The notion of Infinity within the Zermelo system and its relation to the Axiom of Countable Choice 1. Abstract. SD is strictly weaker than

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1 The otio of Ifiity withi the Zerelo syste ad its relatio to the Axio of Coutable Choice 1 Chailos George Departet of Coputer Sciece-Divisio of Matheatics Uiversity of Nicosia, Cyprus e-ail: chailosg@uicaccy Abstract I this article we cosider alterative defiitios-descriptios of a set beig Ifiite withi the priitive Axioatic Syste of Zerelo, Z We prove that i this syste the defiitios of sets beig Dedekid Ifiite, Cator Ifiite ad Cardial ifiite are equivalet each other Additioally, we show that assuig the Axio of Coutable Choice, AC, these defiitios are also equivalet to the defiitio of a set beig Stadard Ifiite, that is, of ot beig fiite Furtherore, we cosider the relatio of AC (ad soe of its special cases) with the stateet SD A set is Stadard Ifiite if ad oly if it is Dedekid Ifiite Aog other results we show that the syste Z SD is strictly weaker tha Z AC 1 Itroductio I this sectio we briefly preset the ais of this article ad we provide a overview of its ai results Furtherore, we provide the ecessary defiitios ad stateets that appear i the developet of our arguets The cocept of fiiteess, defied as doe via atural ubers, is categorical, thus ot probleatic If, however, the cocept of beig fiite is cosidered to be ore fudaetal tha that of a uber, as strogly advocated, eg, by Frege ad Dedekid (where atural 1 Matheatics Subject Classificatio 21: 3E3, 3C62, 3A5 Keywords: Axioatic Set Theory ad its Fragets, Foudatio of Matheatics, Ifiity, Philosophy of Matheatics 1

2 ubers are defied as the cardials of fiite sets) probles arise First of all the cocept of beig fiite loses its absoluteess Secodly, how should fiiteess be defied? Dual probles arise i relatio to the cocept of beig ifiite There are several alterative differet defiitios-descriptios of a set beig ifiite; the oldest oe is due to Dedekid (see [3]) I our presetatio we cosider Dedekid Ifiite ( DIf ), Cator Ifiite (CIf ) ad Cardial ifiite ( W ) alterative defiitios of a set beig ifiite These defiitios ad the Stadard oe of beig Ifiite ( SIf ) are all equivalet to each other withi the axioatic syste ZFC which icludes the Axio of Choice, AC, i full stregth (for details we refer to the classical book i Set Theory by Jech [8]) I this paper we show that this is also true eve i the weaker syste Z AC that it does ot iclude the Axio of Replaceet, the Axio of Foudatio ad i which the Axio of Choice (i full stregth) is replaced by the Axio of Coutable Choice, AC Additioally we show that if we drop AC, the alterative defiitios DIf, CIf, W, are all equivalet each other i Zerelo syste Z 2, but they are ot equivalet to SIf Furtherore, we ivestigate how the stregth of Z AC is affected if we substitute AC with the stateet SD : A set is Stadard Ifiite iff it is Dedekid Ifiite, that is, if there is a oe to oe correspodece oto oe of its proper subsets 3 I this lie of thought, we show that assuig the cosistecy of Z ad usig results fro Set Theory ad Model Theory, the syste Z SD is strictly weaker tha Z AC Thus, i atheatical areas that avoid ay versio of the Axio of Choice we could cosider Z SD istead of Z AC Here it is worth etioig that K Gödel [4] ad P Cohe [1] showed the idepedece of the Axio of Choice fro the syste ZF ad hece fro Z 4 Particularly, assuig ZF is cosistet, K Gödel showed that the egatio of the Axio of Choice is ot a theore of ZF by costructig a ier odel (the costructible uiverse) which satisfies ZFC ZF AC ; thus, showig that ZFC is cosistet Assuig ZF is cosistet, P Cohe eployed the techique of forcig, developed for this purpose, to show that the Axio of Choice ( AC ) itself is ot a theore of ZF by costructig a uch ore coplex odel which satisfies ZF AC ( ZF with the egatio of Axio of Choice added as axio) ad 2 See Sectio 2 for the Axios of the origial Zerelo syste Z 3 See Chailos [2] where a logical paradox is resolved withi Z SD 4 The Axio of Choice is ot the oly sigificat stateet which is idepedet of ZF For exaple, the Geeralized Cotiuu Hypothesis ( GCH ) is ot oly idepedet of ZF, but also idepedet of ZFC However, ZF plus GCH iplies AC, akig GCH a strictly stroger clai tha AC, eve though they are both idepedet of ZF (for details see Jech[8]) 2

