集合・位相入門/松坂和夫 3.2 節：整列集合とその比較定理

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#7 3 ! Zorn "#$ M1

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§ 1. 6!5 Ø6!3 Ø6!5:26!5 Ø& 89 :'& 89 : 4:4 Ø6! Ø(":+ § 2. $5)0+ Ø$5 Ø1/4 ,* Ø$56! Ø$5)0+ § 3. Zorn -7:$#+ Ø $53 7 ØZorn -78.%9 1

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3,( ℕ !8 ≤ :#9&=$5' - 1?ℕ %0 69*!' Ø# ℕ 0 69 =(."2+/ Ø# $ 3,(' - 1 &1 ü $ = 1 =1 = min ℕ = min # &1 ü $ + 1 ∈ # = # $ 3,(' # *!'># $ 3,(' min # = $ + 1 3 )9 (ℕ, ≤) ';74 <$5

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$). ! * ) - ' " () %# -! $) &+!, . * ) " -#, % ∈ ", # < % # < ( < % " ( - # % & -% # &! 4 $) $)

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$). ! * ) - ' " () %# -! $) &+!, . * ) " -#, % ∈ ", # < % # < ( < % " ( - # % & -% # &! 5 $) $) 3 4 &

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%*- ! + * , ( # )* &$ ,! %* %* # )*%* %*"& " ∈ ! ,∃% ∈ !, " < % " '! 6 %* %*

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%*- ! + * , ( # )* &$ ,! %* %* # )*%* %*"& " ∈ ! ,∃% ∈ !, " < % " '! 7 %* %* !( = {%|% ∈ !, " < %} min !′ " '!

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2(1)! % +$&, + 0 -*' 1 %-*'+& .(1 +& )# " 8 (1 -*'.(1/

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" ! $" ∈ !, ! " = {'|' ∈ !, ' < "} " = min ! ! ! " = . " "∗ #"∗ = max ! " " #" = sup ! " 10 !

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$0 !2.0 !" !" ! # ⊂ !" ⟹ # ∈ !"",1 # 2!" = ! Ø!" ≠ ! 2! − !" ≠ * Ø$0.0%# # = min ! − !" Ø# ! − !" % ! # ∩ ! − !" = *3! ! # ⊂ !" Ø1", # ∈ !" 2# = min ! − !" ∈ ! − !" *) 11 (-/ +& ",' 2$0.0

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%9 C2,7D- ,7DE1$A( ! !? *B " ∀$ ∈ ! & < $ ! '! & " /: , C" $ /: 4 C" ! " ! /: ,7D<50? 0+8.;3=&C >@.;3) 12 #6>@.;3 >@.;3

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-356! "2 " 02 # ∈ !, & ∈ ", & < # ⇒ & ∈ !'4 $ ! = " ∃+ ∈ ", ! = " + Ø! ≠ " 4" − ! ≠ . ∃+ ∈ " − !, + = min " − ! Ø" + ⊂ ! .# ∀# ∈ " + , # < + # ∉ " − !4 # ∈ ! Ø! ⊂ " + .# ∃# ∈ !, + ≤ # 4% + ∈ !4+ ∈ " − ! +*4 ∀# ∈ !, # < + # ∈ " + 13 (/1,& ()!-3

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) !, ", $ ∈ ! " ≤ $ ⇔ ! " ⊂ ! $ ): ! → {! " }.∈/ , ) " = ! " ) * ", $ ∈ !, " < $ ! $ " = ! " 14 "%($! "& #'

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38;>! ,6! " 24 7) ( " # ' # ≤ !(#) +1 38<>,6! * . - 7)"$ 38=>,6!", "( 7)"$9 " * . " ) ' 7)"$ "( )( %#9 )( ) ' *0& &/<>,6!", "( 7)"$9 " "( 7)"$*0&: 16 ,6! 7)"$ ,6! 538 &/

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+012! %. " *-/#" " # ! # ≤ !(#) $) Ø∃# ∈ ", !(#) < # Ø+ = # # ∈ ", ! # < # ≠ . min + = #2 Ø#2 ∈ + !(#2 ) < #2 Ø! /#" !(!(#2 )) < !(#2 ) ! #2 ∈ + Ø#2 = min + (' 17 %./# +01,&

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-356&1%)( 2$ Ø! ! " 2$ # !" Ø# ! ,02$#. Ø#(") ∈ ! " #(") < " Ø-34+* 18 &12$ -35/'

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7;>?/ 9!!, !# :+"$< ! .1 ! $ ) :+"$ !# $# '#< $# $ ).2( Ø% ! !# :+"$ Ø0 % ! $ = !# %($) Ø% (6% ! $ 4*<39! !# %($) & ! $ !# %($) :+"$, Ø$# = %($) 5 ! $ ≃ !# $# Ø.-7;=0 19 / 9!:+"$ 7;>80

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#,.!'!, !# (* ! !# ( $ + Ø! ≃ !# *%, & ! !# ( Ø∀( ∈ !, ! ( ≃ !# % ( , ! ( ≃ !# & ( Ø%)- % ( = &(() % = & 20 !' ( #, &"

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! !, !# 2 3 1 1 1. ! ≃ !# 2. ∃&′ ∈ !#, ! ≃ !# 3. ∃& ∈ !, ! & ≃ !# 22

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1 2(1 3 " #') 2, 3 (#') !, !# 2, 3 " $ Ø∃!′ ∈ '#, ' ≃ '# !# ∃! ∈ ', ' ! ≃ '# Ø' ≃ '# !# #'* ∃)#∈ '# !# , ' ! ≃ '# !# )# = '# )# Ø' ! ≃ '# )# ' ! ≃ '# " #')! 23 &% % 3 " $

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! = # # ∈ %, ∃#( ∈ %(, % # ≃ %( #( !( = {#(|#( ∈ %(, ∃# ∈ %, % # ≃ %( #( } (-/$ .# ∈ ! ! #( " % #( ∈ !( ! # " ! ! !( - . - 24 #, &+' &+' 3 * )$

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! ∈ #, % ∈ &, % < ! ⇒ % ∈ # Ø! ∈ #, ) ! = !′ # & ! ≃ &- !- Ø% < ! & % & ! ! Ø &' ∃%/ - ∈ &-, & % ≃ &- !- %/ - = &- %/ - Ø# % ∈ # !- = ) ! , %- = ) % %′ < !′ Ø) ) % = %- = %/ - Ø& % ≃ &- !- %/ - = &- %/ - %/ - = %- < !- 25 %$ $ 3 # "

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! ∈ #, % ∈ &, % < ! ⇒ % ∈ # # !) = + ! , %) = + % %′ < !′ %) a. & ./ # *12$' b. !, % ∈ #; % < ! +(%) < +(!) b + 0 a # = & ∃1 ∈ &, # = & 1 26 !/&-( &-( 3 ,+"

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" ! = # ∃% ∈ #, ! = # % . $+ !( = #( ∃%( ∈ #(, !( = #( %( 7) ! !( 4($% ,2'- 3 0* ! = #, !( = #( # ≃ #( 51 ! = #, !( = #( %( # ≃ #( %( 52 ! = # % , !( = #( # % ≃ #( 53 27 )!3#,2'- ,2'- 3 1/*

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! = # $ , !& = #& $& Ø! = # $ , !& = #& $& Ø# $ ≃ #& $& ! ! $ ∈ ! Ø! = # $ 28 % $ $ 3 # "

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Zorn Statement 3. Zorn 29

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#" ! % # " ! ! % 30 Zorn $ #"

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31 Zorn Zorn