[[기하학,geometry]]의 5공리
1
2
3
4 까지는 [[유클리드_기하학,Euclidean_geometry]]과 공통이지만
5 [[평행선공준,parallel_postulate]]

5를 부정해도 consistent ? 이걸 부정하면 : [[비유클리드_기하학,non-Euclidean_geometry]]

= Sub =
[[순서공리,order_axiom]] - 이미있는페이지 [[순서공리]] - axiom of order ? - rel. [[순서,order]]
[[선택공리,choice_axiom]] axiom of choice, AoC
 MKLINK
 [[Zorn_lemma]] maybe [[초른_보조정리,Zorn_lemma]] =,Zorn_lemma {
  https://artofproblemsolving.com/wiki/index.php/Zorn%27s_Lemma
  [[WpEn:Zorn's_lemma]] = https://en.wikipedia.org/wiki/Zorn%27s_lemma
  [[WpKo:초른_보조정리]] = https://ko.wikipedia.org/wiki/초른_보조정리
   "[[부분순서집합,partially_ordered_set,poset]]이 [[극대원소,maximal_element]]를 가질 필요충분조건을 제시하는 lemma이며, [[선택공리,choice_axiom]]와 동치이다."
  [[WpJa:ツォルンの補題]] = https://ja.wikipedia.org/wiki/ツォルンの補題
  }
 [[Banach-Tarski_paradox]] [[Banach-Tarski_paradox]]? =,Banach-Tarski_paradox . Banach-Tarski_paradox
  {
  바나흐-타르스키 역설 ? Ggl:"바나흐-타르스키 역설" Ndict:"바나흐-타르스키 역설"
  바나흐-타르스키 역설 ?
  https://artofproblemsolving.com/wiki/index.php/Banach-Tarski_Paradox
  [[역설,paradox]]?
  }

 [[WtEn:axiom_of_choice]] = https://en.wiktionary.org/wiki/axiom_of_choice

 https://artofproblemsolving.com/wiki/index.php/Axiom_of_choice
 https://encyclopediaofmath.org/wiki/Axiom_of_choice
 [[Namu:선택공리]]
 WpKo:선택_공리

분리공리? separation_axiom
 https://artofproblemsolving.com/wiki/index.php/Separation_axioms
 [[위상,topology]] [[separation]] [[공리,axiom]]

[[귀납공리,induction_axiom]] - [[귀납,induction]] - curr at [[VG:귀납,induction]]
{
[[귀납,induction]] [[공리,axiom]]

[[귀납,induction]] - [[VG:귀납,induction]]
[[수학적귀납법,mathematical_induction]] - [[VG:수학적귀납법,mathematical_induction]]

https://mathworld.wolfram.com/InductionAxiom.html
https://encyclopediaofmath.org/wiki/Induction_axiom

... Google:귀납공리 Naver:귀납공리
}

[[completeness_axiom]] =,completeness_axiom =,completeness_axiom . completeness_axiom
{
완비성공리 ?

[[실수,real_number]]
[[완비성,completeness]]

WtEn:completeness_axiom
... Google:completeness.axiom Naver:completeness+axiom Naver:완비성공리 Google:완비성공리 
}

[[확률공리]] =,probability_axiom =,probability_axiom . probability_axiom
{

by 안드레이 콜모고로프 ?

probability axioms

[[확률,probability]]

[[https://terms.naver.com/entry.naver?docId=4125166&cid=60207&categoryId=60207 수학백과: 공리적 확률]]에서 설명함.


https://en.wikipedia.org/wiki/Probability_axioms

Ggl:확률공리
Ggl:"probability axioms"
확률공리
"probability axioms"
}

[[Playfair_axiom]] =,Playfair_axiom =,Playfair_axiom . Playfair_axiom
{
[[geometry]]에서 [[parallel_postulate]]관련 

[[WpEn:Playfair's_axiom]]
Ggl:"playfair axiom"
}

[[확률공리,probability_axiom]] =확률공리,probability_axiom =,probability_axiom 확률공리 probability_axiom
{
확률공리

[[확률,probability]]
probability_axiom
axiom_of_probability ?
axiom of probability ?
https://www.google.com/search?q=Axiom+of+probability
하면 
[[WpEn:Probability_axioms]] = https://en.wikipedia.org/wiki/Probability_axioms
} // 확률공리

[[countability_axiom]] =,countability_axiom =,countability_axiom . countability_axiom
{
가산성공리 ?
axiom of countability
axiom_of_countability
WtEn:axiom_of_countability = https://en.wiktionary.org/wiki/axiom_of_countability x [[Date(2023-09-06T23:02:39)]]

mkl [[countability]](curr . 가산집합)
 [[가산성,countability]] ? https://en.wiktionary.org/wiki/countability
  rel [[가산집합,countable_set]]

[[WpEn:Axiom_of_countability]] = https://en.wikipedia.org/wiki/Axiom_of_countability

Ggl:"axiom of countability"
Naver:"axiom of countability"
"axiom of countability"
}

ADDHERE........