3 thus showig that ZFAC is cosistet Together these results establish that the Axio of Choice is logically idepedet fro ZF It is worth otig that the assuptio that ZF is cosistet is harless because addig aother axio to a already icosistet syste caot ake the situatio worse Because of idepedece, the decisio whether to use (ay versio of) the Axio of Choice (or its egatio) i a proof caot be ade by appeal to other axios of set theory The decisio ust be ade o other grouds, soe of which are discussed i the sequel For the followig recall that two sets XY, are called equipotet if ad oly if there exists a bijectio fro X toy ad we also write card card Y We give the foral stadard defiitio of a set to be fiite Defiitio 11 (Fiite Set) A set X is fiite if (ad oly if) there is a atural uber such that there is a oe to oe correspodece betwee X ad Thus, X is fiite iff it is equipotet to soe Defiitios 12 (a) We set the least iductive set guarateed by the Axio of Ifiity (see sectio 2) (b) A set X is coutable if ad oly if there is a bijectio fro X to, thus if X is equipotet to Observe that accordig to the above defiitio is the least coutable ordial The cardiality of ad hece by 12 (b) of ay coutable set X is deoted by Thus, card ( X ) Defiitios 13 (Various defiitios of Ifiite Sets) 1 SIf : A set X is Stadard Ifiite iff it is ot Fiite (as i Defiitio 11) 2 W : A set X is Cardial Ifiite iff it has cardiality at least, card ( X ) 3 CIf : A set X is Cator Ifiite iff it cotais a coutable subset 4 DIf : A set X is Dedekid Ifiite iff it cotais a proper subset B X equipotet to it Hece, card ( B) card ( X ) The followig stateets shall be used extesively i the developet of our ai arguets 3

4 Stateets 14 1 SD : If a set is Stadard Ifiite, the it is also Dedekid Ifiite 2 SC : If a set is Stadard Ifiite the it is also Cator Ifiite 3 SW : If a set is Stadard Ifiite the it is also Cardial Ifiit Here we suarize our ai results (a) I Theore 31 we show that i Z AC the differet defiitios of Ifiite sets, SIf, DIf, CIf, W are equivalet each other (b) I Theore 35, which its proof is a variatio of the proof of Theore 31, we show that i Zerelo syste Z (without ay versio of the Axio of Choice) the differet defiitios of Ifiite sets DIf, CIf, W are equivalet each other Heceforth, the Axioatic Systes Z Z the defiitios SD, Z SC ad Z SW are equivalet each other It is worth to ote that i DIf, CIf, W are ot equivalet to SIf, sice there is a odel of Z such that there exists a Stadard Ifiite set that is ot Dedekid Ifiite Such a odel is the Cohe s First Model A4 (see [6]) (c) Lastly, we exaie the relatio of Z Axio of Coutable Choice o Fiite Sets, result (Theore 38) we show that Z equivalet systes Z SC, Z SW SD with Z AC fi AC, is a theore of Z SD At first we show that the I our ai SD (ad hece, accordig to Theore 35, ay of its ) is strictly weaker tha Z AC Specifically, assuig cosistecy of Z, we show that there is a odel of Z i which all SD, SC, SW hold, but AC fails Fro (b), (c), as above, we coclude that sice the equivalet defiitios of Ifiite sets DIf, CIf, W i Z are ore ituitively obvious tha AC, ay proof that adopts ay of these stateets ad avoids the use of AC is preferable This is iportat i ay areas of Matheatics, such as Foudatio of Matheatics, Philosophy of Matheatics, as well as i Set Theory that avoids the Axio of Choice We ote that the syste Z cosists of 4

