- Email: [email protected]
*ZFC: An axiomatic * approach to nonstandard methods
C. R. Acad. Logique/Logic
Sci.
Paris,
t. 324,
Sbrie
I, p. 963-967,
1997
*ZFC: an axiomatic *approach to nonstandard methods Mauro
DI
NASSO
Dipartimento di Matematica, 15, via de1 Capitano. 53100 E-mail: [email protected]
Abstract.
LJniversitA di Siena, Italy.
Siena.
An axiomatic nonstandard set theory *ZFC is presented, where all axioms of ZFC without foundation are assumed and a strong form of saturation holds. The class of standard elements is a model of ZFC; *ZFC and ZFC are equiconsistent. “ZFC
: une
* approche
axiomatique
des
mhthodes
non
standard
R6sum&
Version
Nous pksentons une the’orie non stundard des ensembles od tous les axiomes de ZFC sont valides saris fondement et ok l’on suppose une forme forte de saturation. Lu classe des e’lkments standard est un mod2le de ZFC; *ZFC et ZFC sont &quiconsistantes.
fraqaise
abrkgke
Soit L, le langage qui consiste en deux symboles de relation binaires E et * ; *ZFC est la theorie de L* qui a pour axiomes les cinq groupes de formules suivantes : 1. ZFCOn admet tous les axiomes de la theorie de Zermelo-Fraenkel y compris l’axiome de choix, mais sans fondement. En outre, on consider-eles schemasde separation et de remplacement pour toutes les L,-formules. On definit la classe S des ensemblesstandard via la hierarchic cumulative WF = UaE~,vVa(0) des ensembles bien fond&‘. 2. * est une fonction de domaine S :
{VlzVyVz
Note
pdsentke
071%4442/97/03240963
par
[*5
Jacques
= y A *z
= z -+
(y
=
2 A z E S)]}
A
[Vx
E s
3y
*5
= y].
TITS.
0 Acadtmie
des Sciences/Elsevier.
Paris
963
M.
Di
Nasso
Bien que * soit partiellement definie, La classe I des ensembles internes Un ensemble externe est un ensemble cps et cp’ sa relativisation a la classe
3. SchCma du Principe
on l’utilise comme symbole de fonction. est definie comme la classe {y : y E *z pour un certain z E 5’). non interne. Pour chaque E-formule cp, on note respectivement des ensembles standard et a celle des ensembles internes.
de Transfert
*:s + I est une immersion E-Clementaire, la formule suivante est un axiome :
4. L’univers est bien fond6 sur les ensembles descendantes d’ensembles externes : V’zf0 Soit p(x) si on a :
une E-formule
ZFC-
c’est-a-dire
internes,
pour chaque E-formule
c’est-a-dire
il n’existe
pas de chaines
[znI=0-+((3yEzynz=0)].
ayant z pour unique variable libre. On dit que q(x)
k V&y
cp(zlT . . . , z~),
[$(z)
A p’(y)
dtfinit
un cardinal
-+ z = y A <<.?:est un cardinal D]
On ne suppose pas que ZFCk 3~z:(ps(z). Par exemple, il existe des formules qui definissent les cardinaux w, HI, Rx, I,, In,, le premier cardinal inaccessible, le premier cardinal mesurable, etc. Les cardinaux definissables peuvent etre utilises comme des constantes. C’est-a-dire que si g(z) est une formule et k designe le cardinal defini par p(z), nous tcrivons CT(~) comme abreviation pour la formule Vz (p’(z) + g(z).
5. Schema
de Saturation.
