inconsistent countable set in second order zfc and unexistence of the strongly inaccessible...
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Inconsistent countable set in second order ZFC and
unexistence of the strongly inaccessible cardinals.
Jaykov Foukzon1
1Department of athematics Israel Institute of !echnology" #aifa" Israel
$aykovfoukzon%list.ru
Abstract:In this article &e derived an importent example of the inconsistent
countable set in second order ZFC ZFC2 &ith the full second'order semantic.
ain results is()i* ),( 2ZFCCon
)ii* let k be an inaccessible cardinal and Hk is a
set of all sets having hereditary size less then k, then)).(( kHVZFCCon =+
Keywords: +,del encoding" Completion of ZFC2 , -ussells paradox"
'model" #enkin semantics" full second'order semantic
/.Introduction.
0ets remind that accordingly to naive set theory" any definable collection is a set. 0et
be the set of all sets that are not members of themselves. If R 1ualifies as a member of
itself" it &ould contradict its o&n definition as a set containing all sets that are not
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members of themselves. 2n the other hand" if such a set is not a member of itself" it
&ould 1ualify as a member of itself by the same definition. !his contradiction is -ussells
paradox. In /345" t&o &ays of avoiding the paradox &ere proposed" -ussells type
theory and the Zermelo set theory" the first constructed axiomatic set theory. Zermelos
axioms &ent &ell beyond Freges axioms of extensionality and unlimited set abstraction"
and evolved into the no&'canonical Zermelo''Fraenkel set theory ZFC. "But how do
we know that ZFC is a consistent theory, free of contradictions? The short answer is
that we don't; it is a matter of faith (or of skepticism)"''' 6.7elson &rote in his not
published paper 8/9. #o&ever" it is deemed unlikely that even ZFC2 &hich is a very
stronger than ZFC harbors an unsuspected contradiction: it is &idely believed that if
ZFC2 &ere inconsistent" that fact &ould have been uncovered by no&. !his much is
certain ''' ZFC2 is immune to the classic paradoxes of naive set theory( -ussells
paradox" the ;urali'Forti paradox" and Cantors paradox.
Remark 1.1.7ote that in this paper &e vie& the second order set theory ZFC2 under
the #enkin semantics 8
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In this paper &e prove that ZFC2fss
is inconsistent. @e &ill start from a simple naive
consideration.0et be the countable collection of all sets such that
ZFC2fss
!X X, &here X is any /'place open &ff i.e."
Y Y !X X Y X . 1.1
0etX
ZFC
2fss
be a predicate such that
ZFC
2fss
Y ZFC2fss
X Y.0et be
the countable collection of all sets such that
X X X
ZFC2fss
X . 1.2
From )/.
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e.g." that ZFC2Hs
has an omega'model ZFC2
Hs
or an standard model stZFC2
Hs
i.e." a
model in &hich the inteers are the standard inteers 8A9.!o put it another &ay" &hy
should &e believe a statement $ust because theres a FC2Hs
'proof of itB Its clear that if
ZFC2Hs is inconsistent" then &e &ont believe ZFC2Hs 'proofs. @hats slightly more
subtle is that the mere consistency of ZFC2 isnt 1uite enough to get us to believe
arithmetical theorems of ZFC2Hs; &e must also believe that these arithmetical theorems
are asserting something about the standard naturals. It is conceivable that ZFC2Hs
might be consistent but that the only nonstandard models NstZFC2
Hs
it has are those in
&hich the integers are nonstandard" in &hich case &e might not believe an
arithmetical statement such as ZFC2Hs is inconsistent even if there is a ZFC2Hs 'proof
of it.
Remark 1.4.#o&ever assumption MstZFC2
Hs
is not necessary. 7ote thatin any
nonstandard model NstZ2
Hs
of the second'order arithmetic Z2Hs
the terms 0,
S0 1,SS0 2, comprise the initial segment isomorphic to stZ2
Hs
MNst
Z2Hs
. !his initial
segment is called the standard cut of the MNstZ2
Hs
. !he order type of any nonstandard
model of NstZ2
Hs
is e1ual to A for some linear order 8A9"89. !hus one can to
choose +,del encoding inside stZ2
Hs
.
