Leibnizs Principles and Topological Extensions

Leibniz’s Principles and T op ological Extensions Marco F orti Dip art. di Matematic a Applic ata “U. Dini”, Universit` a di Pisa, Italy. forti@ dma.unipi. i t Abstract Three philosophical p rinciples are o ften quoted in connection with Leib- niz: “ob j ects sharing the same prop erties are the same ob ject”, “ e v erything can p ossibly exist, unless it yields contradiction”, “t he ideal elemen t s cor- rectly determine t he real things”. Here we giv e a precise f orm ulation of these principle s within the frame- w ork of the T opo logical Extensions of [8], structures that g e neralize at once compactifications, comple tions, and nonstandard extensions. In this top olog- ical con text, the ab o ve Leibniz ’s principles a ppear as a prop ert y of separation, a prop ert y of compactness, and a pro p ert y of a nalytic it y , resp e ctiv ely . Abiding b y this in terpretation, w e obtain the someho w surprising conclu- sion tha t these Leibnz’s principle s c an b e fulfil le d in p airs, but not al l thr e e to gether . Keywor ds: top ological extensions, nonstandard mo dels , transfer principle, indiscernibles 2000 MSC: 54D35, 54D 8 0, 03A05, 03H05 , 03E65 In tr oduction Three philosophical principles are often quoted in connection with Leib- niz: Iden tity of indiscernibles “obje cts sharing the same pr op erties ar e the same obje ct” There are nev er in nature t w o b eings whic h are p erfectly iden tical to eac h other, and in whic h it is imp ossibile to find an y inte rnal difference . . . ( Monadolo gy ) Pr eprint submitte d to T op olo gy and its Applic ations Novemb er 10, 2018 Tw o indiscernible individu als cannot ex ist. [. . . ] T o put tw o indiscernible things is to put the same thing under tw o names. ( F ourth letter to Clarke ) [...] dans les chose s sensibles on n’en trouv e jamais deux indis- cernables [...] 1 ( Fifth letter to Clarke , [ ? ], p. 132) P ossibility as consistency “everything c an p ossi b l y ex i s t , unle ss it yields c ontr adiction ” - Imp ossible is what yields an absurdit y . - Possible is not imp oss ible. - Ne c essary is that, whose o pposite is imp ossible. - Contingent is what is not necessary . ( unpublishe d , 1680 ca.) [. . . ] nothing is a bs o lute ly necessary , when the con trary is p ossi- ble. [. . . ] Absolutely necessary is [. . . ] that whose o pposite yields a con tra- diction. ( Dialo gue b etwe en The ofi l e and Polydor e ) T ransfer principle “the ide al elements c orr e ctly determin e the r e al things” P erhaps the infinite and in finitely smal l [n um b ers] that we con- ceiv e a r e ima ginary, neverthele ss [they are] suitable to determin e the r e al things , as usually do the imaginar y ro ots. They are sit- uated in the ideal regions, from where t hing s are ruled b y la ws, ev en though they do not lie in the pa rt of matter. ( L etter to Jo h ann Bernoul li , 1698) 1 among sensible things o ne nev er finds t wo [that are] indiscernible. 2 In this paper, w e try and giv e a precise mathematical formulation of these principles in the con text of the T op olo gic al Extensio ns of [8], structures whic h generalize at once compactifications, completions, and nonstandard mo dels (see also [4]). Giv en a set M , a top olo gic al extension o f M is a T 1 space ∗ M , where M is a dense s ubspace and ev ery function f : M → M has a distingushe d con tinuous extension ∗ f : ∗ M → ∗ M that preserv es comp ositions and lo cal iden tities. The op erator ∗ can b e appropriately defined so as to pro vide also all prop erties P and relations R with distinguished extensions ∗ P , ∗ R to ∗ M . F ollo wing the basic idea tha t the elemen t s of the [“standard”] set M are the “real o b jects” of t he “actual w orld”, whereas the extens ion ∗ M con tains also the “ideal elemen ts” o f all “p oss ible w orlds”, a n appropriate in t erpreta- tion of the L eibniz’s principles in t he con text of top ological extensions migh t b e Ind d i ff er ent ele m ents of ∗ M ar e sep ar ate d by the extensio n ∗ P of some pr op- erty P of M ; P oss if the extensions of a family F of pr op erties of M ar e not simultane- ously satisfie d in ∗ M , then ther e ar e finitely man y pr op erties of F that ar e no t simultane ously satisfie d in M ; T ran a statemen t involving eleme nts, pr op e rties and r elations of M is true if and only if the c orr e s p on d ing statement ab out their extensio n s is true in ∗ M . 