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η Ҿ Ǿ. й衤 η ȸȰ ʿ , Ȱ ʼ õ () , ؿԴ. й Ǵ μ ָ ()ô, ü 6̶ ִ. ǿ ε߱ٺδϾơƮ Ͽ ȭ ߴǾ.
1. ٺδϾƿ Ʈ
Ʈ ϰ, ٺδϾ Ƽ, 簭, ε , ߱ Ȳ ϱ ۵Ǿٴ ˷ ִ. Ư ϰ ϹǷ ظ Ͽ ġڴ ̰ Ȯ ʿ伺 ϰ Ǿ. ̷ Ͽ Ȯ ȭ Ÿ , Ʈ κ õ ̹ 1 365ϰ 1/4 ̶ ˰ ־ٰ Ѵ. ڴ ߾ ؾ Ƿ ̷ Ͽ Ǿٰ Ѵ. ˷ ִ ְ м 뿵 ڹ Rhind ǰ ߿ ִ Ƹ Ƿ罺̴. Rhind 1858 ̰ ߴ. Ƿ罺 1877 οÿ Ͽ Ǿ. Ƹ Ƿ罺 ϸ Ʈ ̷ Ǹ, ŷ ű Ⱑ ϹݹĢ . 밳 Ǯ ִ ̴. ۾ ͳ ϹݹĢ ߰ ְ װ ʰ ִ. Ƹ Ƿ罺 ίڷ м ǥϰ ְ, 1 1 2Ŀ ͼӵǴ ٷ ִ. ʴ Ʋ . Ƹ Ƿ罺 Ϲ ְ, μ (16/9)2 = 3.1604 ̿ϰ ִ. 4 絵 ߴ ִ. Ʈ , شǴ ִ. Ƹ Ƿ罺 Ʈ Ƕ̵ Ƹ ǥϰ ִ ְ ǰ ̴. ̽Ű εǾ ִ. ٺδϾ ġ Ǿ ŵ ִ. ߿ ༱, 4 , ϴ ִ. ٺδϾ ȣ * 6а 60 谡 ִ ϴ. ⼭ 6 κ ٺδϾ ˰ ־ٴ 6 Ǿ ־ٴ ؼ ִ. ٺδϾ 1, 2ĵ Ǯ ־.
2.
ε Ʈ , ٺδϾƿ () ˷ ִ. Ż Ÿ, ö浵 Ʈ Ͽ ȭ Ͽ. ̵ ȭ Ƶ鿩 ο ñ⸦ Ͽ. п ̰ Ư п Ҹ ִ. Ŭ п(̾), ƸŰ , δϿ УKonikon biblia, 佺 () װ̴. Ƹڷö ǥǴ ڵ ɻ öа ̾. ̿Ͼƶ Ĺ Ҿƽþ ̴. ö ̿Ͼ âڴ Ż(B.C. 640- 546)̴. Ұ ͵ Ż̸ ״ Ƕ̵ ڸ Ͽ ̸ Ƹý ź״ٰ Ѵ. Ż ־ âߴٰ ִ. ̿ ؼ Ʈ ַ ü ߴٰ ̴. Ż ߰ δ .
² .
̵ﰢ ذ .
ﰢ ־ ̿ ΰ ﰢ յ ̴.
ﰢ ־ ̰ ﰢ յ̴.
ݿ ϴ ̴.
ﰢ 2̴.
ﰢ ־ ϴ ǰ ﰢ ̴.
Ż 鿡 ٿ ̵ ǿ 1ٰ̾ Ѵ. ﰢ յ ̿Ͽ ػ ִ ġ ϴ ̾. Ż й ̾ Ÿ(B.C. 580 - 500 ? )̾. ״ Ͽ Ʈ ߰ Ż ũ濡 б ̿Ͼ ոǸ ö ߰ ȭ, ո ̻μ ǥ Ͽ ̴. ٺ Ͽ. ̶ İ â ̶ ִ. ٿ ̷ Ģ μ , , õ . а縦 ̾ ܺη Ǿ.
(1) Ÿ
Ÿ Ÿ Ư а ̴. ߿ ԵǾ ְ, ߿ ԵǾ ִ. ̿ Ÿ ؼ ﰢ ̸ Ÿ ߰ϴ Ģ ߴ.
Ÿ ĵ ߰ .
ﰢ Ÿ
༱ ̷κ ﰢ 2 ̶ .
۵ ߰
߰
ϽõǴ . װ ƴϸ ߰ . Ÿ Ĵ ¡ ߴ.
ȭ ߰
1, 2/3, 1/2 3 ȭ ̶ Ͽ.
ﰢ 簢 ߰
ﰢ : 1 3 6 10 15 21
簢 : 1 4 9 16 25 36 (ﰢ 簢̴.)
ٸü 5 ̶ ߰
(2) ǽƮ
480 ̽ 丣þ ϰ ؿ Ű ߹ϰ ϰ Ǿ. ׳״ ū ڵ ߽ Ǿ. Ÿ ĵ ̰ , Ƴ ׳ ̿Ͼ ö ̽ߴ. ׳ ùε ϻ 뿹 ñ ǽƮ , Ҹ ̷ ߾. Ǿƴ ϹǷ ǽƮ ǹ̳ ̵ ַ ϰ ǾǷ Ŀ ˺ Ҹ.
Ÿ Ŀ ܵ ̾µ ʰ ȴ. ǽƮ Ǿµ ̳ ڿ Ľ ؼ ۵ϴ .
(i) Ǵ ȣ 3 .
(ii)ü . , ־ ü 2 ü ü .
ʵ۵ Ұϴٴ 1837 Wantzel P.L. (France, 1814 - 1848) Ͽ ̷ ǽƮκ 2000 ̻ ذ ̴.
(iii)ݹ. , ־ Ȱ 簢.
ʵ۵ Ұϴٴ 1882 Lindemann F. (Deutschland, 1852-1938) ʿ Ͽ ϼǾ.
(B.C. 495-435)
: 1 л ð ȿ ٴ ̴.
ų ź̿
*Ÿ κ ִٴ Ʋ ̶ .
