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TRANSCRIPT
Sets
and
Gam
es
14.Septe
mber
2004
Logic
Tea,IL
LC
Am
sterd
am
Ste
fan
Bold
Math
em
aticalLogic
Gro
up
Depart
ment
ofM
ath
em
atics
Univ
ers
ity
ofBonn
Inst
itute
for
Logic
,Language
and
Com
puta
tion
Univ
ers
ity
ofAm
sterd
am
This
rese
arc
hhas
been
support
ed
by
the
DFG
-NW
OBilate
ralCoopera
tion
Pro
ject
KO
1353/3-1
/D
N61-5
32
Dete
rmin
iert
heitsa
xio
me,In
finitare
Kom
bin
ato
rik
und
ihre
Wechse
lwirkungen
(2003-2
006;Bold
,K
oepke,Lowe,van
Benth
em
)
What
isa
Set?
Unte
rein
er
Menge
vers
tehen
wir
jede
Zusa
mm
enfa
ssung
Mvon
best
imm
ten,wohlu
nte
rschie
denen
Obje
kte
nm
unse
r
Ansc
hauung
oderunse
resD
enkens(w
elc
he
die
“Ele
mente
”
von
Mgenannt
werd
en)
zu
ein
em
Ganzen.
Georg
Canto
r
“A
set
Mis
unders
tood
tobe
acollection
ofcert
ain
dis-
tinguishable
obje
cts
mofourperc
eption
orourth
oughts
(whic
hare
called
“ele
ments
”ofM
).”
Nota
tion:
Let
Mand
Sbe
sets
.
•m∈
Mm
eans
mis
an
ele
ment
of
M.
•{a
|P}
isth
ese
tofall
asu
ch
that
pro
pert
yP
hold
sfo
ra.
•M
∪S
isth
eunio
nofth
ese
tsM
and
S.
•M
∩S
isth
ein
ters
ection
ofth
ese
tsM
and
S.
•M\S
isth
ese
twe
get
when
we
take
away
all
ele
ments
inM
that
are
also
inS.
•M
⊆S
means
that
Mis
asu
bse
tof
S.
•and
finally,
Vdenote
sth
euniv
ers
eofall
sets
.
Russell’s
Para
dox
:
Let
Xbe
the
set
ofall
sets
that
do
not
conta
inth
em
selv
es.
X=
{M|M
6∈M}
Then,if
Xconta
ins
itse
lf,it
does
not
conta
initse
lf.
...
On
the
oth
erhand,if
Xdoes
notconta
ins
itse
lf,it
must
conta
in
itse
lf,so
either
way
we
have
acontr
adic
tion.
Som
eoth
erpara
doxes
usin
gself
refe
rence:
Epid
em
es
Para
dox:
“All
cre
tians
are
alw
ays
liars
.”
Barb
er
Para
dox:
Ina
small
tow
nlives
am
ale
barb
er
who
shaves
daily
every
man
that
does
not
shave
him
self,and
no
one
else.
Acondense
dLia
rPara
dox:
“T
his
state
ment
isnot
true”
The
axio
ms
ofSet
Theory
:
•Existe
nz:T
here
exists
ase
tth
at
conta
ins
no
sets
,th
eem
pty
set.
•Exte
nsionality
:T
wo
sets
are
the
sam
eif
they
conta
inexactly
the
sam
eele
ments
.
•Pairs:
Forevery
two
sets
exists
ase
tth
atconta
insexactly
those
two
sets
.
•Unio
n:Forevery
set
Xexists
ase
t⋃ X
thatconta
insexactly
the
ele
ments
ofth
eele
ments
of
X.
•Power
Set:
For
every
set
Xexists
the
power
setP(X
),i.e.th
ese
tth
at
conta
ins
all
subse
tsof
X.
•In
finity:T
here
exists
an
infinite
set.