2 W e shall see that the extended prop erties corresp ond exactly to the clop en subsets of ∗ M , a nd so the ab o ve principles turn o ut to b e respectiv ely a prop ert y of sep ar ation , of c omp actness , and of analyticity of the top ology of ∗ M . Grounding on results of [8, 4], we obtain the someho w surprising consequenc e tha t the Leibnz’s principles c an b e fulfil l e d in p airs, but not al l thr e e to gether . The pap er is organized as fo llo ws. In Section 1, w e giv e t he precise definition of top olo gic al extension and we recall the main prop erties stated in [8]. In particular, in Subsection 1.1, w e in tro duce the S -top olo gy and w e 2 Clearly one has to admit only “first orde r ” sta temen ts, so as to av oid trivial incon- sistencies. 3 determine its connection with the principles Ind and P o ss . In Subsection 1.2 w e study the c an o nic al map from a t o pological extension of X in to the Stone- ˇ Cec h compactification β X of the discrete space X : we obt a in in t er alia that the Stone- ˇ Cec h compactification itself is essen tally the unique top ological extension that satisfies b oth principles Ind and P oss . In Section 2, w e presen t tw o simple pro p erties that c haracterize all those top ological extensions ( hyp er extensions ) tha t satisfy the transfer principle T ran . A complete c hara c terization of the h yp erex tensions satisfying also Ind is deriv ed at once, and with it the imp oss ibility o f satisfying simultane ously the thr e e L eibniz’s principle s . In Subsection 2.1 w e sho w ho w to top ologize arbitrary nonstandard mo dels, so as to obtain also top ological extensions where the principles P oss and T ran hold together. A few concluding remarks and op en questions, in particular the set- theoretic problems originated by t he com bination of Ind with T ran , can b e found in the final Section 3. In general, we refer t o [10] for a ll the t o pological notions and facts used in this pap er, and to [6] for definitions and facts concerning ultrap o w ers, ultrafilters, and nonstandard mo dels. General references for nonstandard Analysis could b e [13, 1]; sp e cifical for our “elemen tary” approac h is [4]. The author is grat eful to Vieri Benci, Mauro D i Nasso and Massimo Mugnai for useful discussions and suggestions. 1. T op ologica l extensions and the Iden tity of Indiscernibles In this section w e review the main features of t he top olo gic al ex tens i ons in tro duced in the pap er [8]; these structures natura lly accomoda t e , within a general unified framew ork, b oth Stone- ˇ C e ch c om p actific ations of dis cr ete sp ac es and nonstanda r d mo dels (see also [4]). The most imp ortan t ch ar- acteristic shared b y compactifications and completions in top ology , and b y nonstandard mo dels of ana ly sis is the existence of a distinguished extension ∗ f : ∗ X → ∗ X for eac h function f : X → X . Given an arbitrary set X , w e con- sider here a top ological extension of X as a sort of “top ological completion” ∗ X , where the “ ∗ ” op erator pro vides a distinguishe d c ontinuous extension of eac h f un ction f : X → X . Definition 1.1. The T 1 top ological space ∗ X is a top olo gic al extension of X if X is a discr ete dense subspace of ∗ X , and a distinguishe d c ontinuous 4 extension ∗ f : ∗ X → ∗ X is asso ciated to eac h f unction f : X → X , so as to satisfy the following conditions: ( c ) ∗ g ◦ ∗ f = ∗ ( g ◦ f ) for al l f , g : X → X , and ( i ) if f ( x ) = x for al l x ∈ A ⊆ X , then ∗ f ( ξ ) = ξ for al l ξ ∈ A . Since a finite set