0.
,
limn
n = .
2
0, pi pi (3.9), , pipi pi pi.
-
66 3. .
3.3.5 (k)kN, pi - 0, Q+0 , pi pi pi mn, pi. pi , -
pi 9.
pi. pipi k (3.4) (3.7) 0 n 1, 0 k < n n N. n pipipi, 1, . . . , n, pipi. pipipi pi , pi pipipi .
pipi pi pi = 0 k = 0, k > . , k = 0, k . pi (3.7) , 0 = kn + 0. pi n 6= 0 k = 0 k . (k)kN, pi pi 0.
pipi, pi pi pipi 1, . . . , n , n 3 ,
n = 1 1 = 0 n = 2 (1 = 1 2 = 0) .
pi, pi pipi , n pipi pi pi n 1 1, . . . , n 1.
, p : 1 < + p n = +p. (k)kN (k)kN, pi pi k = pi.
k = k+p k = k+p, k . (3.10) (3.7)
= +p 10 = 10+p n+ +1 = +pn+ +1+p ( +p)n+ (+1 +1+p) = 0.
,
| +p|n = n, pi N0, n 6= 0|+1 +1+p| < n.
-
3.3. Q+0 67
,
+p = 0 +1 +1+p = 0.
,
= +p +1 = +1+p.
pi k , k = k+p. k = k+p k+1 = k+1+p. pi (3.7)
k = k+p 10k = 10k+p kn+ k+1 = k+pn+ k+1+p (k k+p)n+ (k+1 k+1+p) = 0.
,
k k+p = 0 k+1 k+1+p = 0.
,
k = k+p k+1 = k+1+p.
pi pi (3.10) k . pi, . . . +p1 (k)kN, pi .
p , . . . , +p1 9. 9,
10 = 9n+ +1
10+1 = 9n+ +2...
10+p1 = 9n+ +p = 9n+ .
pi p
10p1, 10p2, . . . , 100
-
68 3. .
pi
10p = 9 10p1n+ 10p1+110p1+1 = 9 10p2n+ 10p2+2
...10+p1 = 9n+ .
pi pipi pi
10p =(9 10p1 + + 9 100)n+
(10p 1) =(9 10p1 + + 9 100)n.
,
9 10p1 + + 9 100 = 9(1 10
p 110 1
)9 10p1 + + 9 100 = 10p 1.
Sp p pi pi, pi 1 10.
,
(10p 1) = (10p 1)n = n.
pi, < n. pi (k)kN,
pi 0, pi 9.2
3.3.6 0 pi a, 123 . . . pi
a < a+ 110
a, 1 . . . k < a, 1 . . . k + 110k
, k N. (3.11)
-
3.3. Q+0 69
pi. (3.4), (3.7) :
=m
n= a+
1n,
10kn
= k +k+1n
k N. (3.12)
,
Yk =kn
k N,
,
= a+ Y1, (3.13)10Yk = k + Yk+1, 0 Yk < 1 k N. (3.14)
pi Yk, k = 1, (3.14) (3.13) pi
= a+110
+Y210. (3.15)
k = 2 (3.15) pi
= a+110
+2102
+Y3102
.
pi:
= a+110
+ + k10k
+Yk+110k
= a, 1 . . . k +Yk+110k
, k N. (3.16)
Yk, k N, pi pi Yk = 0, kk+1k+2 . . . .
, pi pi
= a +110
+ + k
10k+Y k+110k
, pi a N0, k N. (3.17)
1, . . . , k 0 Y k+1 < 1, pi a, pi k - 1, . . . , k pi pi 0, 123 . . . Y k+1. ,
= a + 1 + + k + k+1 + k+2 + , pi k N,
-
70 3. .
k+ = , k, N.pi (3.13), (3.14) (3.16) pi (3.11). -
, pi (3.11) pi a, 123 . . . .
2
3.3.7 110
, pi N, pi , pi :
1
10< , N > .
pi.
=12, pi 1, 2 N.
pi pi 2. > ,
1
10 0, pi 3.3.7, ,
k N : | | > 110k
.
pi, pi (3.18) | | < 110k
, k N.2
3.3 (
-
72 3. .
pi A = a, 12 . . . A
= a, 12 . . . pi pi
, pipi , pi pi pi pi pi, pi
A A A (
-
3.3. Q+0 73
1
10k1
Y k1+1 Y k1+1 < 110k1 .,
sign ( ) = sign (a, 1 . . . k1 a, 1 . . . k1) = sign (k1 k1) .pi,
< , k1 < k1
< , k1 <
k1.
2
pi- ; , : pi pi -, pi pi 9, pi 0 ; pi pi pi pi 0.
pi pi pi pi 3.3.8 0, pi pi -pi pi pi 9, .
3.3.10 pi pi
a, 1 . . . k . . .
pi pi 9 pi 0.
pi. pi pi - , pi 0 8. a, a, 1 . . . 1 2, 1 6= .
1. pi pipipi pi
9a+
9,
-
74 3. .
, (3.4), (3.6) 1 = 2 pi 1 (3.4) (3.6) pi
10 (m an) = n+ (m an)10m 10an = n+m an(10 1)m = n+ (10 1) an(10 1) m
n= + (10 1) a,
m
n= a+
10 1 = a+
9=
9a+
9.
9a+
9= a, .