Pour chaque cardinal definissable
k, la propriete de k-saturation
VF c I [IFI < k A VFo c F (F.
suivante est un axiome :
finie -+ nF, # 0)]
--) nF # 0
Soit k un cardinal definissable. Les proprittes suivantes peuvent Ctre demontrees directement dans *ZFC : (i) Proprie’te’ de Standardisation : VA 3B E S B = {b E S : *b E A}. (ii) Si A c ‘X a un cardinal IAl < k, alors A C B pour un certain B g ‘X *fini. (iii) Proprikte’ de Compre’hension : Si f : A -+ B est une fonction oti B E I et A C ‘X a un cardinal IAl < k, alors il existe une fonction interne F : A’ 4 B avec A C A’ C *X et F(a) = f(a) pour chaque a E A. Si A = {*z : z E X} on peut prendre A’ = *X. En ce qui concerne la structure de l’univers de *ZFC, on a : THEOREME. (i) L’univers des ensembEes est V = U,,,,V,(*V,(rZr)). (ii) Pour chaque ordinal a, *v?(0) est isomorphe ri une ultrapuissance &mite V,(0)6lF 2. (iii) On peut dkjinir we fonction rang R sur 1‘univers V d valeurs totalement ordonne’es qui satisfait : = *p(s) pour chaque s E S, (a) R(z) = Sup{R(:c’) + 1 ., z’ E z} pour chaque z # 0 ; (b) R(*s) 02 p est la fonction rang usuelle & valeurs bien ordonne’es; (c) R(t) = < pour chaque ordinal interne [ E Ua~~~\r*cc.
964
*ZFC: an axiomatic *approach to nonstandard methods
(iv) Chaque structure binaire non cyclique (A, E), ou ]A] < k pour un certain cardinal definissable, est isomorphe a un modele nature1 (B: E) 3. Pour l’usage de la saturation dans *ZFC, le rtsultat suivant est utile. META-THBORBME. - Soit P(x) une ~-form&e ayant .T pour unique variable libre. Supposonsque ‘ZFC k GVcardinalk k-saturation + [Vx E 5’ 1x1< k + P(x)] S. Alors, ZFC t- Vx P(x). Maintenant, nous nous occupons de la puissance relative de *ZFC par rapport a ZFC. Si M est un modele de ZFC-, nous notons (WF) ” le sous-modele de It4 dont l’univers est {x E A4 : A4 + << x est bien fond6 B}. 11est bien connu que (WF)“’ b ZFC. On a le THEOR~ME PRINCIPAL. - Si N /= ZFC, alors N est un sow-modele e’lementairede (WF)“’ pour un certain M b *ZFC. Une des consequencesdu theoreme principal est qu’on peut donner une traduction de E-formules dans les L*-formules. COROLLAIRE. - ZFC est3delement interpretable duns * ZFC en relativisant les quantificateurs a la classe des ensemblesbien fond&. C’est-a-dire, pour chaque E-enonce u,
En particulier, *ZFC et ZFC sont equiconsistantes.
The usual axiomatic approachesto nonstandard analysis are not completely satisfactory when dealing with external sets. For instance, in Nelson’s Internal Set Theory (see [7]), basic collections such as Loeb measuresor nonstandard hulls cannot be elements of the universe. On the other hand, the most popular superstructure approach is not an axiomatic one and only sets of finite rank in the cumulative hierarchy are considered. We overcome these difficulties by introducing an axiomatic nonstandard set theory *ZFC in which the enlarging map * is defined for each well-founded set, not necessarily of finite rank. Every model of *ZFC is a model of ZFC without foundation and the class of its standard elements is a model of ZFC. A strong form of saturation is assumedand the standardization and comprehensivenessproperties are theorems of *ZFC. A consequenceof the main theorem in this Note is that ZFC is faithfully interpretable in *ZFC. In particular, ZFC and *ZFC are equiconsistent. We supposethat the reader knows the basics of nonstandard analysis (we suggest[6] as a reference) and have a basic knowledge of set-theoretic and model-theoretic notions (see [5] and [3] respectively). Let L* be the language which consists of the two binary relation symbols E and * ; *ZFC is the theory of L* having the following five groups of formulas as axioms. 1. ZFCAll axioms of the Zermelo-Fraenkel set theory with choice but without foundation are assumed. Moreover, the Separation and Replacement schemata are considered for all L*-formulas. The class 5’ of all standard sets is defined as the cumulative hierarchy WF = lJaE~,vVn(0) of the well-founded sets 4. 2. * is a function
with domain 5’:
{vxvyvz [*x = y A *x = z + (y = z A x E S)]} A [Vx E s 3y *x = y/l, Although it is only partially defined, we use * as a function symbol.
