inconsistent countable set in second order zfc and unexistence of the strongly inaccessible...

7/21/2019 Inconsistent countable set in second order ZFC and unexistence of the strongly inaccessible cardinals.

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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

inconsistent countable set in second order zfc and unexistence of the strongly inaccessible...

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