•Separa
tion:Forevery
set
Xand
every
form
ula
ϕexists
ase
tth
atconta
ins
exactly
the
ele
ments
of
Xfo
rw
hic
hϕ
hold
s.
•Repla
cem
ent:
Forevery
set
Xand
every
functionalfo
rmula
ϕexists
ase
tY
,s.t.
“Y
isth
eim
age
of
Xunder
ϕ.”
•Foundation:If
afo
rmula
ϕis
true
for
at
least
one
set,
then
exists
ase
tX
s.t.
ϕist
true
for
Xbut
not
for
any
ele
ment
of
X.
Functions
as
sets
:
Nota
tion:
(a,b
):=
{{a},{a
,b}}
,A×
B:=
{(a,b
)|a
∈A
and
b∈
B}.
Let
fbe
asu
bse
tof
A×
B,su
ch
that
•fo
rall
a∈
Ath
ere
isa
b∈
Bs.t.
(a,b
)∈
f,
•and
if(a
,b)∈
fand
(c,b
)∈
f,th
en
a=
c
Then
we
call
fa
function
from
Ato
B.
Now
pro
pert
ies
of
functions
like
surjectivity,
dom
ain
,fu
nction
valu
e,etc
.are
just
pro
pert
ies
ofth
ecorr
esp
ondin
gse
t.
Num
bers
as
sets
:
Let
0:=
{}and
a+
1:=
a∪{a}.
Then
all
natu
ralnum
bers
can
be
vie
wed
as
sets
:
1=
0∪{0}=
{}∪{{}}
={{}}
,2
=1∪{1}=
{0,1},
3=
{0,1
,2},
etc
.
Now
we
can
go
on
and
form
alize
the
inte
gers
asse
ts,th
era
tional
num
bers
,th
ere
als,fu
nctions
on
reals,in
tegra
ls,derivations
...
So
all
math
em
atics
take
pla
ce
inth
euniv
ers
eof
sets
,and
set
theory
giv
es
ita
form
alfo
undation.
Diff
ere
nt
Infinitie
s:
There
are
counta
bly
infinitiv
em
any
natu
ralnum
bers
.T
he
size
ofth
epowerse
tP(N
)ofth
enatu
ralnum
bers
isalso
infinite,but
larg
er
than
counta
ble
infinite:
Ass
um
ewe
have
abijection
f:N
→P(N
).
Let
X:=
{z|z
∈N
and
z6∈
f(z
)}.
Then
X∈P(N
),so
there
isan
Ys.t.
X=
f(Y
).
IfY∈
f(Y
),th
en
by
definitio
nof
Xwe
have
Y6∈
X=
f(Y
),
ifY6∈
f(Y
),th
en
by
definitio
nof
Xwe
have
Y∈
X=
f(Y
),
either
way
we
have
acontr
adic
tion,so
such
abijection
can
not
exist.
The
Axio
mofChoice:
Take
afinite
collection
ofnonem
pty
set.
Easily
one
can
choose
an
ele
ment
from
each
set:
Take
an
ele
ment
from
the
firs
tse
tin
the
collection,th
eone
from
the
second,and
soon.
Take
an
infinite
collection
ofnonem
pty
sets
.Now
this
pro
cedure
willnotwork
anym
ore
,atle
ast
itw
illnotcom
eto
an
end
infinite
tim
e.
The
Axio
mofChoic
est
ate
sth
atone
can
alw
ays
choose
,re
gard
-
less
ofth
esize
ofth
ecollection:
(AC)
For
every
set
C=
{Mi|i
∈I}
of
nonem
pty
sets
exists
a
function
f:C
→⋃ C
with
f(M
i)∈
Mi.
Such
afu
nction
iscalled
achoic
efu
nction.
Wellord
erings:
Awellord
ering
ofa
set
Mis
abin
ary
rela
tion≺
on
M,s.t.
•For
all
x∈
Mhold
sx6≺
x.