, a = 0,
10 1 =
9= 0, . (3.19)
2. pipipi
= a+110
+ + 1101
9
(3.17), pipi k = 1 Y k+1 = 9 . pi pi pi (3.19) pi pi a, 1 . . . 1.
pi . . . +p1 pi - p - p 2. a, 1 . . . p, = 1, a, 1 . . . 1 . . . +p1 1 6= , 2.
pipipi = 1
= a+110
p1 + . . .+ p10p 1 , pi 0 () ( ) . 3.4.6 ( pi) pi , A. ,
A < A A > A. 3.4.8 A, A G A A, A A. ,
A A A A.
-
82 3. .
3.4 G .
, pi pi pi G .
1 A G N0
[A] A < [A] +1
10. (3.26)
pi. A G A = a, 1 . . . . . .. pi ,
[A]0 = a, 0 a, 1 . . . < (a+ 1) , 0 = [A]0 + 1100
[A]0 A < [A]0 +1
100
1 8 ,
[A]1 = a, 10 a, 1 . . . < a, (1 + 1) 0 = [A]1 + 1101
,
1 = 9 ,
[A]1 = a, 10 a, 1 . . . < (a+ 1) , 0 = [A]1 + 1101
.
,[A]1 A < [A]1 +
1
101.
pi k 1
[A]k A < [A]k +1
10k.
k+1 8 ,[A]k+1 = a, 1 . . . k+10 a, 1 . . . k+1k+2 . . .
< a, 1 . . . k (k+1 + 1) 0 = [A]k+1 +1
10k+1,
1 = 9 ,
[A]k+1 = a, 1 . . . k+10 a, 1 . . . k+1k+2 . . .< [A]k +
1
10k= [A]k+1 +
1
10k+1.
-
3.4. G . 83
,[A]k+1 A < [A]k+1 +
1
10k+1.
2
2 A G [A]0 [A]1 . . . [A] [A]+1 . . . (3.27)
[A]0 +1
100 [A]1 +
1
101 . . . [A] +
1
10 [A]+1 +
1
10+1 . . . (3.28)
pi. A G A = a, 1 . . . . . . N. pi 3.3.2, 3.4.2 ,
[A]0 = a a+110
= [A]1 ,
[A] = a+110
+ + 10
a+ 110
+ . . .+10
++110+1
= [A]+1 .
,
[A]0 +1100
= a+ 1 a+ 110
+1
10= [A]1 +
1
10,
[A] +1
10= a+
110
+ + 10
+1
10
a+ 110
+ + 10
++110+1
+1
10+1
= [A]+1 +1
10+1.
2
pi 3.3.9, , Q+0( ), pipi - pi . (3.27), (3.28) pi pi .
4
[A] +
10, N,
pi G 3 A = a, 1 . . . . . . 2, (, ).
-
84 3. .
pi. pipi
[A]+1 +
10+1< [A] +
10.
,
[A]+1 [A] k
[A]k+1 +1
10k+1< [A]k +
1
10k
,
[A]k+1 +1
10k+1+
k
10k+1< [A]k +
1
10k+
k
10k
< [A]k +1 + k
10k [A]k [A]k+1 .
,
[A]k+1 +k + 1
10k+1< [A]k+1 .
2
4 pi
A A [A] [A] , N0, (3.30)
([A] [A] k k N0) A A. (3.31)
-
86 3. .
pi. A A. pi pi (3.30) ,
n N0 [A]n > [A]n .,
[A]n +1
10n [A]n .
, pi 2,
A < [A]n +1
10n [A]n A,
A < A.
pi, pi A A. [A] [A] k k N0. pi -
pi (3.31) A < A. pi 3 ,
n N n k : [A]n +1
10n< [A]n .
pi n k ,
[A]n < [A]n +
1
10n< [A]n ,
[A]n < [A]n .
pi, pi [A] [A] k.2
3.4.11
A1 A2 . . . A A+1 . . . , (3.32) G , pi A G pi pi :
1. A Ak, k N.2. A G A < A, m N Am < A.
-
3.4. G . 87
pi.
Ak = ak, k1 . . . k . . . , k, N.
pi pi (3.30) 4 pi pi (3.32) ,
ak = [Ak]0 [Ak+1]0 = ak+1 (3.33)ak, k1 . . . k = [Ak]n [Ak+1]n = ak+1, k+1, 1 . . . k+1, n (3.34)
k, n N. pi
a1 11 12 1 1, +1 a2 21 22 2 2, +1 ... ... ... ... ... ... ...ak k1 k2 k k, +1 ... ... ... ... ... ... ...
(3.35)
pi pi pi. pi (ak)kN, pi a N0, pi (3.33),(3.34) pi
ak ak+1, k N.
, ak, pi pi pi k0 pi. , pi , pi k0
ak0 = ak0+1 = ak0+2 = ak0+3 = . . . .
a = ak0 ,
ak = ak0 = a k k0 a [Ak]0 [Ak] k N. (3.36)
, pi (3.35), (k1)kN. (k1)kN pi pi k0 pi,
ak, k1 = [Ak]1 [Ak+1]1 = ak+1k+1, 1 (pi (3.34), n = 1)
-
88 3. .
ak = a, k k0.,
k1 k+1, 1, k k0., k1 pi pi pi k1 k0 pi. , , pi k1 k0
k1, 1 = k1+1, 1 = k1+2, 1 = k1+3, 1 = . . . .