(3) ö
ũ(B.C. 469-399) ö ١ ũ Ŀ νκ Żƿ Ÿ Ŀ Ͽ. B.C. 389 ö ׳ б(Academy) ۿ Ͽ. б ڴ ԡ̶ ȭ ϴ.
ũʹ ݴ ö ġ ũ Ͽ. ö ڴ ƴϾ. Դ п â , ϰ п Ǵ ûߴ. ״ ڵ ǽ Ҿ ٲپ. Ҿ ֵ ǿ , ۵Ǿ.
(4) ˷帮
Ŭ(B.C. 330 - 257?) ֿ ؼ κ ˷ ʾ ㅒ(Elements) 13 װ 35 - 40 ȴ. ü ö 佺 쵶ҽ ̿ ڵ ڱ ڽ Ͽ 뼺 ̰ ̰ ö , , , 迭ϰ ̴.
Ŭ
1. κ ٸ ִ.
2. ִ.
3. ־ ߽ ϰ ־ ִ.
4. .
5. ٸ ³ ʿ
̷, Ѿ
ų ִ ʿ ( ) ݵ .
κ ѷƵ ʵп ־ Ŭ常ŭ ٴ ִ ġ ϰ ִ . 緹̿ Ŭ忡 "(Elements) ʰ ?"ϰ . Ŭ N "п յ ϴ." ߴٰ Ѵ. Ŭ İ û " 3 潺 ֶ" ߴٴ ȭ ϴ.
ƸŰ(B.C. 287- 212) ÿ ̾ ǿ빮 ߴ. Ư ߿ ߰ ̷ ü ð ߴٴ ȭ ϴ. ƸŰ ַ µ .
( ǥ )=( 4)
( ü) = (ݰ 3 ڽ 4/3 )
30/71 < < 3/7 : ٰ Ѱν ̶ Ѵ.
߱ ü ǥ ϴ ü ǥ 2/3̴.
ö Ա ڴ ٿٴ ̾߱ ϴ. Ŭ嵵 Ƹڷ ö Ҵٰ ˷ ִ. п ó Ͽ üȭ μ 19 ̰ ־. å Ͽ ʷ () Ͽ üȭϴ ó Ŀ ̹ BC 3濡 ־. Ÿ Ͽ ε () ̷ ִµ 1ǿ 4DZ (), 5 ʷ(), 6 , 7ǿ 9 (ߩ), 10 (), 11ǿ 13 ü(ء)̰, ٸü() Ǿ ִ. ü 13 8 ε, ݿ ִ. ü迡 ó ִ. п ģ Ƹ ũ.
ƸŰ (ڪϴ) () () ѷ ̸ ϴ ε, Ư ν, ʿ õǴ ̰ ִ. հ (Ϲ) Ǹ ̸, () ŵΰ ִ.
δϿ С(8) μ ϰ ִ. п ִ о߷μ ļ ؼ() 2̶ Ҹ κ, ؼ ϴ Ἥ 뼺Ͽ. ̷ ſ پ, 鿡 ū .
佺 ̷ ѷϿ. 10 ε ٵ ϴ ¿. ε 7 ƸƹŸ(ryabhata:475553) ƸƹƼϣryabhattam(499) () õ ()ϰ ִ. ó ƶƼڶ Ҹ ߸ ͵ ε̴. Ż Ǻġ ̰ Ұ Ǿ ִ. 15, 16濡 ⸦ 鼭 鿡 ô볪, 17 Ŀ ̴ ѷ . ٸ, Żƿ 3, 4 ع̶簡, ȣȭ ָ ̴. 17 鼭 öСõС Ҿ ٴ, 뿡 ̾ ̸ ô롯 ϰ ȴ. ̵ Ư¡ ⺰ .
17 С
ٿ ν ߰߰ âǰ ʷ Դ. J.÷, J.Ǿ, P.丣 Ͽ R.īƮ, B.ĽĮ, I., G.W. ο о߸ ôϿ. ̵ ܾ СõСö о߱ Ͽ õ̾, ̷ 鿡 Ĵ ڵ ټ ϰ ִ. â ߰߿ Ư Ÿ ִ. м ö īƮ ؼ âڷμ ̸ ִ. а νѼ âϿ. ̰ ߰߿ ġ ִٰ ִ. ϰ âϿ ٴؼ ߴ . Ͽ ٴؼ ߴ . С 迡 ؼ() Ͽ п ū ƴ. 1671 üȭϿ. ߷() Ģ ߰, ڼ(أ) . ŰǾƣPhilosophiae Naturalis Principia Mathematical 1687 Ǿ. Ŀ â ѷΰ ־ ᱹ ڴ ̷ٴ ظǾ. ȣȭ() ū . ȣ ٰ ũ. Сöп ū .
18 С
18 17 â ؼ ô뿴. ϰ ڵ Ȱ νô. ϰ Բ L.Ϸ â ؼ ϽŽ״. Żƿ ִ J.L.ִ Ϸ Ҿ (ܨ) . ؼп ũ P.S.ö, ȭ() â P.G.ֵ ô ̾.
19 С
17⸦ ο â ô, 18⸦ ô Ѵٸ 19 뿡 ̾ ǰ â Ǵٽ ӵǴ ô ִ. ü ϼ ܰ踦 Ͽ ô ϰڴ. п 18 ε Ȱ ū Ͽ 19 ͼ ϻ ־. J.K.F.콺 Ͽ K.̾Ʈ, G.F.B., J.W.R.ŲƮ, G.ĭ, F.Ŭ, D.Ʈ Ǽ ū Ͽ. 콺 () Ͽ о , A.L.ڽ ؼ , 밡 J. Ŭ , 븣 N.H.ƺ а ؼп , E. ķС() , ̾Ʈ, ؼ, â 19 ٽɺκ̶ ִ.
С
ݼ ͼ о߿ ο Ʈ Ͽ. δ Ͽ о ũ ̷ϰ Ǵ , ̸ ʿ ݼ ü () λ ̴. , ʿ Ǿ Ȯ ū ÷ȴ. ĭ շ() â 19 ε о߿ ħϿ ϽŽ״ٰ ȴ. ŲƮ (Ө:Schnitt)̶ Ͽ ʸ Ȯϴ Ͽ. Ŭ ؼп Ӹ ƴ϶ ϣErlangen Program ǥϿ п ٶ ҷ״. ״ б ǰ âϿ. Ʈ (ͧ) Ǽ() ʸ ̷. ʸ Ȯϴ ۾ ϸ鼭, δ ο س, о ȭ ϴ δ ŵϰ ִ.