cannot ha v e nontrivial top ological extensions, w e are in terested only in infinite sets, and fo r conv enience w e stipulate tha t N ⊆ X . It is easily seen that the op erator ∗ preserv es also constant and c har a c ter- istic functions (see Lemma 1.2 of [8]) . So, b y using the characteristic func- tions, the op erator ∗ prov ides also an extension ∗ A for ev ery subset A ⊆ X , whic h turns out to b e a clop en sup erset of A , and actually the closur e A o f A in ∗ X . Notice that , if the top ological extens ion ∗ X of X is Hausdorff, then ∗ f is the unique contin uous extension of f , b ecaus e X is dense . Therefore prop er- ties ( c ) a nd ( i ) a re auto matically satisfied (see [3], where Hausdorff top ologi- cal extensions ha ve b een in t r oduced and studied). Ho w ev er considering only Hausdorff spaces w ould ha v e turned out to o restrictiv e: we shall see b elo w that the Hausdorff top ological extensions of X are particular subspaces of the Stone- ˇ Cec h compactification β X o f the discrete space X that, in general, are not nonstandard extensions. In fact, the existence of Hausdorff nonstandard extensions, although consisten t, has not y et been prov ed in ZF C alone (see [8, 9]) and Section 3 b elo w. These are the reasons wh y w e only require that top ological extensions b e T 1 spaces. 1.1. The S -top olo gy In o r der to study our ve rsions of the Leibniz’s principle s for top ological extensions, it is useful to consider on ∗ X the so called S -top olo gy , 3 i.e. the top ology generated b y the (clop en) sets ∗ A = A for A ⊆ X . The S -to p ology is ob viously c o arser than or e qual to t he orig ina l to pology of ∗ X , a nd we can c haracterize the resp ectiv e se p ar ation pr op erties as in Theorem 1.4 of [8 ]: Theorem 1.2. L et ∗ X b e a top olo gic al extension of X . Then 1. The S -top olo gy of ∗ X is either 0 -d imensional or not T 0 . 3 The S -to p ology (for Standar d topo logy) is a classical notion of nonstandard Analysis, already co nsidered since [15]. 5 2. ∗ X is Hausdorff if and on l y if the S -top olo gy is T 1 , henc e 0 -dimension al. 3. ∗ X is r e gular if and only if the S - top olo gy is the top olo gy of ∗ X (and so ∗ X is 0 -dimensional). Pro of. 1 . The S -top ology has a clop en basis b y definition. In this top ology the closure of a p oin t ξ is M ξ = T ξ ∈ A A . If M ξ = { ξ } for all ξ ∈ ∗ X , then the S - topolo g y is T 1 , hence 0-dimensional. Otherwise let η 6 = ξ b e in M ξ . Then η b elongs to the same clop en sets as ξ , and the S - topolo gy is not T 0 . In fa c t, giv en A ⊆ X , ξ ∈ A implies η ∈ A , b y the choice of η . Similarly ξ / ∈ A implies ξ ∈ X \ A , hence η ∈ X \ A and η / ∈ A . 2 . By p oin t 1, the S -top ology is Hausdorff (in fact 0-dimensional) whenev er it is T 1 . Therefore also the top ology of ∗ X is Hausdorff, b eing finer tha n the S -top ology . F or the conv erse, let U, V b e disjoin t neigh b orho o ds of the p oin ts ξ , η ∈ ∗ X , and put A = U ∩ X , B = V ∩ X . Then ξ ∈ A , η ∈ B , a nd B ∩ A = ∅ . Therefore η / ∈ M ξ , and the S -top ology is T 1 . 3 . The closure of an op en subset U ⊆ ∗ X is the clop en set U ∩ X . There fore an y close d neighbor ho o d of ξ ∈ ∗ X includes a clopen o ne . Since the clop en sets ar e a basis of t he S -top ology , ∗ X can b e regular if and only if its orig ina l top ology is the S -top ology ( a nd so the latter is T 1 , hence 0-dimensional). ✷ No w the pr inciple Ind simply means that the S -top ology of ∗ X is Haus- dorff. On the other hand, the principle Poss states that ev ery proper filter of clop en sets has nonempty in tersection, i.e. that the S -top ology of ∗ X is quasi-compact. 4 So w e ha ve Corollary 1.3. L et ∗ X b e a top olo gic al extension of X . Then 1. the principle Ind holds if and only if ∗ X is Hausdorff; 2. the principle Poss h o lds if and only if the S -top olo gy of ∗ X is quasi- c omp act; 3. b oth principles Ind a nd P oss hold in ∗ X if and only if the S -top olo gy of ∗ X is c omp act. So either ∗ X is c omp act, or it is not r e gular, but b e c omes c omp act by suitably we akening its top olo gy, stil l maintaining al l functions ∗ f c ontinuous. 