GUSEPPE PEANO

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GUSEPPE PEANO

• ?S????? ??S ??G???S
• ?????????S TO??S
• ?? 3153
• ????S???S? ???? S?????FS???

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???G??F???

• Ge??????e st? Spinetta t?? ?ta??a? t? 1858 ?a?
  p??a?e st? ?????? 1932.
• ?? 1870 e??atast????e st? ?????? ??a ?a f??t?se?
  ta s?????? t?? ?????a ?a? st?? s????e?a sp??dase
  st? pa?ep?st?µ?? µa??µat???? t?? p????.

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• ???a? ???st?? ??a t?? s?µa?t??? d???e?? t?? st?
  s?µß????? ??????, t?? a???µat??? µ???d? a??? ?a?
  ??a t?? s?µa?t???? s?µß???? t?? st?? µa??µat???
  a????s?. ??d??e?t??e ep??? st?? µa??µat???
  ??????. ? Peano ?a?????se t??? s?µß???sµ??? t??
  Te???a? S??????. ??s??a?e t? 1889 s?st?µa
  a???µ?t?? µe t? ?p??? e?s????ta? ?? f?s????
  a???µ??. ?a a???µata a?t? ap?t??esa? ?a? t??
  ßas???te?? ???? d?????s?? t??.
• Sta t??? t?? 19?? a???a ? Guseppe Peano, Georg
  Cantor ?a? Helge von Koch pa???s?asa? ??p??a
  µa??µat??? d?µ??????µata p?? a??f??sa? t?? ?????
  ?????? ?a? a?t?????. ?? 1890 ? Peano pa???s?ase
  µ?a ?aµp??? p?? ??? µ??? µp????se ?a ????se?
  se µ?a pepe?asµ??? pe????? t?? ep?p?d?? a???
  ta?t?????a pe????se ?a? ap? ??a ta s?µe?a a?t??
  t?? pe??????.

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• ? Peano p??sp???se ep?s?? ap? t? ?t?? 1892 ?a
  a???e? sta p?a?s?a e??? e???te??? ????? µe t?t??
  Formulario Mathematic t? ?????? se
  s???e???µ??a a???µata. ?? ???? a?t? ???????????e
  t? 1908 ????? ?µ?? ?d?a?te?? ep?t???a. ?p? t?
  ?t?? 1903 p??sp???se ? Peano ?a d?µ??????se? µ?a
  te???t? ???ssa µe ß?s? t?? ?at????? ?a? st???e?a
  t?? ?a??????, ?e?µa????? ?a? a???????.

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??a pa?at???s? p?? ?a ?p?epe ?a ???e? ?s?? af???
ta a???µata t?? Peano e??a? ? e??? ? R.
Dedekind p?? ?e??????e t? 1831 ?a? p??a?e t?
1916 as???????e µe t?? ?eµe???s? t?? f?s????
a???µ?? ?a? t? 1888 ep??e?e p??te a???µata ??a
t?? ?eµe???s? a?t?. ?a a???µata a?t? ta ?p??a
a???te?a ? Peano ta e??f?ase se s?µß????? ???ssa
????a? ???st? ?? ???O???? ??? PEANO
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• ?a a???µata t?? Peano e??a?
• ?? 0 e??a? ??a? f?s???? a???µ??. (P1)
• G?a ???e n ?s??e? ?t?, a? t? n e??a? ??a? f?s????
  a???µ??, t?te e??a? ?a? ? ep?µe??? t?? ??a?
  f?s???? a???µ??. (P2)
• ?? d?? a???µ?? ????? t?? ?d?? ep?µe?? a???µ? ,
  t?te a?t?? ?? d?? e??a? ta?t?s?µ??. (P3)
• ?? µ?d?? de? e??a? ep?µe??? e??? f?s???? a???µ??.
  ( P4 )
• ?? ??a ??a ?p?s????? ? t?? ? ?s??e? ?t?
• a ) t? (µ?d??) 0 a???e? st? ?
• ß ) a? ??a ???e ? p?? a???e? st? ? s??ep??eta?
  ?t? ? 1 a???e? st? ?. (P5)

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F?S???? ???T???

• St?? ep?st?µ? t?? µa??µat???? ?at? t?? d????e?a
  t?? 1870 1880 ? Cantor ?a????se t?? ?????a t??
  f?s???? a???µ?? µe ß?s? t?? ????? p???????µ? t??
  ?s?d??aµ?? s??????.
• ??????t? ? ?pa??? µeµ???µ???? f?s???? a???µ??
  ap?de????eta? st? ?e???a s?????? s?et??? e????a,
  ??a t?? ap?de??? t?? ?pa???? t?? s?????? ???? t??
  f?s???? a???µ?? apa?t???ta? ta a???µata t??
  ?e???a? s?????? t?? Zermelo Fraenkel.