1 = k1, 1 ,
[Ak]1 = a, 1 k k1 a, 1 [Ak]1 Ak k N. 1, . . . , ,
[Ak] = ak, k1 . . . k = a, 1 . . . , k k (3.37)
a, 1 . . . [Ak] Ak, k N. (3.38) + 2 , pi (3.35), - (k,+1)kN. (k,+1)kN pi pi k pi,
ak, k1 . . . k, +1 = [Ak]+1 [Ak+1]+1 = ak+1, k+1, 1 . . . k+1, k+1, +1(pi (3.34), n = + 1)
ak, k1 . . . k = a, 1 . . . k k (pi (3.37))., k,+1 pi pi pi k+1 k pi. , +2 , pi k+1 k
k+1, +1 = k+1+1, +1 = k+1+2, +1 = . . . .
+1 = k+1, +1 ,
[Ak]+1 = ak, k1 . . . kk, +1 = a, 1 . . . +1, k k+1
-
3.4. G . 89
a, 1 . . . +1 [Ak]+1 Ak, k N.
, pi pi
a, 1 . . . +1 . . . = A
pi (3.36), (3.37), (3.38) :
[A]0 = a = [Ak]0 , k k0 (3.39)[A] = a, 1 . . . = [Ak] = ak, k1 . . . k , N k k , (3.40)[A] [Ak] Ak, k N N0. (3.41)
,
[A1] [A2] . . . [Ak] [Ak+1] . . . , pi N0,
[Ak ] pi pi pi pi . A pi pi pi
pi pi 9 , pi, G .
A pi 9, = 9 pi n, n N. pi (3.39), (3.40), k = kn, :
[A]n = a, 1 . . . = [Akn]n = akn, kn1 . . . kn, n. , kn, n = 9.
pi (3.41) pi :
[Akn]n+1 = a, 1 . . . nkn, n+1 a, 1 . . . n9. , kn, n+1 = 9[Akn]n+2 = a, 1 . . . n9kn, n+2 a, 1 . . . n99. , kn, n+2 = 9[Akn]n+3 = a, 1 . . . n99kn, n+3 a, 1 . . . n999. , kn, n+3 = 9
pi
kn, n+ = 9, N0.
pi Ak pi pi- 9. pi, pi Ak G .
-
90 3. .
pi A pi (1) -. pi (3.41), pi N0, pi
A Ak, pi k N.
, pi A pi (1) . pi A pi (2) -
. pi pi A < A pi 3, pi pipi n N
[A]n +1
10n< [A]n A.
,
[A]n +1
10n< A.
pi (3.40) k = kn, [Akn]n = [A]n.
[Akn]n +1
10n< A.
, pi 1 ,
Akn < [Akn]n +1
10n.
, Akn < A. , pi Akn, kn N, (3.32) pi pi Akn < A. , pi Api (2) .
pi A pi pi (1) (2) , . A, A G , A 6= A,pi pi (1) (2).
A < A, (2) A, pi pi pi Am1 a, 1 . . .
n0 = [A]n +
1
10n> A,
pi > A. pi (3.43), (3.45)
< a, 1 . . . n20 . . . = a
, 1 . . . n +
2
10n+1
= [A]n +1
10n+
2
10n+1< [A]n +
2
10n B
pi < B. pi, A < < B.2
-
94 3. .
3.4.15 pipi A G ,
infN
([A] +
10
)= A.
pi.
[A] +
10, N,
( = 1, 2, 4). pi
infN
([A] +
10
)= A.
A < A, pi (2) 3.4.11 pi A,
A < A pi [A]m +
10m< A.
pi, pi 1 4 ,
A < [A]m +1
10m [A]m +
10m.
A < A pi 1, 2
[A] A < 1 < 2 < A = infN
([A] +
10
) [A] +
10
N,
0 < 2 1 < 10
, N.
pi, pi 3.3.7.2
G pi pi- pi pi .
-
3.4. G . 95
3.4.16 ()N - 0 ()N, 0
0 10
pi N.
pi A G
A , N, (3.46)
infN
= A.
pi. pi 0 G ()N, pi piA = infN .
A < A, pi 3.4.14 pi pi pi 1, 2 G , A < 1 < 2 < A. 1 2 pi > 0, 1 2. 1, 2, pi (3.46) (1) 3.4.11 pi A,
A < 1 < 2 < A = infN
, N.
< 1 < 2 < , N,0 < 2 1 < , N
0 10
, , N,
0 < 2 1 < 10
, N.
pi, pi 3.3.7.
-
96 3. .
A < A, pi (2) 3.4.11 pi
A = infN
,
pi pi m N m < A. , pi pi (3.46) ,
A , N.
pi.pi, A = A.
2
3.4 + , , , /. + : pi pi pi pi pi + , pi pi , .
0,
[] + [] + < [] + [] +
2
10, N.
= [] + [
] +2
10, N,
= [] + [
] , N.
()N, 2, ( ) > 0, + < 0 < =2/10, N. 3.4.16 = 2,
+ = infkN
([]k + [
]k +2
10k
).
pi : pi pi pi- pi pipi , pi pi , .
-
3.4. G . 97
,
= a, 1 . . . . . . 0 = , 1 . . . . . . 0.
[] < [] +1
10 [] < [] +
1
10, N,
pi 3.47
[] [] 0,
< 0 < < a+ + 210
, N.
3.4.16 = a+ + 2,
= infN
= infN
([] +
1
10
)([] +
1
10
).
( ): pi pi pi- pi , , pi pi , .
-
98 3. .
= [] +
1
10 [] , N. (3.47)
, 2, pi pi ( ). () N ( ) > 0, 3.4.11, pi infN . pi 3.4.9, pim N
[] +1
10< [] , m.
pi
m+k = []m+k
([]m+k +
1
10m+k
)> 0, k N. (3.48)
, 1,
m+k < < []m+k +
1
10m+k []m+k = m+k, k N.
pi (3.47) (3.48), k N, 0 < m+k m+k
= []m+k +1
10m+k []m+k []m+k + []m+k +
1
10m+k
=2
10m+k.
