JAIST 北陸先端科学技術大学院大学t-nemoto/summerschool.pdf2M8% TD. Troelstra and van...

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1. =.*tXHO)

2. Z@H BHKra

3. s=.6}

4. BtH"3Xt

5. fVMNj}

6. CantorN&Lt,j}HWeak Koenig’s Lemma

7. Heine-BorelNo$j}H Brouwer’s Fan Theorem

=.*tXHO)

• -B*tX (Kronecker, Hilbert)• DR*tX (Poincaré, Weyl, etc..)• >QgAtX (Brouwer, etc..)• F"*tX (Markov, etc..)• Bishop.=.*tX (Bishop, etc..)

=.*tXHO) 1. -B*tX

-B*tX (Kronecker, Hilbert, etc..)

V+3tO@NnC?bN@,,>OMVNnC?bNG"kW

(Kronecker, 1886)

• qN*K (-B*K)=;k=$@1,tX*P]G"k.

=&7?=$eNi;bBz*JbNG"k.

• 8gd=NeNi;JINj]*J50O,-B*tXNP]GOJ$.

• 3N)lGtXN57b-r(=&H7?N,HilbertNWm0i` (1900/")

• Kronecker O8_Z@OqN*Kror~?9P]r-B9FCWG?(J1lPZ@7?3HK

JiJ$H7,Sf' “A ∨ ¬A”K]j*J+rr(7F$?.

• HilbertOSf'r'aF$?.

Kronecker

Hilbert

=.*tXHO) 2. DR*tX

DR*tX (Poincaré, Weyl, etc..)

dj 2 AH B ⊆ AKP7F B r^`G.N ANt,2,8_9k3Hr(7J5$.

sz: B = {C : C O ANt,2G, B ⊆ C}KP7D =

⋂

B H9k.(]$sH) D rjA9kNKHC? B KP7F

D ∈ B G"k.

• sDR*jAHO, eNh&JV?+rjA9kNK=l+H,09k/i9r2H9kjAWN

3H.

• DR*tXOp\*KOsDR*jAr'aJ$,,+3te@1O'ak3H,?$.

• Sf'OlLK'aF$k.

Poincaré

Weyl

=.*tXHO) 3. >QgAtX

>QgAtX

• [TD]Khkp\}0N^Ha• tXHO:@*:*r7&bNG"j, -fsN`nNfK"kbNGOJ$.

tX*@lO>THNdjhjN?aKQ$k

jJG"j\AGOJ$.

• tX*?jN?6Of9NN1KM89k.?j A,?G"kHO ANZ@,?(ilk3HG,6G"kHO A+i7br31k3HNZ@,?(ilk3HG"k.

• tXHO free creationG"k. c(P,f9,W$D-G 1`:DjA9k5Bsb,tX*

P]HJj&k.

Brouwer

• Sf'OlLK'aJ$.• BrouwerH HilbertOP)7F$?.• “BteNXtO9YF"3G"k”JIr3/C'*Jx}r}D.• >QgA@}O,>QgAtXN@}Nt,r Heytingi,0}7FNO=7?bN.

=.*tXHO) 4. F"*tX

F"*tX

• tX*P]HO"k4j:`G"k. ?@7,~V&uV*)sOJ$bNH9k.

• 8_Z@OqN*Kror~?9P]r=.9k"k4j:`r?(J1lPJiJ$.

• A ∨ B NZ@O, A+ B ,:IAi+l}NZ@r?(J1lPJiJ$.

• “d_7J$H9kH7b9k"k4j:`Od_9k” (MarkovN6})r'ak.

Markov

=.*tXHO 5. Bishop.=.*tX

Bishop.=.*tX

• 8_Z@OqN*Kror~?9P]NBz*K+D1kjJ,?(ilJ1lPJiJ$.

• A ∨ B NZ@O, A+ B NIAi,.j)D+Nps,Bz*K?(ilJ1lPJiJ$.

• Sf'O,lLK'aJ$.• 7+7,tX*P]OoK"k4j:`G?(ilk,WOJ$.

=.*tXHO)

• -B*tX (Kronecker, Hilbert)• DR*tX (Poincaré, Weyl, etc..)• >QgAtX (Brouwer, etc..)• F"*tX (Markov, etc..)• Bishop.=.*tX (Bishop, etc..)