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???S??S ?? p???????µ?? t?? pepe?asµ???? s??????
?????ta? pepe?asµ???? a???µ?? ? f?s????
a???µ??.?? p????µe t??? p???????µ??? 0,1,2,...
???? t?? pepe?asµ???? s??????, p?? ?p?? e?paµe
???µ????ta? f?s???? a???µ??, ?e????µe ?t?
ap?te???? ??a s?????, t? ?p??? t? ???µ????µe ?
(s????? t?? f?s???? a???µ??) ? 0,1,2,3,
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S?????S??S ?O? ???O???O? PEANO

• ?? 0 e??a? ??a? f?s???? a???µ??. (P1)
• ?p?????? ep?st?µe? st?? ?p??e? ???s?µ?p??e?ta?
  t? µ?d?? ?p?? st?? µa??µat??? ??????, t? ?e???a
  s?????? ?a? st?? ep?st?µ? t?? ?p?????st??. ???a?
  p??? s?µa?t??? ?a a?af????µe ?t? ??a p??????
  a???e? ?? µ???? ???st?? f?s???? a???µ?? ?ta?
  a?t?? ????? t? µ?d??. St?? ????p? ? ???s? t??
  µ?de??? ?e????se ap? t?? 13? a???a.

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• P2 G?a ???e n ?s??e? ?t?, a? t? n e??a? ??a?
  f?s???? a???µ??, t?te e??a? ?a? ? ep?µe??? t??
  ??a? f?s???? a???µ??.
• ?? a???µa a?t? d??e? t?? t??p? ?atas?e??? t??
  s?????? ?. ?e afet???a t? µ?d?? ?atas?e?????µe
  ???e f?s??? a???µ?. ?ts? t?? ep?µe?? t?? µ?d??
  t?? s?µß??????µe 1 ( ??a ) t?? ep?µe?? t?? ??a
  µe 2, t?? ep?µe?? t?? 2 µe 3 ?.?.?. ?e a?t?? t??
  t??p? µetas??µat????µe t? s????? ?
  0,1,2,3,?, ? 1.....

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• P3 ?? d?? a???µ?? ????? t?? ?d?? ep?µe??
  a???µ? , t?te a?t?? ?? d?? e??a? ta?t?s?µ??.
• St? a???µa P3 d??eta? ?t? ?1 µ1 . ?p? ed?
  e????a s?µpe?a????µe ?t? ? µ , µe t? ??µ? t??
  d?a??af?? ??a ???e µ, ? p?? a???e? st? s????? ?.
  ?e? ?p?????? d?af??et???? µeta?? t??? f?s???? p??
  ?a ????? t?? ?d?? ep?µe??. Ta p??pe? ?a e??a? ?
  ?d??? f?s???? a???µ??.
• P4 ?? µ?d?? de? e??a? ep?µe??? e??? f?s????
  a???µ??

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• P5 ?? ??a ??a ?p?s????? ? t?? ? ?s??e? ?t?
• a ) t? (µ?d??) 0 a???e? st? ?
• ß ) a? ??a ???e ? p?? a???e? st? ? s??ep??eta?
  ?t? ? 1 a???e? st? ?.
• St? p?µpt? a???µa st????eta? ? µ???d??
  ap?de??e?? t?? t??e?a? epa????? ? a??? t??
  µa??µat???? epa?????. ?? µ???d? a?t? t?
  ???s?µ?p????µe ??a ?a ap?de????µe t?? p??t?se??
  p?? a?af????ta? se f?s????? a???µ???.
• ?e t?? µ???d? t?? t??e?a? epa????? e??a??µaste
  ?? e??? a? ????µe µ?a p??tas? p(?) ??a t???
  f?s????? a???µ??? ?a? ?????µe ?a ap?de????µe ?t?
  e??a? a?????, µp????µe ?a e??ast??µe µe t?? e???
  t??p?
• ?p?de??????µe ?t? ? p??tas? e??a? a????? ??a ?
  0 ( ? p??tas? p(0) e??a? a????? ).
• ?p???t??µe ?t? ? p??tas? e??a? a????? ??a ? ?,
  ( p(?) e??a? a????? ) ?a? ap?de??????µe ?t?
  e??a? a????? ??a ? ? 1. ?e a?t?? t?? t??p?
  ap?de????eta? ?t? ? p??tas? µa? ?s??e? ??a ?????
  t??? f?s????? a???µ???.