3.4.16 = 2 3.4.12, pi-
= infkN
m+k = infN
.
/ ( > 0): pi pi pi- pi pi /, > 0, pi pi , .
,
= a, 1 . . . . . . = , 1 . . . . . . > 0.
-
3.4. G . 99
[] 1/10n > 0 pi n N.
pi pi
n+k =[]n+k + 1/10
n+k
[]n+k, k N,
pi , 2, ( ). / = infkN n+k, pi n+k
n+k =[]n+k
[]n+k + 1/10n+k
0 < n+k n+k =([]n+k + [
]n+k + 1/10n+k
)1/10n+k
[]n+k ([]n+k + 1/10
n+k)
110n+k
a+ + 21/10n 1/10n =
102n (a+ + 2)
10n+k.
n+k
n+k, k N.
3.4.16 = 102n (a+ + 2), pi
= inf
kNn+k = inf
N .
3.4 G .
3.4.17 A + B pi A, B :
A+B = infN
= infN
([A] + [B] +
2
10
). (3.49)
-
100 3. .
3.4.18 ()N
= [A] + [B] +2
10, N,
(pi 2). , pi 3.4.11, pi pi infN pi G .
6 A B pi pi G , pi , Q+0 , A+B pi (3.49) pi + , + 0.
7 m, n N
[A]m + [B]m A+B = infN
([A] + [B] +
2
10
)< [A]n + [B]n +
2
10n. (3.50)
pi. pi 2, pi
[A]m A < [A]n +2
10n, m, n N,
[A]m < [A]n +2
10n, m, n N.
,
[A]m + [B]m < [A]n + [B]n +2
10n.
n
[A]m + [B]m < [A] + [B] +2
10 [A]n + [B]n +
2
10n.
pi 3.4.13 pi
[A]m + [B]m inf>n([A] + [B] +
2
10
)= A+B [A]n + [B]n +
2
10n
pi
inf>n
([A] + [B] +
2
10
)= inf
N
([A] + [B] +
2
10
),
-
3.4. G . 101
[A]m + [B]m infN
([A] + [B] +
2
10
)= A+B [A]n + [B]n +
2
10n. (3.51)
, ([A] + [B] +
210
)N,
pi
[A]+1 + [B]+1 +2
10+1< [A] + [B] +
2
10.
pi, (3.51)
A+B [A]n + [B]n + 2/10n
A+B < [A]n + [B]n + 2/10n.
2
3.4.19 A+B = B + A, pi A, B G .pi. pi 3.4.17,
A+B = infN
([A] + [B] +
2
10
)
B + A = infN
([B] + [A] +
2
10
).
[A] + [B] +2
10
[B] + [A] +
2
10
[A] + [B] +2
10= [B] + [A] +
2
10
pi, A+B = B + A.2
-
102 3. .
3.4.20 A < B A+ < B + , pi A, B, G .pi. A < B, pi 3
nN : [A]n +1
10n< [B]n ,
[A]n +2
10n [B]n ,
pi,
[A]n +1
10n+
1
10n+1< [A]n +
2
10n [B]n [B]n+1 .
pi 2
[A]n+1 +1
10n+1 [A]n +
1
10n,
[A]n+1 +1
10n+1+
1
10n+1 [A]n +
1
10n+
1
10n+1< [B]n+1 . (3.52)
pi pi pi (3.52) pi
[A]n +2
10n+1+ [ ]n+1 < [B]n+1 + [ ]n+1 .
pi pi (3.50)
A+ < [A]n+1 + [ ]n+1 +2
10n+1 [B]n+1 + [ ]n+1 B + .
,
-
3.4. G . 103
3.4.21 A,B, , G , A B < , A+ < B+ .pi.
< A+ < A+ A B A+ B +
(A B < ) A+ < A+ B + A+
-
104 3. .
[A] A < [A] +1
10, N
[B] + [ ] B + < [B] + [ ] +2
10, N,
[A] + [B] + [ ] A+ (B + ) < [A] + [B] + [ ] +3
10, N.
pi, pi 3.4.16 (3.55)
A+ (B + ) = infN
([A] + [B] + [ ] +
3
10
)= (A+B) + .
2
3.4.23 0, 0 = 0 pi pi G . , A+ 0 = A, A G .
pi. A G . pi 3.4.17
A+ 0 = infN
([A] + [0] +
2
10
)= inf
N
([A] +
2
10
).
pi 3.4.15 pi
infN
([A] +
2
10
)= A.
A+ 0 = A, A G .
2
3.4.24 A+B = +B A = , pi A, B, G .
-
3.4. G . 105
pi. pi A 6= . pi, A < < A. , 3.4.20, pi
A < A+B < +B
< A +B < A+B., pi A+B < +B +B < A+B pi pi A+B = +B. A = .
2
3.4 pi G .
3.4.25 AB8 pi A, B :
AB = infN
= infN
(([A] +
1
10
)([B] +
1
10
)). (3.56)
3.4.26 ()N
=
(([A] +
1
10
)([B] +
1
10
)), N,
(pi 2). , pi 3.4.11, pi pi infN pi G .
8 A B pi pi G , pi , Q+0 ABpi (3.56) pi , 0. 9 A = a, 1 . . . . . . B = , 1 . . . . . . A,B G pi (2) 3.4.11 pi m,n N
[A]m [B]m AB ([A]n +
1
10n
)([B]n +
1
10n
)< [A]n [B]n +
a+ + 2
10n. (3.57)
8 AB A B.
-
106 3. .