Z@H BHKra

INAN?jKIsJZ@,?(ilkY-+)

• A ∧B NZ@ (p, q)O ANZ@ pH B NZ@ q NZ";

NZ@ O ^?O H JiP NZ@

JiP NZ@ H$&ANZ"

NZ@O N$+JkZ@ b NZ@ XHQ99k,'

NZ@O N$+JkZ@ +ib7br3/,'

NZ@O jAhbN KP7F NZ@rV9,'

NZ@ O H NZ@ NZ"

3lO raHFPlk

N>QgAtXN@}Nt,r i,0}7?bNG

.>QgAtXdF"*tXN@}H7FQ$ilF$k

J< 1K=.*tXH@C?lg .N=.*tXrX93HK9k

CK djdj}O .N=.*tXK*1kbNG"k3HKmU;h

Z@H BHKra


• A ∧B NZ@ (p, q)O ANZ@ pH B NZ@ q NZ";• A ∨B NZ@ (i, p)O i = 0^?O 1H i = 0JiP ANZ@ p, i = 1JiP B NZ@ pH$&ANZ".

NZ@O N$+JkZ@ b NZ@ XHQ99k,'



NZ@ O H NZ@ NZ"

3lO raHFPlk





Z@H BHKra



• A → B NZ@O, AN$+JkZ@ pb B NZ@ q XHQ99k,';



NZ@ O H NZ@ NZ"

3lO raHFPlk





Z@H BHKra



• A → B NZ@O, AN$+JkZ@ pb B NZ@ q XHQ99k,';• ¬ANZ@O AN$+JkZ@ p+ib7br3/,';


NZ@ O H NZ@ NZ"

3lO raHFPlk





Z@H BHKra



• A → B NZ@O, AN$+JkZ@ pb B NZ@ q XHQ99k,';• ¬ANZ@O AN$+JkZ@ p+ib7br3/,';• ∀xA(x)NZ@O, (jAhbN) xKP7F A(x)NZ@rV9,';

NZ@ O H NZ@ NZ"

3lO raHFPlk





Z@H BHKra



• A → B NZ@O, AN$+JkZ@ pb B NZ@ q XHQ99k,';• ¬ANZ@O AN$+JkZ@ p+ib7br3/,';• ∀xA(x)NZ@O, (jAhbN) xKP7F A(x)NZ@rV9,';• ∃xA(x)NZ@ (d, p)O, dH A(d)NZ@ pNZ";

3lO raHFPlk





Z@H BHKra



• A → B NZ@O, AN$+JkZ@ pb B NZ@ q XHQ99k,';• ¬ANZ@O AN$+JkZ@ p+ib7br3/,';• ∀xA(x)NZ@O, (jAhbN) xKP7F A(x)NZ@rV9,';• ∃xA(x)NZ@ (d, p)O, dH A(d)NZ@ pNZ";3lO Brouwer-Heyting-KolmogorovraHFPlk.

BrouwerN>QgAtXN@}Nt,r Heytingi,0}7?bNG,

Bishop.>QgAtXdF"*tXN@}H7FQ$ilF$k.



Z@H BHKra



• A → B NZ@O, AN$+JkZ@ pb B NZ@ q XHQ99k,';• ¬ANZ@O AN$+JkZ@ p+ib7br3/,';• ∀xA(x)NZ@O, (jAhbN) xKP7F A(x)NZ@rV9,';• ∃xA(x)NZ@ (d, p)O, dH A(d)NZ@ pNZ";3lO Brouwer-Heyting-KolmogorovraHFPlk.

BrouwerN>QgAtXN@}Nt,r Heytingi,0}7?bNG,

Bishop.>QgAtXdF"*tXN@}H7FQ$ilF$k.

J

Z@H?jN?6

ϕr$rhdjH9k.

• !Nh&K “jA”5l?8g ArM(k. (0 ∈ A) ∨ (1 ∈ A)O?+)

{x : (x = 1+D ϕ )∨(x = 0+D ¬ϕ )}

• BHKraGO (0 ∈ A) ∨ (1 ∈ A)NZ@O, ϕNZ@^?O?ZH&K?(ilkO:G"k.

7+7 #NH3mIAib?(ilF$J$NG

O?GOJ$

bAms 6GbJ$

Z@H?jN?6

ϕr$rhdjH9k.

• !Nh&K “jA”5l?8g ArM(k. (0 ∈ A) ∨ (1 ∈ A)O?+)

{x : (x = 1+D ϕ )∨(x = 0+D ¬ϕ )}


• 7+7,#NH3mIAib?(ilF$J$NG, (0 ∈ A)∨ (1 ∈ A)O?GOJ$.

bAms 6GbJ$

Z@H?jN?6

ϕr$rhdjH9k.