Remark 1.5.#o&ever there is no any problem as mentioned above in second order set
theory ZFC2 &ith the full second'order semantics becouse corresponding second
order arithmetic Z2fss
is categorical.
Remark 1.6. 7ote if &e vie& second'order arithmetic Z2 as a theory in first'order
predicate calculus. !hus a modelZ2
of the language of second'order arithmetic Z2
consists of a set )&hich forms the range of individual variables* together &ith a
constant 0 )an element of *" a function S from to " t&o binary operations
and on , a binary relation on " and a collection of subsets of "
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&hich is the range of the set variables. @hen D is the full po&erset of , the model
Z2 is called a full model. !he use of full second'order semantics is e1uivalent to
limiting the models of second'order arithmetic to the full models. In fact" the axioms of
second'order arithmetic have only one full model. !his follo&s from the fact that the
axioms of Eeano arithmetic &ith the second'order induction axiom have only one model
under second'order semantics" i.e. Z2 , &ith the full semantics" is categorical by
Dedekinds argument" so has only one model up to isomorphism. @hen is the usual
set of natural numbers &ith its usual operations"Z2
is called an 'model. In this
case &e may identify the model &ith
D, its collection of sets of naturals" because this set is enough to completely determine
an
'model. !he uni1ue full omega'model
Z2fss
, &hich is the usual set of natural numbers &ith its usual structure and all its
subsets" is called the intended or standard model of second'order arithmetic.
@e assume that( )i* ConZFC2fss
, )ii* ConZFC2Hs
'model of ZFC2Hs
.
ain result is( ~ConZFC2Hs
'model of ZFC2Hs
, ~ConZFC2fss
.
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are function symbols" neg " imp " etc." such that" for all formulae " : S
neg c c, S imp c, c
c
etc. 2f particular
importance is the substitution operator" represented by the function symbol sub, .
For formulae x " terms t &ith codes tc :
S sub x c, tc t c. 2.1
It &ell kno&n 859 that one can also encode derivations and have a binary relation
ProvTh x,y
)read proves or x
is a proof of * such that for closedt1 , t2 : S ProvTh t1 , t2 iff t1 is the code of a derivation in Thof the formula &ith
code t2 . It follo&s that
Th iffS ProvTh t, c
2.2
for some closed term t. !hus one can define
PrTh y xProvTh x,y, 2.3
and therefore one obtain a predicate asserting provability. @e note that is not al&ays
the case that 859(
Th iffS PrTh c
. 2.4
It
&ell kno&n 859 that the above encoding can be carried out in such a &ay that the
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follo&ing important conditions D1,D2 and D3 are meet for all sentences 859(
D1.Th implies S PrTh c
,
D2.S PrTh c
PrTh PrTh c
c
,
D3.S PrTh c
PrTh c
PrTh c
.
2.5
ConditionsD1,D2
andD3
are called the Derivability Conditions.
!emma 2.1.Gssumethat( )i* ConTh and )ii* Th PrTh c
, &here is a closed
formula.!hen Th PrTh c
.
"roo#. 0et ConTh be a formula
ConTh t1 t2 ProvTh t1 , c
ProvTh t2 ,neg c
t1 t2ProvTh t1 , c
Prov Th t2 ,neg c
.
2.6
&here t1 , t2 is a closed term. @e note that Th ConTh ConTh for any closed
. uppose that Th PrTh c
, then )ii* gives
Th PrTh c
PrTh c
. 2.7
From )
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t1 t2ProvTh t1 , c
ProvTh t2 ,neg c
. 2.8
;ut the formula )
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the Th:Ded Th
# DedTh
implies
Ded Th#
DedTh
.
Remark 2.1. @e note that a theory Th# depend on model
Th
or !st.Th
" i.e.
Th#
Th#
MTh
or
Th#
Th# M
Nst
Th
correspondingly. @e &ill consider the case
Th#
Th#
MTh
&ithout loss of generality.