а ȸ
Ұ
1. շ .
շ "ο" ʼμ νĵ, ־ ο . â ô ڵ Ǵ ٸ ϴ ̲ , շ ⺻ ǿ ־ ̴. , 19 濡 μ Georg Cantor (1845-1918) ̷ ֿ μ Ƶ鿩 . ̻ϰԵ, ó Cantor ϰ ﰢ
ſ о ̾. ó, ״ ڽ Ű ؼ Ͼ Ǽ Ư տ ״. Cantor ߰ Ϲ տ ߾. 1872 1897 ̿ ߰ ָ 鿡 ״ տ
ԹŲ ü 鿡 ư ó շ ʰ Ϲ ư. ô Cantor ܰ 뺸 ڿ ʰ ̾. "" ΰ ϳ . ڴ Ʋ 1/2, 1/4, 1/8, 1/16 κб Zeno "Paradox" ˰ ִ. κб - ۵
- ѷ 0 ƴ ̸ ְ, κ ִ. , ܰ ƴ ̴ Բ ̸ ִ ̴. ϱ ؼ, ڵ ÿ ϴ μ ȵǾ "Actual" Ѱ ־ ܼ ʰϴ "Virtual" . "Virtual" ִ μ ֵǾ, "Actual" Ǵ μ ֵǾ.
Cantor ̷- (⼭ и "Actual"ǹ̷μ صǾ) - ô ؼ ʾҴ ƴϴ. װ ó ȸǷ Ƶ鿩, Ƿ Ƶ鿩. , 1890 뿡 Cantor ̷ "Palatable" κ а Ǿµ װ ̷ а پ Ϳ Ǹ 븦 ־ ̾. , Ⱑ ٲ շ ڵ鿡 ؼ Ƶ鿩 . ַ װ Ư ؼп ſ μ ̾. ̿ ڵ , Ư Dedekind շ "Unifying"о߿ μ ҿ ־. â ô,
ڵ Ͽ ü о߸ ɼ ϰ ־. ĵ Euclid оߵ п ԵǾ ִٰ ߴ. ( μ Ǿ ִ). Weierstrass Dedekind, ٸ
ڿ( ) ̲ ִٰ ߴ 19 õ ־. װ Ǽ ("Cauchy sequence" Ҹ)μ ִ . , Ǽ װͿ
ҵȴ. μ ִ. ᱹ ϰ ؼ Ͽ ϴ Ǽ ڿ ʵǾ ִ. ʿ ־ å Was sind and was sollen die
Zahlen (1988) Dedekind ڿ շ ̲ ִٰ . ̰ ִ ;
"0" ̶ . (, Ҹ , ȣ μ ǵȴ);
"1" {}μ ǵȴ. , ϳ ϴ . , "2" {0, 1} ϰ, "3" {0,1,2}μ ǵȴ. .
ڿ ǵ շ Ͽ ɼ ִ. Ⱑ ٱж , շ ȸ īٶ κμ ȵ μ ƴ϶, Դٰ ߿ ġ (Antender). ݾ, Cantor ᱹ Ҵٰ ٷ 뿡, "Paradoxes" ù° ˷ µ, װ ħ շ "Cantorian" Ŀ շ ߰տ ǽμ . Ķ ʹ ģ ݹ̾ ڼϰ װ͵ 캼 ġ ִ.
2. THE PARADOXES.
1895 1910 ̿ շ κп ణ ߰ߵǾ. ó ڵ װ͵鿡 ʾҴ. װ͵ "paradoxes" ҷ, ȣɺ ణ μ . Ķ ߿ 1897 Burali-Forti ؼ ǥǾ
, װ ̹ 2 Cantor ڽſ ؼ ߰ߵǾ . Burali-Forti Ķ ټ շ Ÿ ó ǵ ణ װ ĥ̶ ߾. , 1902 Bertrand Russell շ
ϴ Ķ ǰ ־. Ƿ װ õǾ . ٸ ߰ߵǾ µ, װ "safest" 鿡 . շ "paradoxes" ΰ ִ. ϴ paradoxes Ҹ, ٸ ϳ paradoxes Ҹ. "Logical" "semantic"̶ ̸ ̵ Ķ ణ ν 츮 ϰ ̴. , "logical"Ķ ߸ κ , "semantic"Ķ ߸ 뿡 ´. ̰ κп շ ϴ Ķ ΰ Ұ ̴.
ù° "logical"Ķ̰, ι° "semantic"Ķ̴. Ѵ װ͵ Ǿ . Ķ ܼ Russell's paradoxε, ̰ Ǿ ִ.
A ̶ Ѵٸ, װ Ҵ ڽ . Ȳ п Ͼ. , A µ, μ . ⼭, A ڽ ɼ Ͼ. , ´. S ڽ Ұ ƴ ̶ . S ڽ ϱ? ۽, S S Ҷ Ѵٸ-S ǿ ؼ- S S Ұ ƴϴ. S S Ұ ƴ϶, (ٽ, S ǵ ) S S ̴. S S ʿ S S Ұ ƴҶ Ѵ. - ⺻ ̴. , п ־, 츮 ̷ 츮 ϳ ߸ ִٰ ؾ߸ Ѵ. ̷ 쿡, 츮 ڽ ҷ ϴ ϴµ ǹ̶簡 " ڽ Ұ ƴ յ " ͵ ٵ簡 ̲. 츮 ƿ ̴. Ķ Ͽ . Ķ Berry's paradox̴. Ǹ ϱ Ͽ, ܾ ǥ ϵ ִٰ . 20ܾ Ҽ ִ ڿ T . ܾ ϱ 20 յǴ ŷ ܾ Ѱ ִ. - , T ̴. , иϰ T Һ ڿ Ѵ. , 20ܾ ּ ڿ ִ. ǿ ؼ, Tȿ . 츮 Ű 16 ܾ ߴ. , װ Tȿ ִ. ٽ ѹ, 츮 Ͽ. Ǵ ̱ 츮 T 縦 Ѵٸ 츮 T ܼϰ ´. Ķ , 翡 ʾҾ. Cantor "üμ Ǿ ִ 츮 ѷϰ, Ǵ "μ ""Ͽ. ϰ, ĵͿ ڵ 츮 ִٸ, ϴ ִٴ "command-sense" Ƶ鿴.