• !Nh&K “jA”5l?8g ArM(k. (0 ∈ A) ∨ (1 ∈ A)O?+)

{x : (x = 1+D ϕ )∨(x = 0+D ¬ϕ )}


• 7+7,#NH3mIAib?(ilF$J$NG, (0 ∈ A)∨ (1 ∈ A)O?GOJ$.

• bAms,6GbJ$.

=.*tXHO 5. Bishop.=.*tX

Bishop.=.*tX

• 8_Z@OqN*Kror~?9P]NBz*K+D1kjJ,?(ilJ1lPJiJ$.

• A ∨ B NZ@O, A+ B NIAi,.j)D+Nps,Bz*K?(ilJ1lPJiJ$.

• Sf'O,lLK'aJ$.• 7+7,tX*P]OoK"k4j:`G?(ilk,WOJ$.

-!

• X∗ O X N5+iJk-BsN4N• |s|Gs sN95• snG sN n+ 1V\NWG (D^j s0O sNGiNWG)• s ∗ tO sH trDJ$@s,9JoA |u| = |s|+ |t|,

∀n < |s|(un = sn), ∀n < |u|(n ≥ |s| → sn = t(n− |s|))HJks u• s ⊂ tG sO tNZR,9JoA |s| < |t|, ∀n < |s|(sn = tn)r=9.

• +3tN5Bsr.j7c8zN.8z α, β, γ G=9.• αk O αN k − 1V\NWG• ᾱk O |s| = k, sn = αnJks

Bz*JZ@NC

Bz*JZ@ · · · -B*J}!G-B~VbKT(kZ@

HGb@&+

H9k !NAN?j

O?+)

,?G"kJiP NZ@^?O NZ@,

?(ilk

7+7 + NIAi,.j)D+rN+ak

KOlLKO N9YFN`rA'C/9kJ0K}!OJ/ IA

i,.j)D+Npsb IAi+NZ@b?(ilkHOBiJ$

Bz*JZ@,J$

1MK !NAN?j O)

NGiN`+igVKA'C/r7F$CF qt`G ,PlP

vt`,9YF vt`G ,PlPqt`G9YF HJk

7+7 4FN`, Nlg 3&7?"k4j:`O_^iJ$

7?,CF 3N?jKbBz*JZ@OJ$

Bz*JZ@NC

Bz*JZ@ · · · -B*J}!G-B~VbKT(kZ@ (HGb@&+...)

H9k !NAN?j

O?+)

,?G"kJiP NZ@^?O NZ@,

?(ilk

7+7 + NIAi,.j)D+rN+ak

KOlLKO N9YFN`rA'C/9kJ0K}!OJ/ IA

i,.j)D+Npsb IAi+NZ@b?(ilkHOBiJ$

Bz*JZ@,J$

1MK !NAN?j O)



7+7 4FN`, Nlg 3&7?"k4j:`O_^iJ$


Bz*JZ@NC

Bz*JZ@ · · · -B*J}!G-B~VbKT(kZ@ (HGb@&+...)• α : N → {0, 1}H9k. !NAN?j LPO (limited principle of omniscience)O?+)

∀n(αn = 0) ∨ ∃n(αn = 1)• A,?G"kJiP, ∀n(αn = 0)NZ@^?O ∃n(αn = 1)NZ@,?(ilk.

• 7+7, ∀n(αn = 0)+ ∃n(αn = 1)NIAi,.j)D+rN+akKOlLKO αN9YFN`rA'C/9kJ0K}!OJ/,IAi,.j)D+Npsb,IAi+NZ@b?(ilkHOBiJ$.

(Bz*JZ@,J$)

1MK !NAN?j O)



7+7 4FN`, Nlg 3&7?"k4j:`O_^iJ$


Bz*JZ@NC

Bz*JZ@ · · · -B*J}!G-B~VbKT(kZ@ (HGb@&+...)• α : N → {0, 1}H9k. !NAN?j LPO (limited principle of omniscience)O?+)

∀n(αn = 0) ∨ ∃n(αn = 1)• A,?G"kJiP, ∀n(αn = 0)NZ@^?O ∃n(αn = 1)NZ@,?(ilk.

• 7+7, ∀n(αn = 0)+ ∃n(αn = 1)NIAi,.j)D+rN+akKOlLKO αN9YFN`rA'C/9kJ0K}!OJ/,IAi,.j)D+Npsb,IAi+NZ@b?(ilkHOBiJ$.