4 F ollowing [10], we call compact o nly Hausdorff s pa ces. 6 Pro of. W e only need to prov e the last assertion of P oint 3. If ∗ X is reg- ular, then its top ology agrees with the S -top ology . If not, then all functions ∗ f ar e con tin uous also with resp ect to the coarser S -top ology , b ecause the in ve rse images of clop en sets are clop en. ✷ 1.2. The c anonic al map and the principle I n d An y top ological extension of X is canonically mappable into the Stone - ˇ C e ch c omp actific ation β X of the discrete space X . 5 If X is a discrete space, iden tify β X with the se t of all ultrafilters ov er X , endo w ed with t he to pology ha ving as basis {O A | A ⊆ X } , where O A is the set o f all ultrafilters con taining A. So the em b e dding e : X → β X is giv en by the principal ultr a filte r e ( x ) = { A ⊆ X | x ∈ A } , and the uniq ue con tinuous extens ion f : β X → β X o f f : X → X can be defined b y putting f ( U ) = { A ⊆ X | f − 1 ( A ) ∈ U } . Giv en a top ological extension ∗ X of X and a p oin t ξ ∈ ∗ X , put U ξ = { A ⊆ X | ξ ∈ ∗ A } , whic h is an ultrafilter ov er X , and define the c anonic al m ap υ : ∗ X → β X b y υ ( ξ ) = U ξ . Then we can reformulate Theorem 2.1 o f [8] in terms of the Leibniz’s princi- ples Ind and Poss . Namely Theorem 1.4. L et ∗ X b e a top olo gic al extension of X , and let β X b e the Stone- ˇ C e ch c omp actific ation of X . The n 1. The c anonic al map υ : ∗ X → β X is the unique c ontinuous extension to ∗ X of the emb e dding e : X → β X , and υ ◦ ∗ f = f ◦ υ for al l f : X → X. 5 F or v arious definitions and prop erties of the Stone- ˇ Cec h compactification s ee [10]. 7 2. The map υ is inje ctive i f and o n ly if Ind holds in ∗ X . 3. The map υ is surje ctive if an d only if Poss holds in ∗ X . Pro of. 1. F or all x ∈ X , U x is the principal ultrafilter generated by x , hence υ induces the canonical em b edding o f X in to β X . If O A is a basic op en set of β X , then υ − 1 ( O A ) = A , hence υ is con tin uous w.r.t. the S -to pology , and a fortiori w.r.t. the (not coarser) top ology of ∗ X . On the other hand, let a con tinuous map ϕ : ∗ X → β X b e giv en. Since O A is clop en , also ϕ − 1 ( O A ) is clop en and so it is the closure B of some B ⊆ X . If ϕ is the identit y on X , then B ∩ X = A , hence B = A (see Lemma 1.2 of [8]). Therefore all p oin ts of M ξ = { η ∈ ∗ X | ∀ A ⊆ X ( ξ ∈ ∗ A = ⇒ η ∈ ∗ A ) } are mapp ed b y ϕ on to υ ( ξ ), and so υ = ϕ . Moreo ve r, for all ξ ∈ ∗ X , o ne has ξ ∈ A ⇔ ∗ f ( ξ ) ∈ f ( A ), or equiv alen tly A ∈ U ξ ⇔ f ( A ) ∈ U ∗ f ( ξ ) (see Lemma 1.3 of [8 ]); hen ce f ◦ υ = υ ◦ ∗ f , and P oiny 1. is completely prov ed. 2. The map υ is injectiv e if and only if the S - t opolog y is T 1 , and this fact is equiv alent to ∗ X being Hausdorff, b y Theorem 1.2, or to I nd , by Corollary 1.3. Moreov er in this case υ is a homeomorphism w.r.t. the S -top ology , whic h is the same as the top ology of ∗ X if and only if the latter is regular (hence 0-dimensional). 3. The map υ is surjectiv e if a nd only if ev ery maximal filter in the field of all clop en sets o f ∗ X has nonempt y in tersection. This is equiv alen t to ev ery prop er filter having nonempt y inte rsection, whic h in turn is equiv alen t to the S -top ology of ∗ X b eing quasi-compact, i.e. to the principle P oss , by Corollary 1.3. ✷ Notice t ha t the map υ induces a bijection b et w een the basic op en sets O A of β X and the clop en subsets ∗ A o f ∗ X . Therefore υ is op en if and only if ∗ X has the S - topolo g y . Call invariant a subspace Y of ∗ X (resp ectiv ely of β X ) if ∗ f ( ξ ) ∈ Y (resp. f ( ξ ) ∈ Y ) for all f : X → X and all ξ ∈ Y . It is easily seen that an y in v ariant subspace Y of ∗ X is itself a top ological extension of X , and it is mapp ed by υ onto an inv ariant subspace of β X . If ∗ X is homeomorphic to a subspace