•For
all
x,y
∈M
either
x≺
yor
y≺
xor
x=
y.
•If
x≺
yand
y≺
zth
en
x≺
z.
•Every
subse
tof
Mhas
a≺-m
inim
alele
ment.
So
there
are
no
infinitiv
ely
desc
endin
gchain
sin
awellord
ering.
Wellord
erings
ofnum
bers
:
The
natu
ralnum
bers
are
wellord
ere
dby
<,
0// 1
// 2// 3
// 4// 5
// 6// 7
// 8// 9
// 10
// ···
where
as
the
inte
gers
are
not
wellord
ere
dby
<:
···
// −5
// −4
// −3
// −2
// −1
// 0// 1
// 2// 3
// 4// 5
// ···
But
we
can
wellord
er
the
inte
gers
,using
anoth
er
ord
er≺:
0≺
1≺−1≺
2≺−2≺
3···
···
−3
��−2
##−1
%%0
// 1cc
2aa
3\\
···
AW
ellord
ering
ofth
ere
als
under
AC:
We
start
with
one
num
ber,
say
a.
' &
$ %
aX
XX
XX
XX
XX
XX
XX
XX
XX Xz
x
AW
ellord
ering
ofth
ere
als
under
AC:
Then
we
choose
anum
ber
bfrom
R\{
a},
' &
$ %
a
xb
A A AA A A
A A AA A A
A A AUx
≺
AW
ellord
ering
ofth
ere
als
under
AC:
then
cfrom
R\{
a,b},
...
' &
$ %
a
xb
x
cH
HH
HH
HH
HH
HH
HH
HH
HH
HH
HH
HH
HHHj
x
≺≺
AW
ellord
ering
ofth
ere
als
under
AC:
Inth
eend
we
willhave
ord
ere
dall
reals.
a≺
b≺
c≺
d≺
e≺
f≺···
Ord
inals:
We
desc
ribed
the
natu
ral
num
bers
as
sets
,w
here
n∈
Nwas
form
alized
as
n=
{m|m
<n}
and
m<
nm
eant
m∈
n.
Ifwe
genera
lize
this
idea
we
get
the
Ord
inalnum
bers
.Form
ally
an
ord
inalis
atr
ansitive,
wellord
ere
dse
t.T
he
collection
of
all
ord
inals
isnot
ase
t,it
is“to
big
.”
Apic
ture
ofth
euniv
ers
e:
A AA A A
A A AA A A
A A A A���
������
������
V
ON
We
write
ON
for
the
cla
ssof
ord
inals
and
for
our
purp
ose
swe
need
only
toknow
that
the
ord
inals
are
wellord
ere
dby
<.
AW
ellord
ering
ofth
ere
als
under
AC:
Let
f:P(R
)→
Rbe
achoic
efu
nction
for
the
power
set
of
the
reals.
Define
the
function
F:O
N→
Rby
transfi
nite
recurs
ion
(here
α
and
βdenote
ord
inals):
F(α
):=
f(R\
⋃ {F(β
)|β
<α} )
,
undefined
if⋃ {F
(β)|β
<α}=
R.
Ifwe
now
say
a≺
biff
a=
F(α
),b=
F(β
)and
α<
β,th
en≺
is
awellord
ering
on
R.
Infinite
Gam
es:
Let
Xbe
acollection
ofse
quences〈x
i|i∈
N〉ofnatu
ralnum
bers
.
Call
Xth
epayo
ffse
t.
Aro
und
inth
egam
eG
Xis
ase
quence
x=
〈xi|i∈
N〉
ofnatu
ral
num
bers
,w
here
the
ele
ments
with
even
index
xI(
i):=
x2ire
pre
-
sentth
em
ovesofpla
yerIand
those
with
odd
index
xII(i
):=
x2i+
1
the
moves
of
pla
yer
II.
Ifx∈
Xth
en
pla
yer
IIw
ins,
oth
erw
ise
pla
yer
I.