5 costructive axios 5 ad it is idispesable fro ay set theoretical developet that does ot ivolve atos This is aother reaso that ay of the systes Z Z SW is preferable copared to Z AC SD, Z SC, This is of special iterest i discrete structures sice i Z without assuig ay versio of the Axio of Choice the followig is true Lea 15: Coutable sets have the property of beig Dedekid Ifiite (o use of AC ) Proof: If X is a coutable set, by defiitio 12(b), ii Y X a X Take ow the set a \{ } which is a proper subset of X with cardiality To see that X is Dedekid Ifiite cosider the fuctio h: X Y X, where h( a i ) a i 1 for every i Clearly, h is oe to oe, sice if ai a, h( a ) a a h( a ) i i1 j1 j I additio, h is also oto, j sice Rage( h) Y Thus, h: X Y X is a bijectio fro X oto a proper subset of it, ad thus X is a Dedekid Ifiite set 2 The Set Theoretical Fraework Here we preset ad develop the ecessary set theoretical fraework required for provig the ai theores of the paper that are preseted i sectio 3 The Axios of Z AC Suppose that ( x1, x2, x ) is a place well-fored forula (wff) i the foral laguage of set theory L, which is a first order laguage that i additio to the usual sybol of equality it ivolves a sybol for a biary predicate called ebership 1 Axio of Extesioality: If two sets have the sae uber of eleets, the they are idetical xy z ( z x z y) x y This axio does ot tell us how we costruct sets, but it tells what sets are: A set is deteried by its eleets (ad it is atoless) The Axios 2-7 are all costructive axios 2 Null Set Axio: There is a epty set, oe which cotais o eleet xy y x 5 Apart fro the Axio of Extesioality which tells us what sets are (see sectio 2) 5

6 Usig the Axio of Extesioality oe ca easily see that the epty set is uique We write for the epty set 3 Pair Set Axio: If a ad b are sets, the there is a set a whose oly eleet is a ad there is a set ab, whose oly eleets are a ad b xy zw w z w x w y 4 Uio Set Axio: If a is a set, the there is a set a, the uio of all the eleets of a, whose eleets are all the eleets of eleets of a xyz z y w w x z w 5 Axio of Ifiity: There is a set which has the epty set,, as a eleet ad which is such that if a is a eleet of it, the a set is called a iductive set: a, a (or a x x y y x z z x w wz wy w y a ) is also a eleet of it Such Cosequetly, this axio guaratees the existece of a set of the followig for:,,,,,,,, 6 Power Set Axio: For ay set x there exists a set y P( x), the set of all subsets of x That is, xyu u y z z u z x 7 Axio Schea of (restrictive)coprehesio: If a is a set ad ( x, u1, u2, u ) is a wff i L, where the variable x is free ad u1, u2, u are paraeters (i a odel of set theory fro the axios stated up to ow), the there exist a set b whose eleets are those eleets of a that satisfy x y z z y z x z u1 u2 u (,,, ) What this essetially eas is that for ay set a ad ay wff Hx with oe free variable x ad with paraeters i a odel of set theory (by the axios defied as of ow), there exist a set, uique by the Axio of extesioality, whose eleets are precisely those eleets of a that satisfy H We deote this set byb x a : H[ x] Hece, this axio yields to 6

7 subsets of a give set Therefore, as a cosequece of it, if f : a b is a fuctio with doai a, ad codoai b, the Rage( f ) is a well-defied set that is a subset of b The above seve axios costitute the syste Z (Zerelo) 8 Axio of Coutable Choice: AC : Suppose b is ay set ad P b ay biary relatio betwee atural ubers ad ebers of b, the: b y b P(, y) f : b P(, f ( ) Essetially, the above axio states that every Coutable Faily of oepty sets has a choice fuctio That is, if A F= is a coutable faily of oepty sets, the there is i i f : F F such that f ( A i ) A i for every A i F I cotrast to the full Axio of Choice ( AC ) that deads the existece of choice fuctios f : F failies of oepty sets, the Axio of Coutable Choice, F for arbitrary AC, justifies oly a sequece of idepedet choices fro arbitrary coutable failies of oepty sets (see Ch8 i [1]) This is equivalet to the stateet If X i i is a coutable idexed set of oepty sets, the there exists a (choice) fuctio f : Xi such that i, f () i Xi That is X i (See Ch8, Theore 13 of [7]) I light of the above, i i AC is ore ituitively atural ad is i cocord with the techiques ad the geeral spirit of atheatics that are based o Discrete/Coutable structures (see Lea 15) Now we preset ad i soe cases we prove ecessary results fro set theory At first we itroduce the Hartogs uber of ay set Defiitio 21 The Hartogs uber of ay set A is the least ordial uber ha ( ) which is ot equipotet to ay subset of A That is, there is o ijectio f : A For clarificatio ad further details see Ch71 of [7] There, it is show that the Hartogs uber exists for ay set ad it is a iitial ordial (ad thus it is a cardial) Reark 22: Note that if ha ( ) 6, the there is a ijectio g: A Hece, if A is well-orderable set, ha ( ) is the least cardial greater tha card ( A ) ad thus, card ( A) 6 where is the stadard orderig o ordials 7