965
M.
Di
Nasso
The class I of internal sets is defined as the class {y : ?/ E *x for some x E S}. An external set is a set which is not internal. For every E-formula ‘p, denote respectively by (ps and cp’ its relativizations to the class of standard sets and internal sets. 3. Transfer
Principle
schema
* : S -+ 1 is an E-elementary is an axiom: VXI,...
embedding,
>x, ES
i.e. for every E-formula ,x,)
pyc,....
c--)
cp(zl,. . . ~x,),
the following
(pl(*xl,...,*x,)
Notice that, contrarily to the usual superstructure approach, the transfer principle is not limited to the bounded quantifier formulas 4. The universe is well-founded over the internal sets, i.e. there are no descending chains of external sets. VX#~ Let p(s) be an E-formula if the following holds: ZFC-
[znI=0
+(~zJEzE~J~x=~)]
where z is the only free variable. I- VxVy [(p’(x)
A (p’(y)
We say that P(X) defines a cardinal
--) z = y A “2 is a cardinal”]
We do not require that ZFC k 3x (p’(x). For instance, there cardinals w, N1, Nz, &,, IN1, the first inaccessible cardinal, the first The definable cardinals can be used as constants. Namely, if O(X) cardinal defined by p(x), we shall write a( k-) as a short-hand for We now turn to the last group of axioms. 5. Saturation
are formulas which define the measurable cardinal, and so on. is a formula and Ic denotes the the formula Vx (ps( x) --) cr( x).
schema
For every definable cardinal k, the following
k-saturation
VF c I [IFI < /CA VFO c F (FO finite
property
-+ nF, # 0)]
is an axiom: -+ nF # 0.
Let k be any definable cardinal. The following facts can be straightforwardly proved from *ZFC: (i) Standardization Property: V.4 3B E S B = { 13E S : “b E A}. (ii) If A C *X has cardinality IAl < k-, then A 2 B for some *finite B C *X. (iii) Comprehensiveness Property: If f : A --+ B is a function, where B E 1 and A g “X has cardinality IAl < k, then there exists an internal function F : A’ -+ B with A 2 A’ C *X and F(a) = f(u) for every a E A. If A = {*IC : :c E X} one can take A’ = *X. The above properties are fundamental in the practice of nonstandard analysis. We remark that in the various axiomatic approaches, the standardization property is usually postulated rather than proved. With regard to the structure of the universe of *ZFC, the following holds. THEOREM
(i) The universe of sets is the union V = UaE~a\~l~(*Va (0)). (ii) For each ordinal Q, *V, (0) IS isomorphic to a limit ultrapower Va(0)6 IF 5. (iii) A totally-ordered-valued rank function R on the universe V is dejinable which satisfies: (a) R(z) = Sup{R(z’) + 1 : x’ E x} for every x # 0; (b) R( *s) = *p(s) for every s E S where p is the usual well-founded rank function; (c) R(c) = cfor every internal-ordinal < E UnE~Lv*a. (iv) Every acyclic binary structure (A, E), where IAl < k f or somedefinable cardinal k, is isomorphic to a natural model (B, E) 6.
966
*ZFC:
an axiomatic
*approach
to nonstandard
methods
Two fundamental consequences of the axiom of foundation are the existence of a well-orderedvalued rank function and Mostowski’s collapse theorem. (iii) and (iv) show that similar results can be saved in our non-well-founded context. When dealing with saturation in *ZFC, a useful tool is the following: META-THEOREM. - Let P(x) be any E-formula having exactly one free variable and suppose that *ZFC 1 “Vcardinal k k-saturation + [Yx E 5’ 1x1< k --+ P’(x)]“. Then ZFC !- Vx P(x). Let us turn to the relative strenght of *ZFC with respect to ZFC. If A4 is any model of ZFC-, denote by (WF)‘l the submodel of 1M whose universe is {x E A4 : A4 b “x is well-founded”}. It is well-known that (WF)‘%t /= ZFC. The following holds: MAIN THEOREM. - If N k ZFC, then N is an elementary submodel of ( WF)“’ for some h/r /= ‘ZFC. The basic idea in the proof of the Main theorem is that the constructions given in [l] and [4] within an anti-foundational context can be “simulated” in ZFC. In those papers, Boffa’s Superuniversality axiom (see [2]) is used to construct transitive classeswhich are models of weakened versions of *ZFC. As a consequenceof the Main Theorem, a translation of E-formulas into &-formulas can be given by relativizing to the well-founded (standard) sets. COROLLARY. - ZFC is faithfully interpretable in * ZFC by relativizing quantzj’iersto the class WF of well-founded sets. That is, for every E-sentence o
ZFCta
a
*ZFCtc?“F.