Ķ "" Ķ ̲ٸ Ƶ ҹ ־. 1900 շ ʸ Ͼ ߽ ȭ 翴. ϰ ϴ°? Ͽ ϴ°? ϴ κ ο ִ°?
3. .
Ķ ó ϰ Ǯ ʰ ִ ʿ ǥߴ. Cantor "" üȭŴ ó տ շп ʸ Ȯϰ ߴ. - üμ п Ͽ . Ư Ÿ Ǹ μ Ķ ַ µ ߴ. ʿ ߴ.
1905 ٷµ ־ ߰, ״. κ κ з ִ. , װ "aximatic", "logistic", "intuitionist" Ҹ , ֿ ̴. 츮 ҰѴ. , ó ַ ϴµ ȴ. 300 뿡 Euclid Element , Դ. Euclid ؼ ˷ ó о
Ư¡ Ǿ. ̿ܿ ֱ ʹ. , ü迡 츮 ؿ Ϲ κ о ԵǾ. , ڱ п ־ ֿ ڴ.
Euclid ô뿡, "truths" ǥѴ. , ų Euclid "parallel postulate" õ ְ ̴. ϸ A B 3 C A B ݵ . "obviously true", װ ٸ ŭ Ἲ ʾұ , Euclid 1700 ڵ Ἥ ̰ ҷ 翴
. 19 ߿, Bolyai Lobachevski Ͽ ذǾ. , λ ༱ شſ װͰ Ǵ μ "non-Euclidean" ״. Euclid Euclid ŭ ո̶ Ǿ. , װ Euclid ؼ Ͽ ̴. (, "point", "line", "angle" ٽ ؼν Bolyai Ǵ Labachevski Euclid п Ѵ) ༱ Euclid ü ӿ ٸ ؿ
Ӹ ƴ϶, ݸ Euclid ŭ 鼭 츮 ϻ Ȱ ϴ ϴ ƴ ϰ ȴ. , "universal truths" ƴ϶ 츮 ̷п μ ϰ ϴ ̶ νĿ ߴ. Euclid ־ ִ Euclid Ͽ ƴ, Ͼϸ ̴. ΰ ٸ ܽ ΰ ٰ , ̷ 縦 ʾҴ. ٸ Euclid ; ״ "betweenness" ʾҰ ʾҴ. ٸ 忡 rigid motion ϰ , ʾҰ ʾҴ. Euclid ߷ Ⱑ ߴܵǴµ, ȣϱ ̴.
19 ̸ ߰ߵǾ ̷ Ǵ Ÿ Ű ġ ʰ ɼ ־ ϰڴٴ ذ Ǿ. , ( , "", "", "" ) " " νĵǰ õǾ ־ ϸ ̾ Ѵ. 1882 M. Pasch ó ǥϿ, ȣҸ ϴ ̶ ϰ ü õϿ.
19 ߴ. ־ Euclid ٸ ٰ . , "" äõǰ ̷п ߷ Ͽ ̷ ""̶ Ѵ. "" ݿ ο ذ ־. ڼϿ ϸ, Ģ Ͽ, ̳ Ұ ʿ䰡 Ѵ. ̻ Ѵٸ
¾Ҵ Ʋȴ ϴ 谡 ټ ־ Ѵ.
ϻ Ǵ , ΰ ظ Ͽ ؼ ߰ ʿϴ. , ŵǷ ʿϰ ȴ. "Ģ" ϰ ȭǾ ָ ʰ ϰ Ǿ Ѵ. ȣ â п ־ ߿
̴. ϴ ⼭ . κ ϰ .
"X is parallel to Y,"
"y lies between x and z,"
"X is an open set," etc
̰͵ Ǵ ¦ Ǵ Ϲ n 鿡 ̴. ⺻ ϰ X,Y,x,y,z ڴ Ѵ. Ÿ Ͽ ڵڿ ϴ ϴ. , "X Y
" A(X,Y) "y x z̿ ִ" B(x,y,z) Ÿ Ͱ . ߷ "" ƹ ǹ̰ ٴ ƴ л ˰ ִ. , ""̶ "" ƹ ۿ뵵 ʴ´. ۿϴ "X Y
" ٸ ̴. ⼭ ٸ "X V "簡 "Y Z ̴"簡 ϴ . ʺ ̶ Ѵ. װ "ٽ " ̾ ̻ м . װ ٸ ٸ Ͱ ̴.
оߴ Ѱ( ſ Ҽ) ٸ ⺻ ʿ Ѵٴ ̴. Euclid ǥȴ.
P(x): x is a point.
L(x): x is a line.
(1) B(x,y,z): y is between x and z.
E(x,y): x equals y.
I(x,y): x belongs to y.
C(u,v,x,y): the segment uv is congruent to the segment xy.
D(u,v,w,x,y,z): the angle uvw is congruent to the angle xyz.
շ ٽ ϴ ٿ , ϳ ȴ. , xA (x A Ҵ)̴. δ Ÿ ʴ. 縸 Ͱ . " x y ̰, y z ̸ x z ̴" Ѵٰ . ̷ ̾ ִ. , P Q 츮 ̶ .
P: not P.
PQ: P and Q.
PQ: P or Q.
PQ: P im;ies
PQ: P if and only if Q.
"x "̰ "y "̸ "z x y̿ ִ" ϴ " z ִ" Ϸ . ⼭ ȣ ʿ Ѵ. P(x) x .
x, P(x): for every x, P(x).
xP(x): there exits an x such that P(x).
̰ 츮 о δ. ˷ δ ⺻ и ȣ Ÿ ִ. Ǵ ϴ ϱ Ͽ . " x y ٸ ̶ x y̿ z ִ" ȣ Ÿ.