10 A, B G A = a, 1 . . . . . . B = 0 = 0, AB = 0.
pi. B = 0, pi (3.57) m = n
0 = [A]m [0]m A 0 < [A]m [0]m +a+ 0 + 2
10m= 0 +
a+ 2
10m=a+ 2
10m. (3.58)
, (3.58), 3.4.16 - 3.4.15
A 0 = infmN
(a+ 2
10m
)= 0,
A 0 = 0, A G .2
3.4.27 AB = BA, pi A, B G .
pi.pi 3.4.25
AB = infN
(([A] +
1
10
)([B] +
1
10
))
BA = infN
(([B] +
1
10
))([A] +
1
10
).
[A] +1
10 [B] +
1
10
([A] +
1
10
)([B] +
1
10
)=
([B] +
1
10
)([A] +
1
10
).
pi, AB = BA.2
3.4.28 A,B, G , (A > 0 B < ) AB < A .
-
3.4. G . 107
pi. pi, pi (3.4.11 - 3.4.16),
AB ([A] +
1
10
)([B] +
1
10
) [A] [ ] A, N.
pi pi m N ([A]m +
1
10m
)([B]m +
1
10m
)< [A]m [ ]m ,
[A]m [B]m +1
10m
([A]m + [B]m +
1
10m
)< [A]m [ ]m . (3.59)
pi pi, A > 0, pi
n N : [A]m 1
10n> 0, N 3 m n.
(3.59) [A]m
[B]m +[A]m + [B]m + 1/10
m
10m [A]m< [ ]m , m n. (3.60)
A = a, 1 . . . . . . B = , 1 . . . . . . A,B G
[A]m + [B]m + 1/10m
10m [A]m n
[B]m +(a+ + 2) 10n
10m [ ]m . (3.61)
(3.61) pi pi B < , pi- 3.4.10.
2
3.4.29 A,B, , G , 0 < A B < , A < B.
-
108 3. .
pi. A > 0 > 0, pi pi < A B pi,
A < A A B.
, A < B.2
3.4.30 A,B, G , ( 6= 0 A = B ) A = B.
pi. pi pi A 6= B. pi, A < B B < A.
A < B, pi 3.4.28 pi A < B . pi, pi pi pi A = B .
B < A, pi 3.4.28 pi B < A . pi, pi pi pi A = B .
pi, A = B .2
3.4.31 (AB) = A (B ), pi A,B, G .
pi.
A = a, 1 . . . . . . ,
B = , 1 . . . . . . ,
= , 1 . . . . . . .
= 0, 10, (AB) = A (B ). > 0, (1), (2) 3.4.11
1,
[A] [B] AB ([A] +
1
10
)([B] +
1
10
), N
[ ] < [ ] +1
10, N.
-
3.4. G . 109
pi, 3.4.28 3.4.29, :
= [A] [B] [ ] (AB) 0.
0 < =
1
10
([A] [B] + [A] [ ] + [B] [ ] +
1
10
([A] + [B] + [ ] +
1
10
)) 0 pi 7
[A] + [B] A+B < [A] + [B] +2
10, N,
pi 1
[ ] < [ ] +1
10, N.
pi 3.4.28
= ([A] + [B]) [ ] (A+B) 0.pipi
0 < = 110
([A] + [B] + 2
([ ] +
1
10
)) a = .
3.5.12 pi x pi pi pi x, x 6= 0, pi pi x. ,
0 6= |x| = |x| signx 6= signx, x 6= 0.
0 0.
3.5.13 (a, R a < ) a > < a, pi a, a, .
pi. < >, - , , pipi : pipi :
1. a < < 0,
-
124 3. .
2. a 0, 0 a < ,3. 0 < a < .
pi pipi pi 3.5.8, 3.5.10
a < < 0 |a| > || > 0 a > < a.
pipi pi 3.5.9, 3.5.11 -
a 0 a < 0 a = 0, 0 > 0 = 0.
pi pi a < a 6= 0 6= . pi pi 3.5.8, 3.5.10
a < < a a > .
pipi pi 3.5.8, 3.5.10
0 < a < 0 < |a| < || < a a < .
2
3.5 R. 3.5.14 a1, . . . , a , pi 2, pi R.
a1 + + a a1, . . . , a R pi pi :
pi a1, . . . , a pi pi 0, pi pi pi pipi pi pi 0, ap (1 p ), = ap , pi < 0, = 0. , a1, . . . , a pi pi< 0, pi pi A pi pipi pi pi < 0, an (1 n ),
-
3.5. R . 125
A = |a |. , pi 0, A = 0.
pi pi A pi , , pi max(,A), pi , pi min(,A). :
= (max(,A)min(,A)) G .
a1, . . . , a = + A < A.
19 pi 0 pi 0.
20 pi 0 pi 0.
21 0 pi pi R. ,
a+ 0 = a 0 + a = a, a R.
22 x + x x x = 0., a, R a + = 0, a : a = a = .
23 a1 + . . . + a pi- . a + = + a a, R. pi pipi, pi 3.5.14, pi pi pi A pi pi .
3.5.15 (a1 + . . .+ a) pi (a1 + . . .+ a
) pi.
,
(a1 + . . .+ a) = a1 + . . .+ a
(3.84)
-
126 3. .
pi. a1 + . . . + a , 3.5.14, pi , A , a1 + . . .+ a pi
= A, A = = .
pi
A A A A.