8 (where here we cosider card ( A ) as the ordial with which the cardiality card ( A) is idetified) A iportat cocept for liearly (or totally) ordered sets, eeded for the sequel, is that of cofiality With A, we deote a liearly ordered set Defiito 23: Let A, be a liearly ordered set The set B, A, is called a cofial subset i A,, if it is well ordered 7 ad such that a A b B with a b The order type of a well ordered set X,, type( X ), is the uique ordial uber, which is isoorphic to X, Defiitio 24: The cofiality of A, 8 is the sallest possible order type of such cofial set B, i A, That is, cf ( A) i type( B) : B A ad B cofial i A The followig theore is cetral for the developet of our results ad shall be used extesively Theore 25: (Cetral theore for existece of cofial sets ): I ZFC, for every liearly ordered set A, there exists a trasfiite icreasig sequece ˆB, where its rage B, is cofial i A, Furtherore, type( B) card ( A) (Where here we cosider card ( A ) as the ordial with which the cardiality card ( A) is idetified) Proof of Theore 25: We costruct a trasfiite icreasig sequece B ˆ a : for soe ha ( ), where type( B) card ( A) Fix b A (such a eleet exists, sice A is a set) By the Axio of Choice there exists a choice fuctio g for the power set of A, PA ( ) We defie the sequece a : h( A) by trasfiite recursio (see Ch 6, Theore 44 i [7]) as follows: Give a :, we cosider two cases: 7 Ad hece it ca be viewed as a icreasig sequece 8 For a proof, usig the Axio of Replaceet, that such a ordial exists see Theore 631 i [7] 8

9 Case1: If b a g( A ) a, for all Case 2: Otherwise we let b ad A a A: a a holds for all a, we let Observe that fro the recursive costructio of a : this sequece is icreasig Clai: There exists ha ( ) such that a b Proof of Clai: If for all ha ( ) the ters a of the icreasig sequece, as costructed above, are such as a b, the a A Hece : h( A) A will be a ijectio This cotradicts the defiitio of Hartogs uber (see defiitio 21) Thus, there exists ha ( ) such that a b This proves the clai Now let S h( A) : a b By clai, S ad sice S is a well-ordered set, it has a least eleet i S Fro this ad the recursive costructio of the sequece Bˆ a : we get that A ad thus, B Rage Bˆ A Hece, B, A, is a well-ordered set with type( B) card ( A) (**) (See reark 22) Moreover, sice ha ( ) with A, there is o a A such that for all, a a Sice A, is liearly ordered, a A with a B such that a a Thus, Bˆ a : is a icreasig sequece such that B Rage Bˆ A with type( B) card ( A) This cocludes the proof of the theore is cofial i A, It is worth to restate the upshot of the above proof: Corollary 26: For every liearly ordered set A, cofial i, sequece B ˆ a : Note that B ˆ a : there exists a trasfiite icreasig A, such that ha ( ), which is cofial i A, fuctio f :, A, ca be viewed as a icreasig such that f( ) for I such a case we say that A is a cofial ap fro, f :,, a to A, I particular, for ordials 9