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T?O???? ????????S

• ?? ?p???s??µe ?t? t? ( ??, 0 , S ) e??a? s?st?µa
  Peano t? ? e??a? s????? a e ? ?a? ? h ? ?
  e??a? s????t?s? ?p???e? t?te µ?a ?a? µ??? µ?a
  s????t?s? f IN E p?? ??a??p??e? t??
  ta?t?t?te?
• f(0) a
• f(Sn) h(f(n))
• ?? ?e???µa t?? ??ad??µ?? d??a?????e? t??
  s?????sµ??? t??p? µe t?? ?p??? ??????µe
  s??a?t?se?? st??? f?s????? a???µ??? a?ad??µ??? (
  ? epa??????) d??ad? ??a ?a ???s??µe t?? h E
  E p?? p??sd?????e? t?? t?µ? t?? f(Sn) t?? f se
  ???e ep?µe?? a???µ? Sn ap? t?? t?µ? f(n) t?? f
  st? p??????µe?? t?? n f(Sn) h(f(n)).?
  d?a???d?s? e??a? te?e??? d?af??et??? d??t?
  e?a?t?ta? ap? t?? ??e??? µ?a? s??????? .?
  d?a???d?s? ß?a??e? µ?sa ap? t?? a?ad??µ? .?
  a?ad??µ? e??a? p??? p?? s?ßa??.

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S?S????S? ??S ????????S ??? ??S ???GOG?S

• ?? Te???µa ??ad??µ?? ?a? t? a???µa t?? epa?????
  e??a? ta ßas???te?a e??a?e?a t?? ???????.
• G?a ?a µp???s? ?a ap?de??? t? Te???µa ??ad??µ??
  p??pe? ?a ??? t? a???µa t?? epa????? ?a?
  a?t?st??fa ??a ?a ap?de??? t? a???µa t?? epa?????
  p??pe? ?a ??? t? Te???µa t?? ??ad??µ??.
• G?a ?a ap?de??? t?? ap??? ?d??t?te? t?? f?s????
  a???µ?? ??e????µa? ?p?sd?p?te t? Te???µa
  ??ad??µ?? ? t? a???µa t?? epa?????.

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S???????? ??G???

• ? a???µ?t??? Peano pe????e? p???? ???s?a
  ?p?s?st?µata.
• ? ???ssa L st?? ?p??a d???e???µe e??a? ?
  p??t?ß??µ?a ???ssa t?? a???µ?t???? d??ad? ?
  ???ssa µe µ? ?????? s?µß??a ?, , ,lt , 0
• G?a s??t?µ?a PA e??a? ? p??? ???st? ßas???
  ?e???a p?? pe?????fe? t?? s?µpe??f??? t??
  sta?e??? 0, t?? s??a?t?se?? t?? ep?µe???, t??
  p??s?es?? ?a? t?? p???ap?as?asµ?? ?a? t?? s??s??
  lt .

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????S ???? ? ??S???S ??G?S ??? ?S????T??? ? PEANO
?? ??? S???????? ??G???.

• T???se ?a ap???ste?se? s??se?? µe p??????
  s?µß???sµ???. ?e a?t? t?? t??p? ?at?fe?e ??
  p???s???ete? p??t?se?? a???e?a? ?a d?aß????ta?
  p??? p?? ??????a ?a? ap??.
• S??????
• ?a??????? s?µß???sµ??. G?a t??
  ?a?????? s?µß???sµ? ???s?µ?p???se t? ?n
• ?pa???a??? s?µß???sµ??. G?a t??
  ?pa???a?? s?µß???sµ? ???s?µ?p???se t? Sn
• (?p?? ? e??a? t? p????? t?? ?a??????? ?a?
  ?pa???a??? s?µß???sµ?? p?? eµfa?????ta?.)

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X1 , X2.. p?s?de??te?

• ??S???S ????????S
• S1 ?? ?µ?a?ad??µ???? s??se??
• ?? S? ?? a???se?? t?? s??se?? st? S?
• S?1 ?? ?? s??se?? p?? ??a??p????? µ?a
  ?s?d??aµ?a P( x ) ( y)Q( x , y ) ?p?? Q(
  x , y ) e??a? ??
• ?? S? ? ?? ?? s??se?? p?? e??a? S? ?a? ?? .

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???????G???? ????O S??? S???????? ??G???

• S1 ( y ) Q( x , y )
• ?1 ( y )Q( x , y)
• S2 ( y1 )( y2 )Q( x , y1 , y2 )
• ?2 ( y1 )( y2 )Q( x , y1 , y2 )
• S3 ( y1)( y2)( y3)Q( x , y1 , y2 , y3
  )