of β X , then it is 0 -dimens io nal, hence it has the S -top ology , by Theorem 1.2. Con v ersely , if ∗ X has the S -top ology , 8 then υ is injectiv e. Moreov er, for all A ⊆ X , υ ( A ) = O A ∩ υ ( ∗ X ), hence υ is a homeomorphism b et w een ∗ X and its image. On the other ha nd, if ∗ X is Hausdorff but not regular, then υ is injectiv e a nd contin uous, but not op en. Whenev er ∗ X v erifies the principle Ind , the map υ can alw ays b e turned in to a homeomorphism, either by endowing υ ( ∗ X ) with a suitably finer top ol- ogy , o r b y taking on ∗ X the (coarser) S -t opolog y . So a ny suc h extension mak es use of the s a me “function-extendi n g me ch a nism” as the Stone- ˇ Cec h compactification. Moreov er, if a ls o P oss holds, then ∗ X can b e tak en t o b e β X itself, p ossibly endow ed with a suitably finer top ology . More precise ly , the ab o v e discussion pro vides the same c hara c terization of all top ological extensions satisfying the principle Ind that has b een giv en in Corollary 2.2 of [8], namely: Corollary 1.5. A top olo gic al extens i on ∗ X of X satisfies Ind if an d only if the c a n onic al map υ pr ovides a c ontinuous bije ction ( a home om orphism when ∗ X is r e gular ) onto an invariant subsp ac e of β X . Mor e over ∗ X satisfies al s o Poss if a n d only if υ is onto β X . ✷ 2. T op ologica l h yp erextensions and t he T ransfer Pr inciple The T ransfer Principle T r a n is the v ery ground of the usefulness of t he nonstandard metho ds in mathematics. It a llows for obtaining correct results ab out, sa y , the real num b ers b y using ideal elemen t s like actual infinitesimal or infinite num b ers. In fact, both pro p erties ( c ) and ( i ) o f Definition 1.1 are instances of the transfer principle, for they corresp ond to the statemen ts ∀ x ∈ X . f ( g ( x )) = ( f ◦ g )( x ) and ∀ x ∈ A . f ( x ) = x, resp e ctiv ely . So all top ological extensions a lready satisfy se v eral imp ortan t cases of the transfer principle. E.g. , if f is cons tan t, or injective , or surjec- tiv e, then so is ∗ f . More importa nt, w e ha v e already used the fact that the extension of the c hara c teristic function of any subset A ⊆ X is the c harac- teristic function of the closure A of A in ∗ X , th us w e can put ∗ A = A and obtain a Bo olean isomorphism betw een the field P ( X ) o f all subsets of X and the field C ℓ ( ∗ X ) of all clop en subsets of ∗ X (see [8], L emmata 1.2 and 1.3). On the other hand, man y basic case s of the transfer principle may fail, be- cause top ological extensions comprehend, besides nonstandard mo dels , a ls o 9 all inv ariant subspaces of the Stone- ˇ Cec h compactifications o f disc rete spaces. In order to obtain t he f ull principle T r a n , w e p ostulated in [8] t wo additional prop erties, namely Definition 2.1. T he top olo gic al extension ∗ X of X is a h yp erexten sion 6 if ( a ) for al l f , g : X → X f ( x ) 6 = g ( x ) for al l x ∈ X ⇐ ⇒ ∗ f ( ξ ) 6 = ∗ g ( ξ ) for al l ξ ∈ ∗ X ; ( p ) ther e exist p, q : X → X such that for al l ξ , η ∈ ∗ X ther e exists ζ ∈ ∗ X such that ξ = ∗ p ( ζ ) and η = ∗ q ( ζ ) . The prop ert y ( a ), called analyticity in [8], isolates a fundamen tal feature that marks the difference b et w een nonstandar d and or dinary c ontinuous ex- tensions of functions: “d i s joint functions have disjoi n t extensions” . It is obtained by T ran f r o m the statemen t ∀ x ∈ X . f ( x ) 6 = g ( x ), and it can b e view ed as the empty set c ase of a general “principle of pr eservation of e qualizers” : ( e ) { ξ ∈ ⋆ X | ⋆ f ( ξ ) = ⋆ g ( ξ ) } = ⋆ { x ∈ X | f ( x ) = g ( x ) } . The prop ert y ( p ), called c oher enc e in [8], pro vides a sort of “in ternal co ding o f pairs”, useful fo r extending m ultiv ariate f unctions “pa r ame trically”: this p oss ibilit y is essen tial in order to get the full pr inciple T ran , whic h in volv es relations of a ny arit ie s. 7 Notice that the prop ert y ( p ) could seem prima fa cie an illegal instance of the T r ansfer Principle , for it is giv en in a second order fo rm ulation. On t he con trary , a str ong uniform version of that prop ert y can b e obtained b y fixing p, q as the comp ositions of a given bijection δ : X → X × X with the ordinary pro jections π 1 , π 2 : X × X → X , and then applying T ran to the statemen t ∀ x, y ∈ X . ∃ z ∈ X . p ( z ) = x, q ( z ) = y . 