10 , (ad i geeral for well-ordered sets) the above aalysis justifies the followig defiitio Defiitio 27: (cofial aps): Let, ordials with A ap f : is cofial i if it is icreasig ad is such that such that f ( ) Accordig to all of the above (see 24, 25, 26, 27) we have the followig well defied equivalet defiitio of cofiality of ordials Defiitio 28 (cofiality of ordials): The cofiality of a ordial is defied to be the iial ordial such that there exists a cofial ap g : That is: cf ( ) i : g : which is cofial i Havig these equivalet defiitios of cofiality 9 i our disposal, we ca easily obtai results for the cofiality o ordials that are required i the sequel Facts about the otio of cofiality (a) cf () (Iediate fro defiitio 24) (b) Sice for ay ordial, id : is a cofial ap i, we get that cf (c) A ordial is a successor if ad oly if ad hece if ad oly if cf ( ) 1 To see this cosider the cofial ap f :1 such that f () (d) If is a successor ordial, the ( ) The proof is a cosequece of the cf fudaetal theore of cardial arithetic For details see Th 924 i [7] Lea 29: Let be a coutable liit ordial The cf ( ) Proof: Sice is coutable liit ordial, there is a eueratio of That is, Recursively defie a ap f : as: f () i ad f ( ) ax f ( 1), The, by defiitio 27, it is easy to coclude that f : is a cofial ap i Sice is the least coutable liit ordial, by defiitio 28 cf ( ) cf ( ) Accordig to the above lea we have, cf ( ) cf ( ) cf ( ) cf ( ) 9 For a alterative, but accordig to our above aalysis, a equivalet approach to cofiality of ordials we refer to Ch 9, Sec 2 i [7] 1

11 Lea 21: If,, ordials ad f :, g : are cofial aps i ad respectively, the f g: is a cofial ap i Proof: Observe that sice f, g are icreasig, f g is icreasig Now, sice f : is cofial, for every there exists such that f ( ) Sice g : is cofial, there exists such that g( ) Because f is a icreasig ap, f ( ) f ( g( )) Puttig everythig together, f ( ) f g( ) ad thus, for every, there exists, such that f g( ) Therefore, f g: is a cofial ap i Corollary 211: If f : is a cofial ap i, the cf ( ) cf ( ) (To see this, set cf ( ) i the above lea 21 ) Corollary 212: For every ordial, cf cf Proof: I corollary 211 set cf ( ) For the reverse iequality, set cf ( ) ( ) cf ( ) The cf ( ) cf cf ( ) i cf to get cf cf ( ) cf ( ) Lea 213: If is a liit ordial, the cf ( ) cf ( ) Proof: Sice is a liit ordial, by defiitio sup : ad hece The icreasig ap f : such that f ( ) 1, is cofial i Now by corollary 211, cf ( ) cf ( ) (1) Fro (1) ad fact (b), cf ( ) Now let f : with be ay cofial ap of i Cosider g : such that g( ) ax : f( ) Clai: g : is cofial i Proof of Clai: We first show that g : is icreasig Ideed, let, ad let such that f ( ) Sice f : is icreasig, f( ) f( ), ad sice, by defiitio of g we coclude that g( ) g( ) Thus, g : is icreasig Furtherore, if, the ad sice f : is cofial i, there is a such that f ( ) Hece, by the axiality i the defiitio of 1 Note that for every we have 11

12 g : we coclude that g( ) Therefore, for every, there is such that g( ) Hece, g : is cofial i This proves the clai Note that by (1) we get that cf ( ) cf ( ) Sice f : with is ay cofial ap of i, by settig cf ( ) i g : we coclude by corollaries 211 ad 212 that cf ( ) cf cf ( ) cf ( ) (2) Fro (1), (2), cf ( ) cf ( ) ad the proof is ow coplete The last defiitio of this sectio shall be used i the proof of Theore 38 Defiitio 214: A ordial is called sigular iff cf ( ), otherwise it is called regular 3 The Mai Results I this sectio we prove i detail the ai results of this paper ad we discuss their cosequeces I Theore 31 we show that i Z AC the differet defiitios of Ifiite sets, SIf, DIf, CIf, W are all equivalet each other I Theore 35, which its proof is a variatio of the proof of Theore 31, we show that i Z (without ay versio of the Axio of Choice) a set is Dedekid Ifiite iff it is Cator Ifiite iff it is Cardial Ifiite Heceforth, the Axioatic Systes Z Z SW SD, Z SC ad are equivalet It is worth to ote that i ZF (ad hece i Z ) the defiitios DIf, CIf, W are ot equivalet to SIf, sice there is a odel of ZF such that there exists a Stadard Ifiite set that is ot Dedekid Ifiite Thus, SD (ad hece SC, SW ) does ot hold Such a odel is the Cohe s First Model A4 (see [6]) We further study how SD is related to AC ad we show that Z AC SD 11 That is, the stateet SD is a theore of the syste Z AC Furtherore we show that Z SD AC fi, where fi AC is the Axio of Coutable Choice o Fiite Sets 11 X Y deotes that the stateet Y is a logical cosequece of X That is, there is a proof of Y fro X, ad hece Y is a theore of X 12