(2) [P(x)P(y) E(x,y)] [z(P(z)B(x,z,y))],
⼭ (1) δ. 뿡 ϳ ؼ Ȯϰ ϰ Ÿ ִٴ ̴. ణ ϰ ָ ʴ "Ģ" T S ߷еǴ Ѵ. ణ Ģ̶ ̴.
(3) Ģ A: from P and -> Q we may infer Q.
Ģ B: from p and Q we may infer P ^ Q.
Ģ C: from ( P) we may infer P.
Ģ D: from P(c) we may infer x -> P(x).
Ģ ణ ٸ Ģ ʷ Ģ̶ Ѵ. ־ ü ϴ ǥ ̸, ǥ ü ְ, ǥ(Ǵ ǥ) ʷ Ģ Ͽ ִ.
() ü L(x,y) ľ . L(x,y) x
Premises
1) a < b.
2) b < c.
3) [(a (a
x[(a
. ȣ ǥ
1. a < b (i)
2. b < c (ii)
3. (a
4. [(a(a
5. a < c ȣ 3,4; Ģ A
6. (a
7. x[(a
⼭ ߷ Ģ ǥ Ǿٴ ָϿ. ǵ Ͽ, ǥ ȣ ǹ ̶ ȴ. װ 츮 Ͽ ǹ̰ ִٴ ϴ ϰ 谡 . 븦 ̴.
̷п ־ ٲ ȭߴٶ ϴ. ̷ Ǵ ó ̻, װ ʹ ŷӴ. ȣ صϱ ư ô .
ڵ ̷ ȭ ִٴ ϰ ׳ ȭ ʴ Ű ̴. 츮 ⼭ ̴.
4. շ.
1900 ʱ ڵ ϴ շ ̶ Ͽ. ""̶ , " " ⼭ ̴. װ ġ "" "" Ͽ Ͱ . ⼭ ƹ ǹ̰ , μ ־ Ѵ. Ư, Cantor շп ǵ ϰ õǾ Ѵ.
շ ù ȭ 1908 Zermelo Ͽ ־. Zermelo ü Skolem Fraenkel ణ ó θ ǰ ִ. Zermelo θ صǰ DZ . շ ʾҴ. ü п ־ ̾. Zermelo ü迡 ϳ 谡 ִ. ȣδ . xY ǥ "x Y Ҵ" д´. x,y,z,X,Y,... ȣ
ʿ ġŰ ε װ 츮 ""̶ θ ̴.
ڵ Ƹ ΰ , , հ Ұ ־ Ѵٰ ̴. ʿϴ. ̷. ҿ ƴϴ ( ҿ Ȯϰ ). ٸ ϸ, п
̴. , ؼϿ ̴. Ǽ ¦
( ǥ)̴. Ǽ ( )̴ . շп ̷ ܼȭ, Ҹ ٽ ̶ ϴ ̴. ٽ ؼ ̶ ϴ ̴. ̷ ܼȭϸ طο ? ʴ. ̰ շ ̴ ȿ ִ.
ȣ ʿ䰡 ִ. ڿ ڸ Ἥ xY , ʿ䰡 . xy Ǵ XY ִ. ⼭ Ÿ.
츮 ϴ δ Ư η Ǿ ִ. , ־ ϴ η Ǿ ִ. ڿ ̴̰. 츮 x ִ. S(x) ȣ . S(x) ϴ x ϰ ȴ.
() "x ̰, 0x1̴" ϴ x (ٽ Ѵٸ 0 1 ü ) "x ̴" Ÿ ִ xü .
̰ ڿ ̱ , װ ϰԲ շп ϳ Ģ ־ ̴. S(x) ־ , S(x) ϴ xü ̴. 츮 2 ٿ ̷ Ģ ƹ ȴٸ
츮 Ǵ ̴( , ڱڽ Ұ Ǿ ü ִ.) ִ ȵȴ. Zermelo ߴ. S(x) x ̶ . S(x) ϴ x . , A ־ ̶, Aȿ ִ s(x) ϴ x ְ ȴ. ؼ, "" ϴ ٸ ̹ 簡 Ȯ
Ǿ ִ A ־ ϴ θ ϴ ̴.
Zermelo Ģ ü ϳ äߴ. װ վȿ Ҹ ϴ ϱ " " ҷ. װ .
A ̶ . A x Ͽ ǹ̰ ִ ϳ S(x) . S(x) ϴ A x鸸 Ѵ.
簡 ´.
{xA|S(x)}
[̰ "S(x) ϴ A x "̶ д´.]
⼭ Zermelo ü迡 {x|S(x)} [S(x)
x ] ʴٴ ̴. ٸ A
{xA|S(x)} ִ ̴.
Ͽ ִ°? Russell ĸ . Russell "ڱڽ Ұ Ǿ "̾. װ ȣδ {x|xx} ִ. ̹ ٿ Zermelo ü迡 . . 츮 ִ ⲯؾ{xA|xx} ̰ ⼭ A 縦 ִ ̴. {xA|xx} Russell {x|xx} ġѴٸ . , S {sA|xx} ϰ ܰ踦 ʷ . SS Ұϴ. ֳ ϸ SS̸ SS̴ϱ ̴. SS̴. ֳϸ SS̶(SS ƿ) SS, װ ̴.
, Russell A {xA|xx} A ٴ ϰ ̴. Ÿ غ ִ. ִ. , װ͵ ʹ ũ ʹ ϰ ִ. Russell "ڱڽ Ұ Ǿ ʴ ü "̾ Cantor (װ Russell Ͱ ϰ Ǿ ) " "̾. ̷ ū ϴ κ ȴ.
ǹ̷ ϴ ƴ. Berry ʹ ū ƴϾ. Ÿ , Ѵ S(x)ȿ ִ ϴ. ѵ ȿ 輱 Ѵ. S(x) "x 20 ̸ ܾ ִ" ̶ Berry {xN|S(x)}̴. ⼭ N ڿ ̴. S(x) x Ƶ帱 ִ ̶ Ѵٸ Zermelo ü迡 ִ. ǹ̷ ִ "" S(x) ŸԿ Ͽ ȴ. Zermelo S(x) A x Ͽ ǹ̰ Ͽ {xA|S(x)} ִٰ ϸ ̶ Ͽ. ᱹ ״ ϳ ο ǹ
̾. "ǹ̰ " ̳? S(x) ǹ̰ ִ ¾ ΰ? ϳ ִ. ̹ ڴ . " S(x)" , " x " Ѱ? "X " ִٰ ؼ ȵȴ. ֳϸ 츮 շ ȭϷ ϰ ִ ̰ عǷ ϰ ֱ ̴. Zermelo 亯 ߴ. ״ ü踦 ľ ʾұ ̴. 1922 Skolem Fraenkel շ ȭ ۾ ϸ鼭 ߰Ͽ. "x " ܼ ľ ϳ "" x . Zermelo ü迡 ⺻ ϳ̰, ȣ ´. ľ xY, uV , ȣ Ἥ Ÿ ִ.