(Bz*JZ@,J$)

• 1MK,!NAN?j LLPO (lesser limited principle of omniscience)O)

∀m∀n(αm = αn = 1 → m = n) → ∀m(α(2m) = 0)∨∀n(α(2n+1) = 0)

• αNGiN`+igVKA'C/r7F$CF,qt`G 1,PlPvt`,9YF 0,vt`G 0,PlPqt`G9YF 0HJk.

• 7+7,4FN`, 0Nlg,3&7?"k4j:`O_^iJ$.7?,CF,3N?jKbBz*JZ@OJ$.

MarkovN6}JI

1MK α : N → {0, 1}H9k. !NAN?jO?+)• ¬∃n(αn = 1) → ∀n(αn = 0)

• ¬∃n(αn = 1)NH-,$UN nKP7F, αn = [email protected]?,CF$UN nKP7F αn = 0. hCF B O?.

eH1MK? },dj

^?O

N6}

Gb HJk N?(}Oo+iJ$NG

?GOJ$ .

F"*tXGO3N6}O'ailF ?H5lF $?

MarkovN6}JI



• ∀n(αn = 0) → ¬∃n(αn = 1)• eH1MK? (},dj).

^?O

N6}

Gb HJk N?(}Oo+iJ$NG

?GOJ$ .

F"*tXGO3N6}O'ailF ?H5lF $?

MarkovN6}JI



• ∀n(αn = 0) → ¬∃n(αn = 1)• eH1MK? (},dj).

• ¬∀n(αn = 0) → ∃n(αn = 1)^?O¬¬∃n(αn = 0) → ∃n(αn = 1) (MarkovN6})• ¬∀n(αn = 0)Gb, αn = 1HJk nN?(}Oo+iJ$NG?GOJ$ (Bishop.)

• F"*tXGO3N6}O'ailF (?H5lF)$?.

+3t+i-}t,BtX

• =.*tX?/N)lGO,+3tH=NeNp\*Ji; (+, ·)dgx

BtCF)

BtN “jA”

• fX;db;G,&bN• -}tO[D9k5B.t (c: 13 = 0.3333333333 · · · )• 5}tO[D7J$5B.t (c:

√2 = 1.41421356237 · · · )

• Bt =-}t +5}t• gXG,&bN)

1. -}t4NN8gN DedekindZG

2. -}tN Cauchys

=.*tXGb,eN 1, 2KP~9k}!GjAG-k,,#sO Cauchy

sGjA9k3HK9k.

BtNjA

jA

• -}tNs 〈rn〉n G!Nror~?9bNr5'sH$&:

∀mn(|rm − rn| < m−1 + n−1).

5'sNVN1MX8 r!Gjak

5'sN4NN Khk&rBt Hjak

Je 5'sH=l,=9Btr1lk7FBt JIH=9

$/D+NXt }, 3li, G"k3HrN+ah

BtNjA

jA


∀mn(|rm − rn| < m−1 + n−1).

• 5'sNVN1MX8 =R r!Gjak:

〈rl〉l =R 〈sl〉l ⇔ ∀m(|rm − sm| ≤2

m).

5'sN4NN Khk&rBt Hjak

Je 5'sH=l,=9Btr1lk7FBt JIH=9


BtNjA

jA


∀mn(|rm − rn| < m−1 + n−1).


〈rl〉l =R 〈sl〉l ⇔ ∀m(|rm − sm| ≤2

m).

• 5'sN4NN =R Khk&rBt RHjak.Je,5'sH=l,=9Btr1lk7FBt x = 〈xn〉n JIH=9.


BtNjA

jA


∀mn(|rm − rn| < m−1 + n−1).


〈rl〉l =R 〈sl〉l ⇔ ∀m(|rm − sm| ≤2

m).

• 5'sN4NN =R Khk&rBt RHjak.Je,5'sH=l,=9Btr1lk7FBt x = 〈xn〉n JIH=9.

• $/D+NXt (},: 3li, well-definedG"k3HrN+ah)

(x±R y)n = x2n ±Q y2n |x|n = |xn|max{x, y}n = maxxn, yn min{x, y}n = min{xn, yn}(x ·R y)n = x2kn ·Q y2kn, where k = max{|x|1 + 2, |y|1 + 2}

BteNgxHfSD=- 1. x

BtNfSD=- 2. x ≤R y ∨ y ≤R x

jA ∀n(− 1n≤Q xn)HJkBt x = 〈xn〉n rsi (non-negative)H9k.