= 공리관련 주제들 =
[[공리계,axiomatic_system]] ... 보다는 [[공리체계,axiomatic_system]] =,axiomatic_system 공리체계 axiomatic_system (KMS)
{
공리적 체계 ?
KmsK:공리계
KmsE:axiomatic

Sub:
categorical_axiomatic_system
 categorical axiomatic system
 https://mathworld.wolfram.com/CategoricalAxiomaticSystem.html
[[페아노_공리계]] Peano_axiomatic_system ? Peano_axioms ?
 MKLINK: [[자연수,natural_number]] 자연수의 formal_definition 에 쓰임. see [[WpEn:Natural_number#Formal_definitions]]
 [[WpKo:페아노_공리계]] = https://ko.wikipedia.org/wiki/페아노_공리계
 ... Google:Peano+axioms
 rel
 [[페아노_공리,Peano_axiom]] =페아노_공리,Peano_axiom =,Peano_axiom 페아노_공리 Peano_axiom 
  [[WpEn:Natural_number#Peano_axioms]]
  [[WpEn:Peano_axioms]]
  = https://en.wikipedia.org/wiki/Peano_axioms
  [[WpJa:ペアノの公理]]
  https://ja.wikipedia.org/wiki/ペアノの公理
  ...
  "페아노 공리"
  Ndict:"페아노 공리"
  Naver:"페아노 공리"
  Ggl:"페아노 공리"
 rel
 [[페아노_산술,Peano_arithmetic]]

Z공리계 ?

ZF공리계 ZF_axiomatic_system ?
 {
 9개의 공리로 구성.
 https://artofproblemsolving.com/wiki/index.php/Zermelo-Fraenkel_Axioms
 Up: [[axiomatic_system]]
 } ... Google:zf+axioms

[[ZFC공리계]] ZFC_axiomatic_system ?
 {
 9개의 공리 + [[선택공리,choice_axiom]] 해서 10개의 공리로 구성.
 [[Namu:ZFC%20공리계]]
 https://ncatlab.org/nlab/show/ZFC
 Up: [[axiomatic_system]]
 } ... Google:zfc+axioms

NBG공리계
NBG공리체계
 https://www.britannica.com/science/set-theory/The-Neumann-Bernays-Godel-axioms
 ...
 Ggl:"폰 노이만-베르나이스-괴델(von Neumann-Bernays-Gödel)"
 Google:nbg+axioms
  ZFC의 보존적 확장? chk // [보존적확장,conservative_extension] =,conservative_extension . conservative_extension
   {
   '''conservative extension'''

   opp. : [[non-conservative_extension]] { '''non-conservative extension''' } // non-conservative extension Ggl:"non-conservative extension"

   WtEn:conservative_extension
   https://ko.wikipedia.org/wiki/보존적_확장
   https://en.wikipedia.org/wiki/Conservative_extension
   https://ja.wikipedia.org/wiki/保存拡大
    '보존확대'.
    Namu:拡 Naver:拡 은 擴 의 약자.... Namu:擴 -> 넓힐 확. 확대할때 그거.. 
   
   } // conservative extension Ggl:"conservative extension"

MK공리계
MK공리체계
 Ggl:"모스-켈리(Morse-Kelley) 공리계"


... Google:zfc+nbg+mk

...


Twin

[[MW:AxiomaticSystem]]
= https://mathworld.wolfram.com/AxiomaticSystem.html

https://www.gabormelli.com/RKB/Axiomatic_System

Up: [[공리,axiom]] [[system]]

... Google:공리계 Naver:공리계 Google:axiomatic_system
} // axiomatic system Ggl:"axiomatic system"

[[공리화,axiomatization]]
{
=공리화,axiomatization =,axiomatization . axiomatization 공리화
[[axiomatization]]
WtEn:axiomatization = https://en.wiktionary.org/wiki/axiomatization
https://en.wikipedia.org/wiki/Axiomatic_system#Axiomatization
... Google:Axiomatization Naver:Axiomatization
공리화 pagename due to KmsE:axiomatization
}

[[axiom_schema]] =,axiom_schema =,axiom_schema . axiom_schema (w)
aka [[axiom_scheme]] =,axiom_scheme =,axiom_scheme . axiom_scheme (via wt)
{
Twins:
https://foldoc.org/axiom+schema
https://mathworld.wolfram.com/AxiomSchema.html
https://en.wikipedia.org/wiki/Axiom_schema
https://en.wikipedia.org/wiki/Axiom_schema

Semi-twins:
https://planetmath.org/logicalaxiom 에서 "Any such a set of axioms is called an" 검색
 ... 하이라이트 링크 : https://planetmath.org/logicalaxiom#:~:text=Any%20such%20a%20set%20of%20axioms%20is%20called

… https://www.google.com/search?q=axiom+schema
}

[[axiomatic_set_theory]]

....

[[공준,postulate]]이라는 용어도 있고 옛날엔 axiom과 구분했지만 요새는 별로 구분하지 않는다는 걸 본 기억이 있는데... 암튼 compare.

같은 단어 고유명사 [[액시엄,Axiom]]
{
Axiom Maths Program - Martin Baker
https://www.euclideanspace.com/prog/scratchpad/axiom/index.htm

}

= Twins =
https://artofproblemsolving.com/wiki/index.php/Axiom
https://mathworld.wolfram.com/Axiom.html


https://en.wikipedia.org/wiki/Category:Mathematical_axioms
https://www.proofwiki.org/wiki/Category:Axioms