ִ "" S(x) ľ ǥ ִ Ѵٸ ǹ̷ ְ ȴ. , "x 20ܾ ̸ ִ" ľ . ̰ Ͽ ǹ̷ ϴ ٶ ̴. ö ̻ ذå̶ , δ п ʿ ִ ̴. 忡 츮 " " "
" , ذå õ ʾҴ.
Von Neumann Ͽ и ´ٴ ָߴ. ù° 츮 ̹ ʹ ũٴ ̴. ° "ū" Ұ ִٰ Ͽٴ ̴. Zermelo ù° Ǹ Ͽ. , ״ ʹ ū ٰ μ ̴. von Neunmann ° Ͽ. ״ Ͼ 縦 Ϸ Ѵ. ٸ װ Ұ ɼ ٰ Ͽ.
ؼ von Neumann ü ǥ ִ. Zermelo ⺻ ϳ̴. , xY̴. x,y,X,Y 츮 (Class) θ Ÿ.
ִ. , - װ Ұ Ǵ ǵȴ - ҵ . Zermelo "" ġ ִ.
S(x) x , S(x) ϴ "" x Ѵ. ٽ ؼ, S(x) x , 츮 {x|x ̰ S(x) Ѵ} ִ.
ü迡 russell ʴ´ٴ ϱ Ͽ Russell ܰ踦 . S {x|x ̰ xx}̴. SS Ұϴ. ֳϸ SS SS̴ϱ , SS̴. S Ұ ƴϴ. ֳϸ S Ҷ SS ̱ װ ̴. Russell S Ұ ƴ϶ ϰ ̴. ǹ̷ Zermelo ü迡 ľ Ͽ ǥ ִ "" μ, ִ.
von Neumann ü Godel Bernays Ͽ Ǿ. װ Zermelo ü躸 շп ٴ ̴. S(x) x , S(x) ϴ x ϴ 縦 Ѵ. п
, von Neumann ü " ҵ " Ͽ ְ ְ Ұ ƴ ϰ ش. ü ū Ұ ִ Ұ ɼ -װ ΰ - ϰ ־ Ѵٴ ̴. Ҹ von NeumannŸü ū 뼺 ڿ ǽɹ ġ ִ. å 츮 von Neumann ü ణ ̴.
5. OBJECTIONS TO THE AXIOMATIC APPROACH. OTHER PROPOSALS
շ ֵ ̰, ̵ Ǿ Դ°?
ϱ Ͽ, 츮 20 ʿ ڵ շ ʸ ϴ ȯ ȸؾ Ѵ. cANTOR ̹ ȿ ħϰ ־. ڵ鿡Դ Ǿ ־. а ؼ շ Ʋ ȿ Ǿ. ȭϰ Cantor İڵ Ἥ ̷. ߰ߵǰ Cantor ü Ǽ ǽ , κ ڵ ü踦 ⸦ ȴٴ 츮 ִ. ̰ܳ ߰ߵǾ δ ƴϴ Cantor κ ϰ ־. Hilbert ִ. "Cantor 츮 츮 Ѱܳ
̴." ο ߰ߵǰ, װ Ϸ ó õ շ ״ ٴ иϰ Ǿ. ΰ ȵ ̾. 븦 ɾ ּ ְ
ŭ շ ƾ ϰڴٴ ̾.
Zermelo ü von Neumann ü ѵ ϴ ̴. ű⼭ շ з Ҵ. ٸ Zermelo ü迡 ̹ ٿ ϴ ʴ´. װ ġǾ.
ű⼭ "" κ ϴ ְ Ͽ. ư ִ ϰ ȣ x,y,z,... Ÿ ִ ̾. 츮 ̶ ϴ κ- " " ֿ ִ " " - շп ̶ Ѵ. շ ϴ "" 켱 , ִ {},{,{}},...
̴. ü ̿ ô ִٴ ̴. շ 츮 ٰ ϴ ݴ ߿ ö Դ. װ "" κ̴. Ǵ ߽ ϰ ִ. ̶ ΰ һ(, ߸)̳?
װ 츮ʹ ö ϰ ִ ̾ ڵ鿡 Ͽ ߰
Ǵ ٸ ִ ̳? ظ "ö "̶ Ͽ п ش. 츮 븳ϴ ذ ϳ ü - ڿ - Ͽ Ǵ° ϴ Ͽ . ö 忡 "ϳ", "", "" ڿӿ ϰ װ ù ΰ ϱ Ѵ.
̿ 簡 ִٸ 츮 ٸ ڿ ߰ ̰, 츮 ߰ Ͱ ̴. ݸ, ؿ , , Ǵ ڿ ӿ , ڿ 3 츮 â̸ 츮 ڿ ϴ ߸ ̴(0 , ȸ 1 Ͽ μ 1,2,3,... ). ̷ Ͽ 츮 α ִ ̴.
ö շ ¿ ִ°? ö 츮 Ư¡ Ҿ ⼺ǰ ־ ̴. ""̶ ٸ ǥߴٴ ȴ. ٸ, 2+2=4 ƴ϶, װ " " ϰ ִ ̴. - 츮 Ҿ 츮 ־ٰ
Ѵٸ, Ư"" ǵ ̰ 츮 ̴. Zermelo von Neumann ʴ ǥϴ ϴ ȴ. ᱹ 츮 Ǯ Ƶδٸ Zermelo von Neumann ü踦 ȿ ̶ ô ȴ.