,
a1 + . . .+ a = +
= + a1 + . . .+ a
= = .
pi (3.84).2
3.5.16
x+ y + z = (x+ y) + z, pipi x, y, z R. (3.85)
pi. pi x, y pi, x 0 y 0 x 0 y 0, 3.5.1 pi . ,
x+ y = , x 0 y 0 x+ y = , x 0 y 0.
(3.85) pipi + z = + z, pipi + z = + z.
pi x, y pi, x > 0 y < 0 x < 0 y > 0, pipi pi x < 0 y > 0 pipi pi x > 0 y < 0 (pi 23). pi, (3.85) , pi pi pipi.
x < 0 y > 0, z pi 0 > 0 < 0. z = 0 (3.85) , pi 21,
x+ y + 0 = x+ y = (x+ y) + 0.
-
3.5. R . 127
x = X < 0, y = +Y > 0, z = +Z > 0
(3.85)
x+ y + z = (Y + Z)X, Y + Z X (3.86)x+ y + z = (X (Y + Z)), X Y + Z, (3.87)
(3.85)
(x+ y) + z = (Y X) + Z Y X Y + Z X, (3.88)(x+ y) + z = Z (X Y ) X Y Z X Y (x+ y) + z = Z (X Y ) X Y Y + Z X, (3.89)(x+ y) + z = ((X Y ) Z) X Y X Y Z (x+ y) + z = ((X Y ) Z) X Y X Y + Z. (3.90)
(3.88), (3.89, (3.90) (3.86), (3.87) , pi 3.4.36 - 3.4.38, (3.85).
x < 0, y > 0, z < 0
x y x = y < 0, y = x > 0 z = z > 0, pipi pi pi pi pi. , x, y, z
x + y + z = (x + y) + z,
, pi 23,
x + y + z = (x + y) + z,
pi 3.5.15
x+ y + z = (x + y + z) = ((x + y) + y) = (x + y) + z = (x+ y) + z.
2
3.5.17 (a+ ) + = a+ ( + ), pi a, , R.
-
128 3. .
pi.
(a+ ) + = a+ + (pi 3.5.16)= + + a (pi 23)= ( + ) + a (pi 3.5.16)= a+ ( + ) (pi 23).
2
3.5.18 a+ = + a = , pi a, , R.
pi. , , pi a+ = +
a+ + = + + a+ ( + ) = + ( + )
a+ 0 = + 0a = .
2
3.5.19 x < y x+ z < y + z, pipi x, y, z R.
pi. pipi :
1. 0 x < y,2. x < 0 < y,
3. x < y 0.
1. pipi 0 x < y, x = +X, y = +Y , pi 0 X < Y . z pi 0, < 0 |z| X, < 0 X < |z| Y , < 0 |z| > Y .
() z = +Z 0, x + z = X + Z y + z = Y + Z. G X < Y X + Z < Y + Z, pi
x+ z < y + z.
-
3.5. R . 129
() z = Z Z X < Y , x + z = X Z y + z = Y Z. , X Z < Y Z X < Y , pi x+ z < y + z.
() z = Z 0 X < Z Y , x+ z = (Z X) |y+ z|, x+ z < y + z.
2. pipi x < 0 < y,
x < 0 x+ z < z z R (3.91)0 < y z < y + z z R. (3.92)
pi, x+ z < z < y + z x+ z < y + z.
3. pipi x < y 0, 0 y < x piy, x y, x . , pi (1) y + z < x + z z R. pi, pi 3.5.13, (x + z) < (y + z). , x+ z < y + z.
2
24 pi pi pi pi a, , , R :
a < < , a+ < + .
a < , a+ < + . a , a+ + .
3.5 R. 3.5.20 R a+ x = x. a + x = pi a pi a. ,
a+ ( a) = ( a) + a = .
-
130 3. .
3.5.21 a + x = x a,, x R x = +a, pi a a. , pi x ,
a + a+ x = a + ,
0 + x = x = a + = + a.
, x = + a,
a+ ( + a) = a+ (a + ) = (a+ a) + = 0 + = .
, pi x = + a pi pi .
3.5.22 pi, pi (, a) - a, a pi .
a = + a + a = a. 3.5.23 R pi- pi , pi 3.5.20,
a = + a + a = a. , a a
a a a = + a = + (a).
pi, , a a + a pi a + a = a = (a).
pipi a+ , a+ = a + = + a = a.
25 pi G , 3.4.34, pi pi pi , pi pi
AB + E = (((AB) ) +) E,
-
3.5. R . 131
,
A B(AB)
((AB) ) + E.
R, 3.5.20, 3.5.22, pi pi pi pi
a + = (((a ) ) + ) a, , , , R, (3.93)
pi pi pi .pi 3.5.23
a + = a+ () + () + + () . (3.94)
pi,
a+ + +
pi pi pi (3.93), pi 3.5.15 (3.94)
(a+ () + () + + ()) = (a) + + + () + = a+ + + .
pi (3.94) pi pi
(a + ) = + (a + )= a+ + + .
26 pi (3.93), (3.94), . pi, pi 23, pi pi- pi (pi pi: + a + ) .
3.5.24 a < ( a) > 0, pi a, R.
-
132 3. .
pi. a < ( a) > 0. pi , pi 3.5.19
( a) 0a+ ( a) a+ 0
a.
pi, pi pi a < . ( a) > 0 a < . pi ,
pi 3.5.19
a a+ (a) + (a)
0 a,
pi, pi pi 0 < ( a).2
3.5.25 (a+ ) ( + ) = a , pi a, , R.
pi.