In particular, *ZFC and ZFC are equiconsistent. The results of this Note were presented at the International Applications “, Edinburgh, August II- 17, 1996.
Symposium
on “Nonstandard
Analysis
and its
’ La hierarchic cumulative sur un ensemble quelconque X est definie de la maniere suivante : VO(X) = X; T;,+,(X) = Vu(X) U P(V,(X)); V,(X) = U,,
1997.
References [I] Ballard D. and Hrbacek K., 1992. Standard Foundations for Nonstandard Analysis, J. Symb. Logic, 57, pp. 741-748. [2l Boffa M., 1972.. Forcing et Negation de I’Axiome de Fondement, M&n. Acad. Sci. Brig., Tome XL, no. 7. [3] Chang C.C. and Keisler H.J., 1990. Model Theory, 3rd edition, North-Holland. [4] Di Nasso M.. Pseudo-superstructures as Nonstandard Universes, J. S.wnbolic Logic, to appear. [5] Kunen K., 1980. Set Theory, an Introduction to Independence Proofs, North-Holland. [6] Lindstrtim T., 1988. An Invitation to Nonstandard Analysis, in Nonstandard Ana(wis and ifs Applicarions (N. Cutland,
ed.), Cambridge University Press, pp. l- 105. [7] Nelson
E., 1977. Internal
Set Theory:
a New Approach
to Nonstandard
Analysis,
Bull. Amer.
Moth.
Sot.,
83,
pp. 1165-l 198.
967
Sci.
Paris,
t. 324,
Sbrie
I, p. 963-967,
1997
*ZFC: an axiomatic *approach to nonstandard methods Mauro
DI
NASSO
Dipartimento di Matematica, 15, via de1 Capitano. 53100 E-mail: [email protected]
Abstract.
LJniversitA di Siena, Italy.
Siena.
An axiomatic nonstandard set theory *ZFC is presented, where all axioms of ZFC without foundation are assumed and a strong form of saturation holds. The class of standard elements is a model of ZFC; *ZFC and ZFC are equiconsistent. “ZFC
: une
* approche
axiomatique
des
mhthodes
non
standard
R6sum&
Version
Nous pksentons une the’orie non stundard des ensembles od tous les axiomes de ZFC sont valides saris fondement et ok l’on suppose une forme forte de saturation. Lu classe des e’lkments standard est un mod2le de ZFC; *ZFC et ZFC sont &quiconsistantes.
fraqaise
abrkgke
Soit L, le langage qui consiste en deux symboles de relation binaires E et * ; *ZFC est la theorie de L* qui a pour axiomes les cinq groupes de formules suivantes : 1. ZFCOn admet tous les axiomes de la theorie de Zermelo-Fraenkel y compris l’axiome de choix, mais sans fondement. En outre, on consider-eles schemasde separation et de remplacement pour toutes les L,-formules. On definit la classe S des ensemblesstandard via la hierarchic cumulative WF = UaE~,vVa(0) des ensembles bien fond&‘. 2. * est une fonction de domaine S :
{VlzVyVz
Note
pdsentke
071%4442/97/03240963
par
[*5
Jacques
= y A *z
= z -+
(y
=
2 A z E S)]}
A
[Vx
E s
3y
*5
= y].
TITS.
0 Acadtmie
des Sciences/Elsevier.
Paris
963
M.