"ro%osition 2.1.Gssume that )i* ConTh and )ii * Th has an 'model Th . !hen
theory Th can be extended to a maximally consistent nice theoryTh
# Th
#M
Th .
"roo#. 0et 1 . . . i. . . be an enumeration of all &ffs of the theory Th )this can be
achieved if the set of propositional variables can be enumerated*. Define a chain
Thi|i ,Th1 T of consistent theories inductively as follo&s( assume that
theory Thi is defined. )i* uppose that a statement )
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Thi PrTh i i c
andThi i MTh
i . 2.11
!hen &e define theory Thi 1 as follo&s( Thi1 Thi i . sing 0emma
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Thi PrTh i i c
andThi PrTh i i c
i. 2.15
@e &ill re&rite the condition )
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satisfied"i.e. Thi PrTh i i c
andThi i M
Th i then clearly
Thi1 Thi i is consistent by 0emma
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Remark 2.3. 7ote thatTh
# y x PrTh # !x x
c
De#inition 2.6.0et be a collection such that: x x xis a Th# 'set .
"ro%osition 2.2. Collection is a Th#'set.
"roo#. 0et us consider an one'place open &ff x such that conditions ) * or ) *
is satisfied" i.e. Th#
!x x . @e note that there exists countable collection
of the one'place open &ffs n x n such that( )i* x and )ii*
Th !x x nn x n x
or
Th !x PrTh x c
nn PrTh x n x c
and
MTh
!x x nn x n x
2.19
or of the e1uivalent form
Th !x 1 1x 1 nn 1x1 n,1 x1
or
Th !x PrTh x1 c
nn PrTh x1 n x 1 c
and
MTh
!x x 1 nn x 1 n x1
2.20
&here &e set x 1x1, n x1 n,1 x 1 and x1 . @e note that any
collection k n,kx n , k 1,2,. . . such above defines an uni1ue set k, i.e.
k1 k2 iff k1 x k2 . @e note that collections k,k 1,2,.. is no part
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of the ZFC2 , i.e. collection k there is no set in sense of ZFC2 . #o&ever that is
no problem" because by using +,del numbering one can to replace any collection
k,k 1,2,.. by collection k g k of the corresponding +,del numbers such
that
k g k g n,kx k n , k 1,2,. . . . 2.21
It is easy to prove that any collection k g k,k 1,2,.. is a Th#'set.!his is
done by +,del encoding 859"8/49 of the statament )
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n,k g n,kx k Frgn,k,v k )see 8/49*. 0et us define no& predicate
gn,k,v k
gn,k,v k PrTh !x k 1,kx1 c
!x kv k x kc
nn PrTh 1,kx k c
PrTh Frgn,k,v k.
2.24
@e define no& a set k such that
k k
gk ,
nn gn,k k
gn,k,v k
2.25
;ut obviously definitions )
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Th#
c c c c 2.22
and therefore
Th#
c c c c . 2.23
;ut this is a contradiction.
"ro%osition 2.6.Gssume that )i* ConTh and )ii * T has an 'model !stTh . !hen
theory Th can be extended to a maximally consistent nice theoryTh
# Th
# MNst
Th .
"roo#. 0et 1 . . . i. . . be an enumeration of all &ffs of the theory Th )this can be
achieved if the set of propositional variables can be enumerated*. Define a chain
Thi|i ,Th1 T of consistent theories inductively as follo&s( assume that
theory Thi is defined. )i* uppose that a statement )
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)ii* uppose that a statement )
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!hen &e define a theory Thi 1 as follo&s( Thi 1 Thi.
)iv* uppose that a statement )
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satisfied"i.e. Th PrTh i c
andThi i M
Th i then clearly
Thi1 Thi i is consistent by 0emma
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Remark 2.4. 7ote that Th#
!x x .
Remark 2.5. 7ote thatTh
# y x PrTh # !x x
c
De#inition 2.(.0et be a collection such that: x x xis a Th# 'set .