Bt ReNX8 ≤R r!Gjak:

x ≤R y ⇐⇒ y − x,si

(mU) x ≤R y ⇔ x

lg,1NF/KC/

3l^GG,!,.j)?J$3H,o+C?:

x

XtN"3-HfVMNj}

jA !r~?9Xt f : R → Rr"3XtH$&:

∀x ∈ R∀p ∈ Q+∃q ∈ Q+∀y ∈ R(|x− y| < q → |f(x)− f(y)| < p)

?j (fVMNj})

"3Xt f : [0, 1] → RH f(0)

=.*JfVMNj}

j} "3Xt f : [0, 1] → RH f(0)

s,ZH König’s Lemma

jA

• $UN t ∈ T H s ⊆ tKP7F s ∈ T HJk T ⊆ {0, 1}∗ rs,Z (binary tree)H$&.

• t ∈ T r T N^H$&.• T ,5B8gNH-,5Bs,ZH$&.• ∀n(ᾱ ∈ T )HJk α : N → {0, 1}r T NQ9H$&.• s ∈ {0, 1}∗ KP7F Ts = {t ∈ {0, 1}∗ : s ∗ t ∈ T} b^?s,ZG"k.

......

......

......

......

......

e: 04s,Z

Weak counter example

?j ϕ(n)H T ⊂ {0, 1}∗ r!GjA9k:ϕ(n) :=“π N=J.t8+K.th nL+i 99e"37? 9,"k”

T =

{

t ∈ {0, 1}∗ : t = 0n∧(∃k < nϕ(k)Ji=NG.N k Ovt)

t = 1n∧(∃k < nϕ(k)Ji=NG.N k Oqt)

}

• T O@i+K5BZ@,,Q9,"kH, ϕ(n)Ji=NG.NbNOvt(qt)NZ@,"k3HKJk.

E5*JtXGO KOQ9,8_9k, eNcOQ9N8_,=.

*KO(;J$3Hr(7F$k

3&7? ?c O "k$OGiKs2?MN>

0rHCF HFPlk

B] !N3H,NilF$k

j} O r3/



T =

{



}


• E5*JtXGO T KOQ9,8_9k,,eNcOQ9N8_,=.*KO(;J$3Hr(7F$k.

3&7? ?c O "k$OGiKs2?MN>

0rHCF HFPlk

B] !N3H,NilF$k

j} O r3/



T =

{



}



• 3&7? “?c”O weak counter example"k$OGiKs2?MN>0rHCF Brouwerian counter exampleHFPlk.

B] !N3H,NilF$k

j} O r3/



T =

{



}



• 3&7? “?c”O weak counter example"k$OGiKs2?MN>0rHCF Brouwerian counter exampleHFPlk.

B],!N3H,NilF$k:

j} Weak König’s LemmaO LLPOr3/.

CantorN&Lt,j}HWeak König’s Lemma

jA

• BtN8g AG,$UNBt ε > 0KP7FBt y ∈(x− ε, x+ ε)∩A,8_9kH-, x ∈ AHJkbNrD8gH$&.

?j (CantorN&Lt,j})

BteND8gNs 〈Ci〉i KP7F,$UN iK"kBt x ∈⋂

k≤iCi ,8_9kH-,Bt x ∈

⋂

i∈NC ,8_9k.

dj ([0, 1]N4-&-)

$UN x ∈ [0, 1]H k ∈ NKP7F,"k j ≤ 2k ,8_7F, |x− j2k| ≤ 1

2k.

j} CantorN&Lt,j}OWeak König’s LemmaO1MG"k.

Weak König’s LemmaH Brouwer’s Fan Theorem

?j (Brouwer’s Fan Theorem)

s,Z T G,$UN α : N → {0, 1}KP7F ᾱn /∈ T HJk n,8_9lP,"k m ∈ N,8_7F$UN t ∈ T KP7F |t| < mG"k.

mU E5@}eGO O

NPvJNG 1MG"k

.=.*tXGO.j)?J$ F"*tXGO7br3/

>QgAtXGO O'ailF$?

KO ?Z ,?(ilF$?3HKmU

j} O r3/



s,Z T G,$UN α : N → {0, 1}KP7F ᾱn /∈ T HJk n,8_9lP,"k m ∈ N,8_7F$UN t ∈ T KP7F |t| < mG"k.(mU)E5@}eGO, Brouwer’s Fan TheoremOWeak König’s Lemma

NPvJNG,1MG"k.

.=.*tXGO.j)?J$ F"*tXGO7br3/

>QgAtXGO O'ailF$?