Di
Nasso
Bien que * soit partiellement definie, La classe I des ensembles internes Un ensemble externe est un ensemble cps et cp’ sa relativisation a la classe
3. SchCma du Principe
on l’utilise comme symbole de fonction. est definie comme la classe {y : y E *z pour un certain z E 5’). non interne. Pour chaque E-formule cp, on note respectivement des ensembles standard et a celle des ensembles internes.
de Transfert
*:s + I est une immersion E-Clementaire, la formule suivante est un axiome :
4. L’univers est bien fond6 sur les ensembles descendantes d’ensembles externes : V’zf0 Soit p(x) si on a :
une E-formule
ZFC-
c’est-a-dire
internes,
pour chaque E-formule
c’est-a-dire
il n’existe
pas de chaines
[znI=0-+((3yEzynz=0)].
ayant z pour unique variable libre. On dit que q(x)
k V&y
cp(zlT . . . , z~),
[$(z)
A p’(y)
dtfinit
un cardinal
-+ z = y A <<.?:est un cardinal D]
On ne suppose pas que ZFCk 3~z:(ps(z). Par exemple, il existe des formules qui definissent les cardinaux w, HI, Rx, I,, In,, le premier cardinal inaccessible, le premier cardinal mesurable, etc. Les cardinaux definissables peuvent etre utilises comme des constantes. C’est-a-dire que si g(z) est une formule et k designe le cardinal defini par p(z), nous tcrivons CT(~) comme abreviation pour la formule Vz (p’(z) + g(z).
5. Schema
de Saturation.
Pour chaque cardinal definissable
k, la propriete de k-saturation
VF c I [IFI < k A VFo c F (F.
suivante est un axiome :
finie -+ nF, # 0)]
--) nF # 0
Soit k un cardinal definissable. Les proprittes suivantes peuvent Ctre demontrees directement dans *ZFC : (i) Proprie’te’ de Standardisation : VA 3B E S B = {b E S : *b E A}. (ii) Si A c ‘X a un cardinal IAl < k, alors A C B pour un certain B g ‘X *fini. (iii) Proprikte’ de Compre’hension : Si f : A -+ B est une fonction oti B E I et A C ‘X a un cardinal IAl < k, alors il existe une fonction interne F : A’ 4 B avec A C A’ C *X et F(a) = f(a) pour chaque a E A. Si A = {*z : z E X} on peut prendre A’ = *X. En ce qui concerne la structure de l’univers de *ZFC, on a : THEOREME. (i) L’univers des ensembEes est V = U,,,,V,(*V,(rZr)). (ii) Pour chaque ordinal a, *v?(0) est isomorphe ri une ultrapuissance &mite V,(0)6lF 2. (iii) On peut dkjinir we fonction rang R sur 1‘univers V d valeurs totalement ordonne’es qui satisfait : = *p(s) pour chaque s E S, (a) R(z) = Sup{R(:c’) + 1 ., z’ E z} pour chaque z # 0 ; (b) R(*s) 02 p est la fonction rang usuelle & valeurs bien ordonne’es; (c) R(t) = < pour chaque ordinal interne [ E Ua~~~\r*cc.
964
*ZFC: an axiomatic *approach to nonstandard methods
(iv) Chaque structure binaire non cyclique (A, E), ou ]A] < k pour un certain cardinal definissable, est isomorphe a un modele nature1 (B: E) 3. Pour l’usage de la saturation dans *ZFC, le rtsultat suivant est utile. META-THBORBME. - Soit P(x) une ~-form&e ayant .T pour unique variable libre. Supposonsque ‘ZFC k GVcardinalk k-saturation + [Vx E 5’ 1x1< k + P(x)] S. Alors, ZFC t- Vx P(x). Maintenant, nous nous occupons de la puissance relative de *ZFC par rapport a ZFC. Si M est un modele de ZFC-, nous notons (WF) ” le sous-modele de It4 dont l’univers est {x E A4 : A4 + << x est bien fond6 B}. 11est bien connu que (WF)“’ b ZFC. On a le THEOR~ME PRINCIPAL. - Si N /= ZFC, alors N est un sow-modele e’lementairede (WF)“’ pour un certain M b *ZFC. Une des consequencesdu theoreme principal est qu’on peut donner une traduction de E-formules dans les L*-formules. COROLLAIRE. - ZFC est3delement interpretable duns * ZFC en relativisant les quantificateurs a la classe des ensemblesbien fond&. C’est-a-dire, pour chaque E-enonce u,
En particulier, *ZFC et ZFC sont equiconsistantes.