6 T op ological h yp erextensions ar e in fact hyp er-exten si ons in the sense of [4] ( i.e. non- standard mo dels), by Theor em 2 .2 b elo w. 7 The r atio o f considering only unary functions lies in the following facts that hold in every topolo gical h yp erextension ∗ X of X (see Sectio n 5 of [8]): - F or al l ξ 1 , . . . , ξ n ∈ ∗ X ther e exist p 1 , . . . , p n : X → X and ζ ∈ ∗ X such that ∗ p i ( ζ ) = ξ i . - If p 1 , . . . , p n , q 1 , . . . , q n : X → X and ξ , η ∈ ∗ X satisfy ∗ p i ( ξ ) = ∗ q i ( η ) , then ∗ ( F ◦ ( p 1 , . . . , p n ))( ξ ) = ∗ ( F ◦ ( q 1 , . . . , q n ))( η ) for al l F : X n → X . It fo llows tha t there is a unique w ay of assigning a n extension ∗ F to ev ery function F : X n → X in suc h a w ay that all compos itions a re preser v ed. By using the characteristic functions in n v a r iables one can assign an extension ∗ R also to all n -ary re la tions R on X . 10 W e shall see b elo w that there are in v ariant subspaces of the Stone- ˇ Cec h compactification β X where ( a ) holds whereas ( p ) fails and vic e v e rs a , as w ell as inv ariant subspaces where b o th f ail or hold. So the prop erties ( a ) a nd ( p ) are indep en den t, also when Ind holds. W e consider v ery remark able the fa c t that the combination of fo u r natur al, simple instanc es of the tr a ns fer principle, like ( c ) , ( i ) , ( a ), a nd ( p ) , give s to top ological h yp erexten sions the strongest T r ansfer Principle T ran . In reason of its imp ortance, w e ha v e already given three differen t pro ofs of this fact in preceding pap ers of ours: a “logical” and a “logico-algebraic” pro of in [8], and a “purely algebraic” pro of in [1 1 ] (see also the surv ey in [4]). So w e state here without pro of the fo llowing theorem: Theorem 2.2. A top olo gic al ex t ension ∗ X of X satisfies the p rin ciple T ran if and only if it is a hyp er extension. ✷ W e a re no w able to characterize all top ological extens ions satisfying b oth principles Ind and T ran . These extensions a r e spaces of ultrafilters, accord- ing to Corolla ry 1.5. So w e use the reform ulation in terms of ultrafilters give n in [8] for the condition ( e ) ab ov e. Call an ultr afilter U on X Hausdorff 8 if, for all f , g : X → X , ( H ) f ( U ) = g ( U ) ⇐ ⇒ { x ∈ X | f ( x ) = g ( x ) } ∈ U . Call dir e cte d a subspace Y of β X where the prop ert y ( p ) holds, i.e. there exist p, q : X → X suc h that for all U , V ∈ Y there exists W ∈ Y such that U = p ( W ) and V = q ( W ). By com bining Theorem 2 .2 with Corolla ry 1.5 w e obtain Theorem 2.3. A top olo gic al extension ∗ X of X satisfies b oth princi p les Ind and T ran if and only i f the c anon ic al map υ is a c ontinuous bije ction b etwe en ∗ X and a dir e cte d invariant subsp ac e of β X that c ontains only Hausdorff ultr a filters. Mor e over υ is a h o me omorph ism if and only if ∗ X has the S -top olo gy, or e quivalently is r e gular. ✷ No w it is easy to sho w that the prop erties ( a ) and ( p ) are indep end en t. 8 The prop ert y ( H ) ha s b een in tro duced in [7] under the na me ( C ). Haus do rff ultr a filters are studied in [9] and [2 ]. 