KO ?Z ,?(ilF$?3HKmU

j} O r3/




NPvJNG,1MG"k.

• Bishop.=.*tXGO.j)?J$. F"*tXGO7br3/.

>QgAtXGO O'ailF$?

KO ?Z ,?(ilF$?3HKmU

j} O r3/




NPvJNG,1MG"k.

• Bishop.=.*tXGO.j)?J$. F"*tXGO7br3/.• >QgAtXGO, Brouwer’s Fan TheoremO'ailF$?.

(Weak König’s LemmaKO “?Z”,?(ilF$?3HKmU)

j} Weak König’s LemmaO Brouwer’s Fan Theoremr3/.

Brouwer’s Fan TheoremH Heine-BorelNo$j}

jA

• BtN8g AG,$UN x ∈ AKP7F"kBt ε > 0,8_7F(x− ε, x+ ε) ⊂ AHJkbNr+8gH$&.

8g KP7F +8gNs G$UN KP7F"k

,8_7F HJkbNr N+o$H$&

mU +o$H+8gNjA+i +o$O NAN8gN2H7Fh$

dj N4-&- FG

$UN H KP7F "k ,8_7F

?j No$j} N+o$KO-Bt,o$,8_9k

j} O No$j}H1MG"k


jA


• 8g X ⊆ RKP7F,+8gNs 〈Oi〉i G$UN x ∈ X KP7F"k i,8_7F x ∈ Oi HJkbNr X N+o$H$&.

(mU)+o$H+8gNjA+i,+o$O (a, b)NAN8gN2H7Fh$.

dj N4-&- FG

$UN H KP7F "k ,8_7F


j} O No$j}H1MG"k


jA




dj ([0, 1]N4-&-,FG)


2k.


j} O No$j}H1MG"k


jA




dj ([0, 1]N4-&-,FG)


2k.

?j (Heine-BorelNo$j}) [0, 1]N+o$KO-Bt,o$,8_9k.

j} O No$j}H1MG"k


jA




dj ([0, 1]N4-&-,FG)


2k.

?j (Heine-BorelNo$j}) [0, 1]N+o$KO-Bt,o$,8_9k.

j} Brouwer’s Fan TheoremO Heine-BorelNo$j}H1MG"k.

^Ha

CIP=⇒

WKL=⇒

LLPO⇐= ⇐=(1)

⇓6⇑ (1)⇓⇑HBT

=⇒BFT IVT⇐=

⇓BITV

.=.*tX

• CIP: CantorN&Lt,j}• WKL: Weak König’s Lemma• HBT: Heine-BorelNo$j}• BFT: Brouwer’s Fan Theorem• IVT:fVMNj}• BIV:=.*fVMNj}

1. soKe$bN@,,*rx}rW9k(DO#s(5J+C?t,r=9)

^Ha

CIP=⇒

WKL=⇒

LLPO⇐= ⇐=(1)

⇓6⇑ (1)⇓⇑HBT

=⇒BFT IVT⇐=

⇓BITV

Bishop.=.*tX

• CIP: CantorN&Lt,j}• WKL: Weak König’s Lemma• HBT: Heine-BorelNo$j}• BFT: Brouwer’s Fan Theorem• IVT:fVMNj}• BIV:=.*fVMNj}

1. soKe$bN@,,*rx}rW9k(DO#s(5J+C?t,r=9)

2M8%

TD. Troelstra and van Dalen, Constructivism in Mathematics Vol.1,

Elsevier, 1988

BB. Bishop and Bridges, Constructive Analysis, Springer, 1985

BR. Bridges and Richman, Varieties of Constructive Mathematics,

Cambridge University Press, Cambridge, 1987.

I. Ishihara, Constructive reverse mathematics: compactness properties,

In: L. Crosilla and P. Schuster eds., From Sets and Types to Analysis

and Topology: Towards Practicable Foundations for Constructive

Mathematics, Oxford Logic Guides 48, Oxford Univ. Press, 2005,

245-267.

0. 目次構成的数学とは？構成的数学とは？ 1. 有限的数学構成的数学とは？ 2. 可述的数学構成的数学とは？ 3. 直観主義数学構成的数学とは？ 4. 再帰的数学構成的数学とは 5. Bishop 流構成的数学構成的数学とは？証明と BHK 解釈証明と命題の真偽構成的数学とは 5. Bishop 流構成的数学記法実効的な証明の話Markov の原理など自然数から有理数, 実数へ実数って？実数の定義実数上の順序と比較可能性1. x

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