The usual axiomatic approachesto nonstandard analysis are not completely satisfactory when dealing with external sets. For instance, in Nelson’s Internal Set Theory (see [7]), basic collections such as Loeb measuresor nonstandard hulls cannot be elements of the universe. On the other hand, the most popular superstructure approach is not an axiomatic one and only sets of finite rank in the cumulative hierarchy are considered. We overcome these difficulties by introducing an axiomatic nonstandard set theory *ZFC in which the enlarging map * is defined for each well-founded set, not necessarily of finite rank. Every model of *ZFC is a model of ZFC without foundation and the class of its standard elements is a model of ZFC. A strong form of saturation is assumedand the standardization and comprehensivenessproperties are theorems of *ZFC. A consequenceof the main theorem in this Note is that ZFC is faithfully interpretable in *ZFC. In particular, ZFC and *ZFC are equiconsistent. We supposethat the reader knows the basics of nonstandard analysis (we suggest[6] as a reference) and have a basic knowledge of set-theoretic and model-theoretic notions (see [5] and [3] respectively). Let L* be the language which consists of the two binary relation symbols E and * ; *ZFC is the theory of L* having the following five groups of formulas as axioms. 1. ZFCAll axioms of the Zermelo-Fraenkel set theory with choice but without foundation are assumed. Moreover, the Separation and Replacement schemata are considered for all L*-formulas. The class 5’ of all standard sets is defined as the cumulative hierarchy WF = lJaE~,vVn(0) of the well-founded sets 4. 2. * is a function
with domain 5’:
{vxvyvz [*x = y A *x = z + (y = z A x E S)]} A [Vx E s 3y *x = y/l, Although it is only partially defined, we use * as a function symbol.
965
M.
Di
Nasso
The class I of internal sets is defined as the class {y : ?/ E *x for some x E S}. An external set is a set which is not internal. For every E-formula ‘p, denote respectively by (ps and cp’ its relativizations to the class of standard sets and internal sets. 3. Transfer
Principle
schema
* : S -+ 1 is an E-elementary is an axiom: VXI,...
embedding,
>x, ES
i.e. for every E-formula ,x,)
pyc,....
c--)
cp(zl,. . . ~x,),
the following
(pl(*xl,...,*x,)
Notice that, contrarily to the usual superstructure approach, the transfer principle is not limited to the bounded quantifier formulas 4. The universe is well-founded over the internal sets, i.e. there are no descending chains of external sets. VX#~ Let p(s) be an E-formula if the following holds: ZFC-
[znI=0
+(~zJEzE~J~x=~)]
where z is the only free variable. I- VxVy [(p’(x)
A (p’(y)
We say that P(X) defines a cardinal
--) z = y A “2 is a cardinal”]
We do not require that ZFC k 3x (p’(x). For instance, there cardinals w, N1, Nz, &,, IN1, the first inaccessible cardinal, the first The definable cardinals can be used as constants. Namely, if O(X) cardinal defined by p(x), we shall write a( k-) as a short-hand for We now turn to the last group of axioms. 5. Saturation
are formulas which define the measurable cardinal, and so on. is a formula and Ic denotes the the formula Vx (ps( x) --) cr( x).
schema
For every definable cardinal k, the following
k-saturation
VF c I [IFI < /CA VFO c F (FO finite
property
-+ nF, # 0)]
is an axiom: -+ nF # 0.