"ro%osition 2.&. Collection is a Th#'set.
"roo#. 0et us consider an one'place open &ff x such that conditions ) * or ) *
is satisfied" i.e. Th#
!x x . @e note that there exists countable collection
of the one'place open &ffs n x n such that( )i* x and )ii*
Th !x x n n MstZ2
Hs
x n x
or
Th !x PrTh x c
n n MstZ2
Hs
PrTh x n x c
and
M!stTh
!x x n n MstZ2
Hs
x n x
2.34
or of the e1uivalent form
Th !x 1 1x1 n n MstZ2
Hs
1x1 n,1 x 1
or
Th !x PrTh x 1 c
n n MstZ2
Hs
PrTh x1 n x 1 c
and
M!stTh
!x x 1 n n MstZ2
Hs
x1 n x1
2.35
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&here &e set x 1x1, n x1 n,1 x 1 and x1 . @e note that any
collection k n,kx n , k 1,2,. . . such above defines an uni1ue set k, i.e.
k1 k2 iff k1 x k2 . @e note that collections k,k 1,2,.. is no part
of the ZFC2Hs, i.e. collection k there is no set in sense of ZFC2
Hs. #o&ever that is
no problem" because by using +,del numbering one can to replace any collection
k,k 1,2,.. by collection k g k of the corresponding +,del numbers such
that
k g k g n,kx k n , k 1,2,. . . . 2.36
It is easy to prove that any collection k g k,k 1,2,.. is a Th#'set.!his is
done by +,del encoding 859"8/49 of the statament )
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xx c x PrTh # x x c
. 2.38
"ro%osition 2.'.Gny collection k g k,k 1,2,.. is a Th#'set.
"roo#. @e define n,k g n,kx k n,kx kc,v k x k
c. !herefore
n,k g n,kx k Frgn,k,v k )see 8/49*. 0et us define no& predicate
gn,k,v k
gn,k,v k PrTh !x k 1,kx 1 c
!x kv k x kc
n n MstZ2
Hs
PrTh 1,kx kc
PrThFrgn,k,v k .
2.39
@e define no& a set k such that
k k
gk ,
nn gn,k k
gn,k,v k
2.40
;ut obviously definitions )
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Th#
c c PrTh # c c c
. 2.41
From formula )/* and Eroposition
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implies Th .
De#inition 3.2.0et x be one'place open &ff such that the conditions(
Th !x x or
Th PrTh !x x c is satisfied.
!hen &e said that" a set is a Th'set iff there is exist one'place open &ff x such
that x . @e &rite Th iff is a T 'set.
Remark 3.1. 7ote that Th !x x .
Remark 3.2. 7ote that Th y x PrTh
!x x c
De#inition 3.3.0et be a collection such that: x x xis a Th'set .
"ro%osition 3.2. Collection is a T 'set.
De#inition 3.4.@e define no& a Th'set c :
xx c x PrTh
x x c . 3.2
"ro%osition 3.3.)i* Th c, )ii* c is a countable T 'set.
"roo#.)i* tatement Th c follo&s immediately by using statement and axiom
schema of separation 8>9. )ii* follo&s immediately from countability of a set .
"ro%osition 3.4.G set c is inconsistent.
"roo#.From formla )=.
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Th c c c c 3.4
and therefore
Th c c c c . 3.5
;ut this is a contradiction.
>.*onc+$sion.
In this paper &e have proved that the second order ZFC &ith the full second'order
semantic is a contradictory"i.e. ConZFC2 . ain result is( let k be an inaccessible
cardinal and Hk is a set of all sets having hereditary size less then k, then
ConZFC V Hk.
!his result &as obtained in 89"8/=9 by using essentially another approach.
Re#erences.
8/9 6. 7elson.@arning igns of a Eossible Collapse of Contemporary
athematics.https(&eb.math.princeton.eduKnelsonpapers&arn.pdf
89 . hapiro" Foundations &ithout Foundationalism( G Case for econd'order
0ogic. 2xford niversity Eress. I;7 4'/3'5
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niversity of ;irmingham