Ix0
x2
x4
x6
...
(=xI)
IIx1
x3
x5
x7
...
(=xII)
The
Axio
mofD
ete
rmin
acy:
An
infinite
gam
eG
Xis
dete
rmin
ed,
ifone
of
the
pla
yers
has
a
win
nin
gst
rate
gy,
i.e.
ifth
ere
isa
function
that
tells
one
pla
yer
what
todo
next,
dependin
gon
the
pre
vio
us
moves,
s.t.
inth
e
end
that
pla
yer
win
sth
egam
ere
gard
less
of
the
moves
of
his
opponent.
So
ast
rate
gy
for
pla
yer
Iwould
be
afu
nction
ofth
efo
rm
f:
⋃ {N2n|n
∈N}→
N.
The
Axio
mofD
ete
rmin
acy
now
issim
ply
the
state
ment
(AD)
All
infinite
gam
es
are
dete
rmin
ed.
Counta
ble
Choice
under
AD
:
AD
isnot
com
patible
with
full
choic
e,
but
itim
plies
counta
ble
choic
efo
rcollections
ofse
quences
from
N.
Let
M=
{Sj|j
∈N}
be
acounta
ble
collection
ofnonem
pty
sets
ofsu
ch
sequences:
Counta
ble
many
Sj
︷︸︸
︷S1
S2
S3
······
x1
=〈x
1(i
)|i∈
N〉
y1
=〈y
1(i
)|i∈
N〉
z 1=
〈z1(i
)|i∈
N〉
······
x2
=〈x
2(i
)|i∈
N〉
y2
=〈y
2(i
)|i∈
N〉
z 2=
〈z2(i
)|i∈
N〉
······
. . .. . .
. . .······
xα
=〈x
α(i
)|i∈
N〉
yα
=〈y
α(i
)|i∈
N〉
z α=
〈zα(i
)|i∈
N〉···
. . .. . .
. . .···
Counta
ble
Choice
under
AD
:
We
now
const
ruct
the
payo
ffse
tX
from
M:
Foreach
j∈
Nwe
define
Xjto
be
the
collection
ofall
sequences
xth
atst
art
with
jand
where
xII∈
Sj.
Let
Xbe
the
unio
nofth
e
Xj.
X=
⋃ {Xj|j∈
N}
︷︸︸
︷X
1X
2X
3······
〈1,x
1(1
),�
,x1(2
),�
,···〉
〈2,y
1(1
),�
,y1(2
),�
,···〉
·········
. . .. . .
·········
〈1,x
2(1
),�
,x2(2
),�
,···〉
〈2,y
2(1
),�
,y2(2
),�
,···〉
······
. . .. . .
. . .······
〈1,x
3(1
),�
,x3(2
),�
,···〉〈2
,y3(1
),�
,y3(2
),�
,···〉
···
. . .. . .
. . .. . .
Counta
ble
Choice
under
AD
:
Then
pla
yer
Icannot
have
aw
innin
gst
rate
gy
inth
egam
eG
X:
Anyth
ing
he
does
aft
er
his
firs
tm
ove
jhas
no
conse
quences
on
the
outc
om
e,and
pla
yerII
only
needsto
pla
yone
ofth
ese
quences
from
the
set
Sj
tore
fute
him
.
So
by
AD
pla
yer
IIm
ust
have
aw
innin
gst
rate
gy.
Let
τbe
II’s
win
nin
gst
rate
gy
and
let
(x∗
τ) I
Ibe
the
sequence
of
pla
yerII’s
moves
inth
egam
ew
here
pla
yerIpla
ys
xand
pla
yerII
pla
ys
accord
ing
toτ.
Then
F(S
j):=
(〈j|i∈
N〉∗
τ) I
Idefines
achoic
efu
nction
for
M.