13 Lastly, we show i Theore 38, usig results fro sectio 2 ad odel theory, that the syste Z Z SC, SD (ad hece ay of its equivalet-accordig to Theore 35- systes, Z SW ) is strictly weaker tha Z AC I the followig theore we show that i Z AC the differet defiitios of Ifiite sets, SIf, DIf, CIf, W are all equivalet each other I particular Z AC SD Theore 31: I Z AC a set is Stadard Ifiite if ad oly if card ( X ), if ad oly if it is Cator Ifiite, if ad oly if it is Dedekid Ifiite Thus, SIf, DIf, CIf, W are all equivalet each other Proof: For this, we show i (a)-(c) that SIf CIf DIf SIf ad i (d) that CIf W (a): We show, SIf CIf : (That is, every stadard ifiite set cotais a coutable subset) The proof is rather siple if we assue the Axio of Replaceet ad the Axio of Choice i its full stregth (see Appedix Lea A1) Here we prove it oly fro Z AC Let X be a Stadard Ifiite set For every, let A be the set of all subsets of X with cardiality 2 Thus, the ebers of ifiite set, by defiitios 11 ad 13,, A Apply F= i order to obtai a choice fuctio : F A i i B A are sets of cardiality 2 Sice X is a stadard AC to the faily F, ad via it a subsequece F, where each B is a subset of X of cardiality 2 (It is worth to ote that the Axio of Replaceet is ot required sice B predefied set of a well-defied fuctio o a set 12 ) Observe that the sets F is the iage i a B are also distict We defie recursively the sequece C of utually disjoit sets as follows (this is a stadard techique i aalysis): C B, C1 B1 \ C,, C B \ Ci, Obviously, 1 card ( C ) 2, sice faily C 1 i 1 i1 i Now it is trivial to observe that the S is a faily of oepty ad utually disjoit sets 12 See also sectio 2 the discussio followig the Axio of Replaceet 7 13

14 A secod applicatio of AC to the oepty faily C S guaratees the existece of a choice fuctio : S S, ad through this we obtai a set C c S, such that ci cj, i j utually disjoit sets) This set C c (sice C is a sequece of is the coutable set that establishes X to be Cator Ifiite (As doe earlier i the proof, the Axio of Replaceet is ot required sice C c S ) (b): We show, CIf DIf (that is, if a set is Cator Ifiite the it is Dedekid Ifiite) Sice X is Cator Ifiite it cotais a coutable subset B X That is, there exists a bijectio f : B Defie the fuctio h: X X as follows: h( x) f ( 1) if x f ( ), x B ad h( x) x, if x X \ B (Thus, the actio of h o B is idetified with the actio of f o the successor of each eleet of that leaves the reaiig eleets of X ivariat) Clai 1: The fuctio h: X X is oe to oe Proof of Clai 1: Let x, y X with h( x) h( y) I order for such a thig to hold we ca easily see that either both x, y B, or x, y X \ B I the first case, where x, y B, we have x f ( ), y f ( ) for soe, the defiitio of h, h( x) h( y) f ( 1) f ( 1), ad sice f is oe to oe, 1 1 ad hece Thus, x y I the secod case, where x, y X \ B, we coclude that h( x) h( y) x y This proves clai 1 Fro Let D h( X ) X The the fuctio h: X D fro clai 1 is oe to oe ad sice D Rage h, the h: X D is also oto ad hece a bijectio Thus, X is equipotet to its subset D h( X ) To coplete the proof it is eough to show the followig Clai 2: The set D h( X ) is a proper subset of X Proof of Clai 2: Let x f() We clai that x D Ideed, if we suppose that x D, the x h() z for soe z X, ad we have the followig two cases (i) If z B, the z f ( ) for soe Hece by the defiitio of h, x f () h( z) f ( 1) Thus, f () f ( 1) Sice f is oe to oe, 1 This leads to a cotradictio sice is ot a successor 14