11 F or U ∈ β X let Y U = { f ( U ) | f : X → X } b e the in v arian t subspace generated b y U . Cle arly Y U is directed, so ( p ) holds for all ultrafilters U , whereas ( a ) holds if and only if U is Hausdorff. On the other hand, let U and V b e Hausdorff ultrafilters suc h that neither o f them b elongs to the inv ariant subspace generated b y the other one: then Y U ∪ Y V is an in v ariant subspace where ( a ) holds, but it is no t directed, hence ( p ) fails. W e shall deal in t he final section with the set theoretic strength of the com bination of Ind with T ran . By now w e simply recall that there are plen ty of non-Hausdorff ultrafilters ( e.g. all diag o nal tensor pro ducts U ⊗ U ). Th us w e can easily conclude Corollary 2.4. No top olo gic al extension satisfies at on c e the thr e e L eibniz’s principles Ind , P oss , and T ran . ✷ 2.1. The star top olo gy W e are left with the task of com bining P oss with T ran . T o this aim w e recall that a nonstandard mo del whose S -top ology is quasi-compact is commonly called enlar gemen t . It is well known that ev ery structure has arbitrarily saturated elemen tary extensions (see e.g. [6]), and ob viously a 2 | X | + -saturated extension of X is a n enlarg e men t (see e.g. [1] or [4]). So, if w e can top ologize ev ery nonstandard extens ion of X in suc h a w a y tha t all functions ∗ f b ecome con tin uous, then w e get a lot o f top ological h yp erexten- sions satisfying P oss . This task ha s been already accomplished in [8], where the the c o arsest suc h top ology is defined. Ev ery top ological extension ∗ X should b e a T 1 space, so all sets of the form E ( f , η ) = { ξ ∈ ∗ X | ∗ f ( ξ ) = η } , for f : X → X and η ∈ ∗ X , should b e closed in ∗ X . The (arbitrary) in tersections o f finite unions of suc h sets are the close d sets of a top ology , called the Star top olo gy , whic h is b y construction the c o arsest T 1 top olo gy on ∗ X that makes al l functions ∗ f c ontinuous . When ∗ X is a nonstandard extension of X , the four defining prop erties ( c ) , ( i ) , ( a ), and ( p ) of top ological hyperextensions a r e fulfilled b y h yp othes is. So one has only to pro ve that X is dense in ∗ X in order to obtain Theorem 2.5 (Theorem 3.2 of [8 ]) . A ny nonstandar d extens i o n ∗ X of X , when e quipp e d with the Star top olo gy, b e c omes a top olo gic al hyp er extension of X . C o nversely, any top olo gic al h yp er extension ∗ X of X is a nonstandar d extension, p ossibly endowe d with a top olo gy finer than the Star top olo gy. 12 Pro of. W e hav e only to pr ov e t hat X is dense. If X ⊆ S 1 ≤ i ≤ n E ( f i , η i ), then w e ma y consider w.l.o.g. o nly those com- p onen ts with η i ∈ X , b ec ause eac h ∗ f i maps an y p oin t x ∈ X to the p oin t f i ( x ) ∈ X . Hence the T r ansfer Principle o f the nonstandar d extensions may b e applied to the statemen t ∀ x ∈ X ( f 1 ( x ) = η 1 ∨ . . . ∨ f n ( x ) = η n ) , th us pro ducing ∀ ξ ∈ ∗ X ( ∗ f 1 ( ξ ) = η 1 ∨ . . . ∨ ∗ f n ( ξ ) = η n ) . So the whole space ∗ X is included in S 1 ≤ i ≤ n E ( f i , η i ), and X is dense. ✷ So, in order t o get top ological extensions satisfying b oth principles Poss and T ran , w e ha v e only to put the s tar top olo gy on an y nonstandar d en- lar gement of X . 3. Final remarks and op en questions W e ha ve seen that (at least) one of the three principles that w e ha v e in ve stigated has to b e left o ut. The most reasonable c hoice seems to b e that of dropping Ind . In fa c t, ev en if one neglects the set theoretic problems that will b e outlined b elo w, one should pay at t ention to Leibniz himself. [...] cette supp osition de deux indiscernables [...] pa r o is t p ossible en termes abstraits, mais elle n’est p oin t compatible av ec l’ordre des c hoses [. . . ] Quand je nie qu’il y ait [. . . ] deux corps indiscernables, je ne dis p oin t qu’il soit imp ossible absolumen t d’en p oser, mais que c’est une c hose con trair e la sagesse divine [. . . ] Les parties du temps ou du lieu [...] son t des c hoses ideales, ainsi elles se rassem blent parfaitemen t comme deux unit´ es abstraites. Mais il n’est pas de m ˆ eme de deux Uns concrets [...] c’est ` a dire v eritablemen t a c tuels. 