Fin
ite
gam
es:
Every
finite
exte
nsive
gam
eofperfectin
form
ation
thatis
aW
in/Lose
-gam
eis
dete
rmin
ed.
To
see
whic
hof
the
two
pla
yers
Aand
Bhas
the
win
nin
gst
rate
gy,
we
use
asim
ple
backtr
ackin
gm
eth
od:
Let
this
tree
repre
sent
such
agam
e,th
epla
yers
take
turn
sand
pla
yer
Ast
art
s:
•
������
����
����
����
���
&&NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN
A
•
������
����
����
����
���
��////////////////•
������������������
##GGGGGGGGGGGGGGGGGGGGGGGG
B
•
������������������
��////////////////• ��
•
��������������������
����>>>>>>>>>>>>>>>>>>>
• ��
A
••
••
••
•
Win
nin
gstr
ate
gie
sfo
rfinite
gam
es:
First
we
labelth
eend-n
odes: •
~~}}}}
}}}}
}}}}
}}}}
}}}}
''OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
A
•
~~ ~~~~
~~~~
~~~~
~~~~
~~~~
��0000000000000000•
������������������
$$IIIIIIIIIIIIIIIIIIIIIIIIII
B
•
������������������
��////////////////• ��
•
������
����
����
����
���
����@@@@@@@@@@@@@@@@@@@@
• ��
A
AA
BB
AB
A
Win
nin
gstr
ate
gie
sfo
rfinite
gam
es:
The
last
move
isA’s,we
labelth
epre
vio
us
nodes
accord
ingly
:
•
~~}}}}
}}}}
}}}}
}}}}
}}}}
''OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
A
•
~~ ~~~~
~~~~
~~~~
~~~~
~~~~
��////////////////•
������������������
$$HHHHHHHHHHHHHHHHHHHHHHHHH
B
A
���������������
��3333333333333B ��
A
~~ ~~~~
~~~~
~~~~
~~~~
��!!DDDDDDDDDDDDDDDDD
A ��
AA
BB
AB
A
Win
nin
gstr
ate
gie
sfo
rfinite
gam
es:
Now
itis
B’s
turn
:
•
~~ }}}}
}}}}
}}}}
}}}}
}}}}
''OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
A
B
|| yyyy
yyyy
yyyy
yyyy
yy
��3333333333333A
���������������
%%LLLLLLLLLLLLLLLLLLLLLLLL
A
���������������
��3333333333333B ��
A
~~ ~~~~
~~~~
~~~~
~~~~
��""EEEEEEEEEEEEEEEEEE
A ��
AA
BB
AB
A
Win
nin
gstr
ate
gie
sfo
rfinite
gam
es:
And
inth
eend
we
see
that
pla
yer
Ahas
aw
innin
gst
rate
gie
:
A
~~ }}}}
}}}}
}}}}
}}}}
}}}}
'''g
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'g'g
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'g
A
B
~~}}}}
}}}}
}}}}
}}}}
}}}}
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AAAAAAAAAAAAAAAAAAAAA ���O�O�O�O�O�O�O�O�O
A
AA
BB
AB
A
Sim
ilarities
betw
een
AC
and
AD
:
•Both
axio
ms
gra
nt
the
existe
nce
offu
nctions.
•Both
imply
rest
ricte
dvers
ions
ofth
eoth
er.
•Both
are
exte
nsions
offinite
princip
les.
•Both
are
consist
ent
with
ZF
(under
som
eass
um
ptions)
.
•T
hey
contr
adic
teach
oth
er.
and
Diff
ere
nces:
•Under
AC
exists
awellord
ering
ofth
ere
als,butnotunder
AD.
•Under
AD
every
subse
tofth
ere
als
isLebesg
ue
measu
rable
,
not
sounder
AC.
•Under
AC
every
set
can
be
wellord
ere
d,but
not
under
AD.
•Under
AD
infinite
part
itio
npro
pert
ies
hold
,butnotunder
AC.
•···