15 (ii) If z X \ B, the x f () h( z) z ad thus (fro the defiitio of f : B ) z B This leads agai to cotradictio (sice by hypothesis z X \ B ) Puttig everythig together, we coclude that there exists x X \ h( X ) ad therefore D h( X ) it is a proper subset of X This proves clai 2 The proof of part (b) is ow coplete Reark 32: Observe that i (b) of the above proof we have show thatcif without usig the Axio of Coutable Choice DIf (c): We show, DIf SIf (that is, if a set is Dedekid Ifiite, the it is Stadard Ifiiteequivaletly ot fiite) Suppose, for cotradictio, that X is a fiite, yet a Dedekid ifiite set It is well kow that i Z (ad hece i Z AC ) the Pigeohole Priciple holds: If a set is fiite, the it does ot cotai a proper subset equipotet to it (see Appedix Lea A2) Hece, X does ot cotai a proper subset equipotet to it This cotradicts the hypothesis of X beig Dedekid Ifiite (ad thus cotaiig a proper subset equipotet to it) Therefore, DIf iplies SIf Reark 33: Observe that i (c) we have show that DIf of Coutable Choice SIf without usig the Axio (d): We show, CIf W (that is, a set is Cator Ifiite iff it is Carial Ifiite) (i) CIf W Suppose that X is a Cator Ifiite set The X cotais a coutable subset A with card ( A), ad hece there is a bijectio : A X Thus, card ( A) card ( X ) (see also sectio 1 ) (ii) W CIf Suppose that X is a set such that card ( X ) Hece, there is a ijectio g: X Thus, g[ ] X is a coutable set ad hece X is Cator Ifiite Reark 34: Observe that i (d) we have show CIf W without usig the Axio of Coutable Choice Puttig (a), (b), (c), (d) together we coclude that i Z AC, SIf W CIf DIf The proof of the theore is ow coplete 15

16 Theore 35: I the axioatic syste Zerelo, Z, a set is Cardial Ifiite if ad oly if it is Cator Ifiite, if ad oly if it is Dedekid Ifiite Hece, W,, CIf DIf are all equivalet each other i Z Proof: Fro reark 34, CIf W i Z, ad fro reark 32, CIf DIf i Z It reais to show that i Z, DIf CIf That is, if X is Dedekid Ifiite, the X it is also Cator Ifiite To this ed, sice X is Dedekid ifiite, it cotais a proper subset B equipotet to it Thus, f : X B X which is oe to oe ad oto B X Cosider a X \ B The, a Rage( f ) To show that X is Cator Ifiite we costruct recursively a well-defied coutable (ifiite) subset Y of X as follows: Cosider f : X X as above The by (siple) recursio theore (see 56 i [1]) there is a uique fuctio g: X, such that: g() a ad g( 1) f (( g( )) for all Y Rage( g) g( ) is a well-defied coutable subset Y X Ideed, by The set costructio, for ay, g f a ( ) ( ) ( ) where ( ) f deotes the th recursive iteratio of the actio of f o a If (wlog) i j such that g( i) g( j), the ( i) ( j) f a f a ( ) ( ) ad hece ( ji f ) ( a) a where ji Thus, a Rage( f ), ad this cotradicts the choice of a Hece, g is a oe to oe well-defied fuctio This set Y is the Ifiite Coutable set we are lookig for I the ext theore we prove that i Z AC fi, holds Thus, SD the Axio of Coutable Choice o Fiite Sets, fi AC is a logical cosequece (theore) of Z SD The ai idea for the proof of the ext theore is fro Theore 212 i [5] Theore 36 Z SD AC fi 16

17 Proof: Let X F= be a coutable faily of oepty fiite sets We show that X ad hece fi AC holds 13 For this, defie X X The X is ot a fiite set, equivaletly it is Stadard Ifiite ad thus by SD it is Dedekid Ifiite By Theore 35 it is also Cator Ifiite ad hece there is a ijectio f : is a fiite set, usig proof by cotradictio, we ca easily show that, X M f X is a stadard ifiite set : X Sice Clai 1: M X Proof of Clai 1: Sice f X, for every M defie ( ) i : f ( ) X Thus, for every M there exists a uique x x X such that f ( ) x, Heceforth, there exists a eleet X M Thus, X M M Accordig to the above, i order to coplete the proof it is eough to show the followig: Clai 2: If X F= is a coutable collectio of oepty fiite sets such that it exists a stadard ifiite set M with the property X, the X (Hece, M the Axio of Coutable Choice for Fiite Sets holds) Proof of Clai 2: Cosider the sequece of oepty fiite sets X ad defie Y k X Iductively observe that for every k k the sets Y k are oepty ad are fiite Now cosider the sequece of oepty fiite sets Yk k By hypothesis there exists a fiite set M with the property Y, ad hece, there exists a eleet y M Y M with y 1, 2,, x x x X M 1 i M, x X (*) i i Equivaletly, 13 See the discussio i sectio 2 that follows Axio 8, Axio of Coutable Choice There we refer explicitly to equivalet to AC stateets 17