( + ) + (a ) = + + a+ () (pi 3.5.23)= + () + a+ (pi 23)= 0 + (a+ ) (pi 22)= a+ (pi 21).
,
(a+ ) ( + ) = ( + ) + (a ) ( + )= ( + ) ( + ) + (a )= 0 + (a )= a .
2
-
3.5. R . 133
3.5 pi R. 3.5.26 pi a1, . . . , a ,pi 2, pi a1 a pi pi pi |a1| |a | pi pia1, . . . , a pi + pi pi pi < 0 pi .
27 a1 a - pi pi = 0.
28 pi 0, pi < 0.
29 . pia1a2 = a2a1.
30 pi pi (a1a2) a3 = a1 (a2a3).
pi. pi pi -
(a1a2) a3 = a1a2a3 = a1 (a2a3) .
2
3.5.27 (a+ ) = a + (a ) = a , pi a, , R.
pi. pi pi, a = a + pi (a ) = (a+ ) = a + = a + () = a .
pi a, , = 0, (a+ ) = a + . a 6= 0, 6= 0, 6= 0, pi
(a+ ) = a + (a + ) = a + ,
pi pi , pi
-
134 3. .
pi + a, , :a a + + + + + ++ + + + + +
pi pi .
1. a = +A, = +B, = + ,
a+ = A+B, (a+ ) = (A+B), a = A, = B.
A, B, G , pi 3.4.32,(A+B) = A +B . pi
(a+ ) = a + .
2. a = +A, = +B, = , a+ = A+B,
(a+ ) = (A+B), (3.95)a = A, = B.
(A+B) = (A +B ) (3.96)a + = (a ) = (A +B ) . (3.97)
, pi (3.95), (3.96), (3.97),
(a+ ) = a +
3. a = +A, = B, = + , a = A , = B a+ = AB A Ba+ = (B A) A < B,
(a+ ) = (AB) A B (3.98)(a+ ) = (B A) A < B, (3.99)
-
3.5. R . 135
a + = A B = (AB) A B A B (3.100)a + = (B A )
= (B A) B > A B > A. (3.101)pi (3.98), (3.100) (3.99), (3.101)
(a+ ) = a + .
4. a = +A, = B, = , a = A , = B a+ = AB A > Ba+ = (B A) A B,
(a+ ) = (AB) A > B (3.102)(a+ ) = (B A) A B, (3.103)
a + = B A = (B A) A B (3.104)a + = (A B ) = (AB) A > B.(3.105)
pi (3.102), (3.105) (3.103), (3.104)
(a+ ) = a + .
2
pi 3.5.27, pipi- R pi pi pi .
3.5.28 a < > 0 a < , pi a, , R.pi. 3.5.24,
a > 0. pi 3.5.27 a = ( a) . pi 3.5.24, pi a < pi a > 0. ,( a) > 0 > 0, ( a) > 0. , a > 0.
2
-
136 3. .
3.5.29 a < < 0 a > , pi a, , R.
pi. pi 3.5.27, a = (a ) . pipi a < 0 < 0 (a ) > 0 (pi 28), pi a > 0 pi pi 3.5.24 pi a > .
2
3.5.30 1 pipi- R. ,
1 a = a, a R.
pi. a 6= 0 |1 a| = 1 |a| = |a| sign (1 a) = sign a. a = 0 1 0 = 0.
, 1 a = a, a R.2
3.5 R. 3.5.31 R ax = a 6= 0 x. ax = pi /a pi a. ,
a a=
a a = .
3.5.32 ax = x a 6= 0,pi a, , x R, .
, = 0, x = 0 (pi 27). 6= 0 pi x , pipi |x| 6= 0 |a||x| = ||, signx = sign a. , xpipi pi
|||a|
pi + a pi a pi.
,
a= /a
-
3.5. R . 137
pi pi pi
|||a|
pi + a > 0 a < 0, 3.5.26
a a=
a a = .
3.5.33 pi pi (, a) a 6= 0 /a a.
3.5.34 a = 0 , , pi-pi pi 6= 0, 0 x = R , pipi = 0, 0 x = 0 , pi pi pi x.
31 pi /a a - , pi.
3.5.35 a, R,
a=
a R 6= 0.
pi. pi 3.5.31 pi pi
x1 =
a.
pi 29 , ax1 = a x1. pi pi 3.5.31pi a x1 = , 6= 0. pi a x1 = , a 6= 0.,
a=
a R 6= 0.
2
-
138 3. .
3.5.36 a, R,
a= 1
a a 6= 0.
pi. 1a = || 1a
= || 1|a| = |||a|, 6= 0,
sign
a= sign
( 1
a
)= + a > 0
sign
a= sign
( 1
a
)= a < 0.
a= 1
a, 6= 0.
= 0
a= 1
a,
a=
0
a= 0 1
a= 0 1
a= 0.
2
3.5.37 a, , R, a+
=
a
+
6= 0
a
=a
6= 0.
pi. pi 3.5.36 3.5.27 a+
= (a+ )
1
= a 1
+ 1
=a
+
pi pi 3.5.36 3.5.27 a
= (a ) 1= a 1
1
=a
.
2
-
Dedekind.
pi Dedekind pi - pi , pi 1.
pi pi pi Dedekind, pi pi pi 1.2.
, pi ( ) A B pi , A pi , B pipi, ( .1).
q
A B////////////////// / / / / / / / / / / / / / / / / /
.1: .
pipi :1 pi , Dedekind,
pipi pi pi Rudin (2000, : 24 31).
139
-
140 . DEDEKIND.
pi A. ,
A = {, Q : } ,B = {, Q : > } .