Let k be any definable cardinal. The following facts can be straightforwardly proved from *ZFC: (i) Standardization Property: V.4 3B E S B = { 13E S : “b E A}. (ii) If A C *X has cardinality IAl < k-, then A 2 B for some *finite B C *X. (iii) Comprehensiveness Property: If f : A --+ B is a function, where B E 1 and A g “X has cardinality IAl < k, then there exists an internal function F : A’ -+ B with A 2 A’ C *X and F(a) = f(u) for every a E A. If A = {*IC : :c E X} one can take A’ = *X. The above properties are fundamental in the practice of nonstandard analysis. We remark that in the various axiomatic approaches, the standardization property is usually postulated rather than proved. With regard to the structure of the universe of *ZFC, the following holds. THEOREM
(i) The universe of sets is the union V = UaE~a\~l~(*Va (0)). (ii) For each ordinal Q, *V, (0) IS isomorphic to a limit ultrapower Va(0)6 IF 5. (iii) A totally-ordered-valued rank function R on the universe V is dejinable which satisfies: (a) R(z) = Sup{R(z’) + 1 : x’ E x} for every x # 0; (b) R( *s) = *p(s) for every s E S where p is the usual well-founded rank function; (c) R(c) = cfor every internal-ordinal < E UnE~Lv*a. (iv) Every acyclic binary structure (A, E), where IAl < k f or somedefinable cardinal k, is isomorphic to a natural model (B, E) 6.
966
*ZFC:
an axiomatic
*approach
to nonstandard
methods
Two fundamental consequences of the axiom of foundation are the existence of a well-orderedvalued rank function and Mostowski’s collapse theorem. (iii) and (iv) show that similar results can be saved in our non-well-founded context. When dealing with saturation in *ZFC, a useful tool is the following: META-THEOREM. - Let P(x) be any E-formula having exactly one free variable and suppose that *ZFC 1 “Vcardinal k k-saturation + [Yx E 5’ 1x1< k --+ P’(x)]“. Then ZFC !- Vx P(x). Let us turn to the relative strenght of *ZFC with respect to ZFC. If A4 is any model of ZFC-, denote by (WF)‘l the submodel of 1M whose universe is {x E A4 : A4 b “x is well-founded”}. It is well-known that (WF)‘%t /= ZFC. The following holds: MAIN THEOREM. - If N k ZFC, then N is an elementary submodel of ( WF)“’ for some h/r /= ‘ZFC. The basic idea in the proof of the Main theorem is that the constructions given in [l] and [4] within an anti-foundational context can be “simulated” in ZFC. In those papers, Boffa’s Superuniversality axiom (see [2]) is used to construct transitive classeswhich are models of weakened versions of *ZFC. As a consequenceof the Main Theorem, a translation of E-formulas into &-formulas can be given by relativizing to the well-founded (standard) sets. COROLLARY. - ZFC is faithfully interpretable in * ZFC by relativizing quantzj’iersto the class WF of well-founded sets. That is, for every E-sentence o
ZFCta
a
*ZFCtc?“F.
In particular, *ZFC and ZFC are equiconsistent. The results of this Note were presented at the International Applications “, Edinburgh, August II- 17, 1996.
Symposium
on “Nonstandard
Analysis
and its
’ La hierarchic cumulative sur un ensemble quelconque X est definie de la maniere suivante : VO(X) = X; T;,+,(X) = Vu(X) U P(V,(X)); V,(X) = U,,
1997.
References [I] Ballard D. and Hrbacek K., 1992. Standard Foundations for Nonstandard Analysis, J. Symb. Logic, 57, pp. 741-748. [2l Boffa M., 1972.. Forcing et Negation de I’Axiome de Fondement, M&n. Acad. Sci. Brig., Tome XL, no. 7. [3] Chang C.C. and Keisler H.J., 1990. Model Theory, 3rd edition, North-Holland. [4] Di Nasso M.. Pseudo-superstructures as Nonstandard Universes, J. S.wnbolic Logic, to appear. [5] Kunen K., 1980. Set Theory, an Introduction to Independence Proofs, North-Holland. [6] Lindstrtim T., 1988. An Invitation to Nonstandard Analysis, in Nonstandard Ana(wis and ifs Applicarions (N. Cutland,
ed.), Cambridge University Press, pp. l- 105. [7] Nelson
E., 1977. Internal
Set Theory:
a New Approach
to Nonstandard
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pp. 1165-l 198.
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