Russell Ÿ ̷ ƸԵ ܼ . ϰԵ װ ϱ ؼ Russell ȵǰ Ǿ. ħ ̷̶ ʹ ġ彺 ߰ ʹ Ͽ Ǿ. ù°
""̶ ϴ ̶ ־ ߴ. ǹ̷ ϱ Ͽ ؿ ִ յ "" Ͽ ٽ ߴ. Russell " ɼ " Ǿµ װ ݰ Zermelo ſ ̰ ٰŸ ̾. ̰ ֱ Ÿ ̷ װ ְ о̱ ڵ ̿ θ ߴ.
Ī ڶ ڵ 翡 Ͽ . ڵ Cantor շ κе Ͽ Ģ ϴµ ʸ ΰ ְ װ ߸̶ Ѵ. շп ڵ µ ϴ ſ . װ ô̶ µ. ô Ϸ µ ؾ Ѵ. ڰ ڵ Ģ ִ. ڵ鿡 Ͽ ǰ ǰ, ظ ġ ȿ Ģ̶ ϴ ü ӿ Ģμ ȴ. ̷ Ģ ó 迡 ־ ٸ ̾ װ Ģȭǰ ٸ Ȳ Ͽ ǰ Ǿ. ü غ. 15 ־ ٺ̾ Euclid Ͽ
Ѱ Ѵ. ( ) ۵ Ͽ ־. Euclid ĥ ̷ ÿ Ǵ. ٸ ϴ װ Ű ű⼭ ü ۵ ־ ̰ ű⼭
̵ Ģ ٸ ʴٰ ڵ Ѵ. Law of the excluded middle . Ģ " S SȤ S ̴." ϴ ̴. Ư
A ̰ P(x) A x Ͽ ǹ̰ ִ . ߷ ϸ Aȿ P(x) x ֵ, ƴϸ A X Ͽ P(x) ̴ ̴. A ̾ P(x) ƴ A ҿ Ͽ ִٸ Ģ Ǵٰ ڵ Ѵ. , Ģ ٰ̾ ڵ Ѵ. 츮 迡 ϸ 츮 A x Ͽ P(x) ħ ƴ
Ѵٸ ( Ͽ A ̾ Ѵ) ΰ ɼ ۿ . , x Ͽ P(x) ħ 츮 ߰ߴ, ƴϸ A x Ͽ P(x) ̴.
츮 쿡 ߷ Ȯϰ ȴ. 츦 . װ 츮 ̰ ̴.
ű⼭ ߷ Ѵٴ ٰŰ ̴. ڵ ̰ ߷ źѴ. п ٸ Ģ Ѵ. װ Ѿ ̴.
Euclid Ϲ ʱп . . Pythagoras ϴ κ յ̱ Ǵ ۵ Ͽ Ǿ. ۵ ϼǰ յ κ ϰ Ͽ ϴ ̴. , ڵ Ѵ. Ģ̶ ̰ ۵ Ȳӿ ¸ ϱ ٰ̾. , ߷ Ѱ ־ (ڳ ۽ ٵ簡
Ͽ) P ִ ʴĸ Ʈϴ ־ٸ, ٸ 츮 Ͽ Ʈ Ͽ ϳ 䱸 Ѵٵ簡 Ǵ 䱸 Ѵٵ簡 ϴ ˼ ִٰ Ϸ ̴. ڵ ظ ϸ, . , (۵) Ƿμ ǥ Ұϴ. (۵)̾ Ѵ. 츮
Ѵٰ Ϸ, ϴ μ Ͽ Ѵ. ־ ¦ ̿ 谡 Ѵٰ Ϸ ǰ ִ ¦ Ͽ Ʈ ϴ Ͽ Ѵ. " Ģ"̶ ϴ ܼ Ұϴ. а õ Ѿ Ģ ȴٰ ٰŴ 츮Դ . - ̿ ۿ Ģ Ϸ ǹ ̴. п ־ μ
ƴϴ. ڵ հ 츮 Ͱ ٸٴ иϴ.
Cantor . 츮 ϸ Ѵٴ . -Zermelo von Neumann ǥ ƿ - ̶ ڵ . ϴ װ Ǿ߸ Ѵ.
츮 Ѵ. ̰ ڵ ̴.
ڵ շ Ұϴ 츮 å . п ߿ Ư Ұϴ ʿ ̴. װ spread ϴ ̴. spread Ҹ Ģ Ѵ. spread "̹ " ü ϴ ƴϰ "ϴ " Ѵ. Ҵ 츮 Ģ ϱ⸸ ϸ ִ. Ķ Ķ п ʰ Ǵ ؼ ǥǾ շп Ͼ .
6.CONCLUDING REMARKS
20 ȿ, 츮 ô ó, ̷ ٸ "schools" ߵǰ ȵǾ. 츮 շ ⺻ Ұ Russell Ÿ ̷а տ Ҵ. ۿ Ÿ ª п . ٷ ߿ 屸 ϴٰ ȴ. Zermelo von Neumann ü迡 "" ؼ ̰, ,
ġ ־ װ Cantor հ ٸ ʴ. ȣ м- ΰ ȣѴ. Ư ߿ շ κ ڵ ڿ δٴ ̴.
շ ôϴ ö ̴. շ ϱ⸦ źϴ ö ÿ κ Ÿ缺 ϰ ȴ. ؼ κ װ
ʾҴٰ ؼ ϰ ȴ. ̿ ̰ 츮 Ḧ װ ı ִ. 70 ȿ շ ̰ ϴ ϴ о߶ θ ް Ǿ. 츮 ڿ Ǵ ϴ Ͱ շ ȿ ̹ Ҵ. ű⼭ ϴ ̰ Ǽ Ҽ Cantor "ѱ" ִ. Լ, , 䵵 ǵȴ. о߰ շ ȿ .
翬 ƴ϶ ʿ ̴. "շ ü ü迡 Ͽ 븦 ִ?" Ư շ ߽Ǽ ٰ Ȯ ִ°? , ٸ ȿ Ų - - ߽Ǽ ̴. ʴٸ 츮 Ǽϵ װ ġ ̴.