B. ,
A = {, Q : < } ,B = {, Q : } .
pipi, .
pi, pi pipi pi - :
1. A B.
2. , pi , pi B pi A, .
pi pi pi pi pi pipi . .
, pi - A B , pipi pi pi pipi , - pi pi (pi ), pi .
pi, x2 2 = 0 pi pi pi . - , pi pi x2 2 > 0, pi pi x22 < 0 A B, pi pi , pi 2, pi pi- , , pi 2.
-
141
, pi (1) (2). pi pi pi , , pi pi A, B, ( ), pi pi x2 2 = 0 , ( -) , pi
2.
pi A B, pi (.2).
q2
A B////////////////// / / / / / / / / / / / / / / / / /
.2: 2.
A B, A B. pipi :
1. A pi pi pi .
2. B pi pi - pi . pipi (1) (2), pi (A,B) .
3. pi A, B2 ( ).
2 pi A
-
142 . DEDEKIND.
, A B, A B, , , pi - Dedekind3. Julius Wilhelm Richard Dedekind pi pi- , .
, (, pi- ) pi pipi .
pi Dedekind , Dedekind(pipi 1 2) (pi-pi 3).
pi pi, pi A B, , A B, pipi pi x, x A x B,pi .
, pi A, pi pi A, pi B, pi pi B. A pi pi A Aa pi A B pi pi B Ba pi B.
A B , x x ( ), A x B. : , x , x, A pi y, pi y x ( pipi ) pi .
B, pi +2 .3 pi , Dedekind,
, pi pi (1996, :1 10).
-
143
x, < x, pi A A pi pi B ( A B). , , y B < y, pi > ypi. y A. x B. , x, A B A B.
pi pi- pi ( ).
pi pi : x x pi- . A B pi x, A B pi x ( .3).
qx
A B////////////////////// / / / / / / / / / / / / /
qx
A B////////////////// / / / / / / / / / / / / / / / / /
.3: x, x.
1, 2, 1, 2 1 A, 2 B, 1 A 2 B. pi
1 < x < 2,
1 < x < 2.
pi 1 +
1 < 2 +
2.
-
144 . DEDEKIND.
, pi , pi : pi pi pipi 2+
2, 1+1
pipi , 2 + 2. x pi x , pi , ( .4). 2 + 2 ( pi 2 + 2 pi , 2 + 2 pi ) 1 +
1 ( pi 1 + 1 pi ,
1 + 1 pi ).
qx = x+ x
////////////////// / / / / / / / / / / / / / / / / /
.4: x.
x, 2 + 2 1+
1. , 1+1 < x < 2+2. x .
pi y , 1 + 1 < y < 2 + 2 x 6= y, x, y pi . 3, 3 pi 3 < 3 1 + 1 < 3 < 3 < 2 + 2. 3 3 pi ,
3 3 < (2 + 2) (1 + 1) . (.1)
(2 + 2) (1 + 1) , 21 21 . pi (.1). x = y.
x x x pi x = x+ x.
-
145
qx
q1
q2
A B//////////////////////////// / / / / / / /
qx
q1
q2
A B/////////////////// / / / / / / / / / / / / / / / /
qx
q1 + 1 2 +
2
q /////////// / / / / / / / / / / / / / / / / / / / / / / / /
.5: x+ x.
.5 ( 0 < 1 < 2 < 1 < 2) pi x = x+ x .
pi, pipi pi, , pipi- .
pi, ( pi) pi pi Dedekind pi : (A,B) Dedekind R pi pi a, pi - A B, pi : maxA = inf B = a supA = minB = a (pipi 1962, : 29).
-
R .
pi :
R pi pi , pi : pi pi - + pipi pi (pipi1962, : 4).
pi R pi 3, R pi - . R pi pi(+) pipi () , :
1. (R,+, ) .
2. R = R {0} (pi 0 pi), (R, ) .
: pipi R 3 (R,+) (, : () pi pi pi R, ()pi pi pi R, () pi () pi pi ). pi, pipipi R 3, pi
147
-
148 . R .
pipi pi R pi pi pi pi pi R. pipi pipi pi R ( 1). pi, (R,+, ) .
pipi pi R 3 () pi pipi pi, ()pi pi R () pipi- R 3, R . pi, (R, ) .
pi - 0 (pipi 1962, : 5).
pi -, : pi pi R+ R pi pi (pipi 1962, : 7):
1. a R 1 pi :
a = 0 a R+ a R+.
2. a R+ R+, a+ R+ a R+.
pi III, pi pi- G pi 3.
R+ (pipi 1962, :7) G {0}. , pi R+ pi pi G {0}.
IV ( pi ) R pi- 2 pi , : pi M R pi , pi , pi pi3 R (pipi 1962, : 21).
1 pipi.2 pi.3 pi (supremum).
-
149
pi pi : - pi pi M R pi pi R(pipi 1962, : 21).
IV pi, pi pi pi 3, .
IV I, II, III, pi pi .
pi pi - : , pi pi , - pi I, II, III, pi pi pi pi, IV :
IV1 ( - ) pi a pi > a (pipi 1962, : 62).
IV2 pi pi Cauchy4 (pipi 1962, : 62).
pi pi , - IV, pi IV1 IV2 pi. pi , IV1 IV2, pi IV pi. pi-, pi pi , IV1 IV2 IV.
pipi, pi - +.
4 Cauchy: (a)N pi- , pi pi (pipi 1962, : 61):
1. (a)N Cauchy ( , pi - ).
2. pi pi a , a a, : (a)N R.
-
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