13 Je ne dis pas que deux p oin ts de l’Espace son t un meme po int, n y que deux instans du temps son t un meme instant comme il sem ble qu’on m’impute [. . . ] 9 ( Fifth letter to Clarke , [14], pp. 131-1 35) It app ears t ha t Leibniz considered the iden tity o f indiscernibles as a “ph ysical” rather than a “lo g ic al” principle: it is actually t r ue , but its nega- tion is non- c on tra dic tory in principle, so it c ould fail in som e p ossible world . Moreo ve r only “prop erties of the real w orld” M are considered in all these principle: so it seem s natural, and not absurd, to assume that ob jects in- discernible b y these “real” prop erties ma y b e separated b y some abstract, “ideal” prop ert y of ∗ M . On this ground w e finally decide to call L eibnizi an a top ological extens ion that satisfies b oth Poss and T ran , and so necessarily not Ind . Th us the existence of plen t y of L eibnizian extensions is g r a n ted b y the final results of Section 2, without an y need of supplemen tary set theoretic h yp otheses. 3.1. Existenc e of Hausdorff extensions As sho wn by Theorem 2.3 , com bining Ind with T ran requires special ultrafilters, named Hausdorff in Section 2. Despite the apparen t we ak eness of their defining prop ert y ( H ), whic h is actually true whenev er an y of the in volv ed functions is injectiv e (or constan t), not mu c h is know n ab out Haus- dorff ultrafilters. On c ountable sets, t he prop ert y ( H ) is satisfied b y sele ctive ultrafilters as w ell a s b y pr o ducts of p airwise noniso morphic sele ctive ultrafilters (see [9]), but their existence in pure ZFC is still unprov ed. How ev er any h yp othesis pro viding infinitely many nonisomorphic selectiv e ultrafilters ov er N , lik e the Con tinuum Hyp othesis CH or Martin Axiom MA , provide s infinitely man y non-isomorphic hy p erex tensions of N that satisfy Ind . 9 . . . this suppositio n of tw o indiscernibles . . . seems abstractly po ssible, but it is incom- patible w ith the order of things . . . When I deny that there are . . . tw o indiscer nible bo dies, I do not say that [this existence] is abso lut ely imp ossible to assume, but that it is a thing contrary to Divine Wisdom . . . The parts of time or place . . . are ideal things, so they perfectly resemble like tw o abstract unities. But it is not s o with tw o concr ete Ones,. . . that is truly actual [things]. I don’t say that tw o p oin ts of Spa ce a re one sa me p oin t, neither that tw o instants of time are one same ins ta n t as it seems that o ne imputes to me . . . 14 On unc ountable sets the situation is highly pro ble matic: it is pro ved in [9] that Hausdorff ultrafilters o n sets of size not less than u cannot b e r e gular . 10 In particular, the existence of a h yp erextens ion satisfying Ind with uniform ultrafilters, ev en on R , w o uld imply that of inner mo dels with measurable cardinals. ( T o b e sure, suc h ultrafilters ha ve b een obtained only b y m uch stronger hy p othese s, see [12]). Be it as it may , as far as w e do not abide ZF C a s our foundational theory , we c annot pr ove that hyp er extensions without indisc ernible s exist at al l . 3.2. Some op en questions W e conclude this pap er with a few op en questions that inv olv e sp ec ial ultrafilters, and so should b e of indep ende n t set theoretic in t e rest. 1. Is t he existence of to p ological h yp erextensions of N without indis- cernibles pro v able in ZFC , or at least deriv able from set-theoretic h y- p otheses w eake r than t ho se pro viding selectiv e ultrafilters? E.g . from x = c , where x is a cardinal inv ariant of the contin uum not dominated b y cov ( B )? 2. Is it consisten t with ZF C that there are nonstandard real lines ∗ R with- out indiscernibles where all ultrafilters are uniform? 3. Is the existence o f coun tably compact h yp erextensions consisten t with ZF C ? (These extensions w ould b e of great in terest, b ecause they w ould v erify Ind , T ran , and the w eak ened v ersion of P oss that considers only se quenc es of prop erties.) References [1] L.O. Arker yd, N.J. Cutland, C.W. He nson (eds.) - Nonstandar d A nalysis - T he ory and Applic ations . NA TO ASI Series C 493 , Klu we r A.P ., Dordrec ht 1997 . [2] T. Bar toszynski, S. 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