շ ߽ ˷ ʾҴ. ̰ ణ ؿ ʹ ƴϴ , 1931 K.Godel ڿ ߽ Ѽ ִ Ұϴ" ߴ. о߿ Ȳ ʹ
. 츮 շ ߽Ǽ ּ ̴. 츮 . 뼺 ߽Ǽ ణ ̰ ִ. ֱٿ Zermelo շ ߽ϴٸ von Neumann Ǿ. Ƹ ؼ ߽Ǽ տ ؾ߸ ̴.ó ̳ ̴ ̷ ٰ Ͽ ƴϴ. ʰ Ǵ , ȮС ̹ƽ(cybernetics) ۷̼ ġ(operations research) , ߸ Ҿ ȮǾ, ڿ ̰, ιСȸС̳ 濵 ġ ִ.
( )
ʿ μ а. ̸ , ǥ һμ Ŭ п(̾) ̸ μ () ü踦 ߰ Ǿ. ̶ Ҹ ü (), () (ު)Ŭ ߰ Ͽ, ٴ () Ͽ. , 19 G.ĭ Ͽ â () ʿ Ǵ ſ ̶ νĵǾ. ̸ ڿ G.Ƴ (ͧ), J.W.R.ŲƮ ڿ, ⺻̰ Ͽ, ư о߿ ħϿ. , 1901 B. ĭ ǿ շ (, Ǵ ) ߰Ͽ ̰ ڿ Ͽ ʸ ݼ, ϴ Ⱑ Ǿ. , ̲, ij ü ʿ ݼ Ͽ бʷ ٰܳ ִ. ̶ ̾ ϴ° 忡 бʷ (:logicism), L.E.J.ο캣 (κ:intuitionism), D.Ʈ (:formalism) Ǿµ, Ȱ о̴.
ǡ ̰ а ̴. 繰 ؼ ϴ ( ) ٷ й̸, ٸ ̾߸ ̶ þ о߰ ƴϰڴĶ ϴ ̴. ûô̸, ô롯 Ѵ. 忡 ̸ ȣ 籸Ϸ ȯ()Ѱ()ð() Ͽµ, ڽŵ Ҹϴٴ ǥϰ ִ.
ǡ ̰ , ϴ Ű ϴ ƴ϶, Ȱ¿ ٷ ̶ ϴ ̴. L.ũγĿ, H.Ǫī, L.E.J.ο캣 Ͽ ǥȴ. , ο캣 ߷() δϴٰ Ѵ. ̸ P ڿ ϰų ʰų ̴١ P ڿ ǰų, Ǵ Ѵٰ ϴ õǾ 쿡 Ǵ. ǵ Ȯ 쿡 ̶ ̶ . ߷ Ϲ Ģ̶ ٴ ̴. ̷ 忡 籸ڸ κп ľ ϰ, ؾ ϴİ ־ δ. ״ Ǹ ǥϰ Ʈ 븳ϸ ִ.
ǡ Ʈ Ͽ ǥǴ ǰ ̿ شȴ. öϰ ȭ () ϰ ϴ ԵǾٰ ִ. ̶, ̹ ̶ ˷ Ȯ ȯϴ ε, ٰŰ ǰ ִ շ ȯ ʶ ΰ, ΰ Ͽ ״ () (ء), ʼ(:metamathematics) âѴ. Ǵ бʷп K. ҿ(), ڿ , Ÿ շ ø ִ.
()
п ̴ ȣ. ľϿ ȣ ǥѴ. ȣ ũ ͿԴ. ǥȭ()ϰ, ȭ()Ͽ йμ Ư ϰ ִ. ֿ бȣ ǥݿ .
(ҡ)
п. ڿ n P(n) P(n) ڿ Ͽ ϴ Ϸ, 2 ϸ ȴ.
P(1) Ѵ.
P(k) Ѵٰ Ѵٸ, P(k1) Ѵ.
̿ , 2ܰ迡 ؼ ־ P(n) ڿ Ͽ ̴ ͳ Ǵ ͳ̶ Ѵ. ̸ , n ڿ , 135(2n1) = n2
ͳ ϸ,
n1 , º и 1̸, 캯 121̹Ƿ, n1 , Ѵ.
nk , Ѵٰ ϸ,
135(2k1)=k2
纯 2k1 ϸ,
135(2k1)(2k1) =k2(2k1)
̸, 캯 ϸ, (k1)2 ȴ. ,
135(2k1)(2k1)=(k1)2
̸, Ŀ nk1 ̸, ⼭ nk Ѵٰ ϸ nk1 Ѵٴ ̴. , 迡 ؼ ڿ n Ͽ Ѵ. ߷ ڿ ü Ƴ (ͧ) 5 ʷ Ͽ ̷ ̴. Ƴ 5 ͳ Ѵ.
()
1910~13 B. A.N.ȭƮ尡 () м. [ڵ:AA, :(ABAB), ߷:A=BAB] հ ȯų ִٴ Ͽ ü踦 籸Ϸ ← å, ټ Ͽ. п ȿ ȯ() Ģκ ٴ 巯, ŷο ʷϿ. ұϰ, ̷(Theory of Types) < շп ־ ð >(35), <Ϲ ü >(35) οǴ å м ڵ οǰ ִ.
(ϰ)
տ ο . Ѵ. , ()ȯ()ü() ̰, ̿ Ǿ , Ģ(ߩЮ) ϰ ִ. Ÿ() ̿ ٰ谡 Ǿ ִ. Ϲ (), ϴ Ϳ Ͽ ־ ִ ̴. ̿Ͱ п ϴ ܼ ڽŸ ʰ, Ǵ ̿ ־ ִ Ϳ Ͽ Ѵ. , տ ־ ִ Ϲ Ѵ. (ϰ), (ϰ), (ϰ) Ǵµ ̵ ϴ ֱ ʷμ ſ ߿ ϳ̴.
()
Ȯ . Ȯ̶ Ѵ. 3 , Ͼ , ո H, T ϸ, {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 8̰, 찡 Ÿ ִ. ̵ ߿ 2 ո, 1 {HHT, HTH, THH} 3μ, Ȯ 3/8̶ ִ. ̿Ͱ Ͼ ִ n ְ, 쵵 ÿ Ͼ ʰ, 찡 Ͼٰ , E Ͼ r̸, p=r/n E Ͼ Ȯ Ǵ Ȯ̶ Ѵ. 0??̹Ƿ 0?? ȴ.
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