18 Let ad cosider the set Z M : Sice M is a Stadard Ifiite set, by defiitio 11 we have that M : Thus, the set Z is a oepty subset of ( ) i Z i{ M : } ad hece it has a iiu eleet Now defie Sice, y ( ) ( ) ( ) ( ) 1, 2,, x x x( ) X ( ) ad ( ), accordig to (*), f ( ) x X ( ) x ( ) X The sequece f : Xi i, is a choice fuctio for the faily ( ) x F=, with X, where X (see footote 13) Therefore X, ad sice X, the X This fiishes the proof of clai 2 The proof of theore is ow coplete I our last result we show, usig results fro sectio 2 ad odel theory, that the syste Z SD is strictly weaker tha Z AC, as oe could aturally cojecture fro the preceedig results (see Theores 31, 35 ad 36) For this we show that there exists a odel M of Z where SD holds, but AC does ot hold Thus, the stateet SD, Every Stadard Ifiite set is Dedekid Ifiite, is eve weaker tha the Axio of Coutable Choice, AC Therefore, it could be aturally added to the axios Z (istead of AC ) i order to address probles i a fraework that does ot assues ay versio of the Axio of Choice Heceforth, i this sese, we could develop a rather powerful axioatic syste i areas that ay versio of Axio of Choice is uatural 14 This is of great iterest i the area of costructive atheatics where the adoptio of ay versio of Axio of Choice is doubtful or uacceptable (see Richa[11]) Recall, as stated earlier i the paper, that the stateet SD caot be proved fro Z or ZF aloe, sice there exists a odel of ZF (ad hece of Z ) i which there exists a stadard Ifiite set that is ot Dedekid Ifiite Such a odel is the Cohe s First Model A4 (see [6]) Stateet Wˆ : For every set X ad a cardial, either card ( X ) or card ( X ) 14 See Chailos [2] for resolutio of Zeo paradoxes where SD is used without ay use of AC 18

19 Observe that Wˆ is the stateet that every set is either Cardial Fiite or Cardial Ifiite (see sectio 1) The followig theore is Theore86 of [9] Theore 37: Let (i) For each, be a sigular aleph The there exists a odel of ZF i which ˆ W holds (ii) Wˆ fails ad ACcf ( ) fails Theore 38: I Z the stateet SD is strictly weaker tha AC Proof: Fro the proof of Theore 31, Z AC SD Thus, by soudess theore ay odel of Z that satisfies Z SD is weaker tha AC it also satisfies SD Hece, i order to prove that the syste Z AC it is eough to show that assuig the cosistecy of Z, there exists a odel M of Z that satisfies SD i which AC fails Sice is a liit ordial, by lea 213 cf ( ) cf ( ) ad hece by defiitio 214 is sigular We set ad i Theore 37 to coclude that there is a odel M of ZF (ad hece of the weaker Z ) i which Wˆ holds, but o the other had AC AC AC fails Now, if a set X is Stadard Ifiite the it is ot fiite ad cf ( ) sice Wˆ holds, card ( X ) Thus, stateets stateet SW holds i M Fro Theore 35 the SW ad SD are equivalet i Z, ad thus i M the stateet SD holds, but the AC fails That is what we wated to prove Appedix I this Appedix we prove auxiliary results etioed i the paper At first we provide a sketch of aother proof of the fudaetal result SIf CIf of Theore 31 This proof is i ZFC ad it uses explicitly the Axio of Replaceet ad the full versio of the Axio of Choice (see Ch 8, Theore 14 of [7]) 19

20 Lea A1: ( SIf subset CIf ): If A is a stadard ifiite set, the A cotais a coutable Sketch of proof ( ZFC ): Let A be a stadard ifiite set Fro the Axio of Choice ( AC AC), equivaletly fro the Well Orderig Theore (see Ch8 i [7]), A is well orderable set ad hece usig the Axio of Replaceet it is isoorphic with a uique ifiite ordial uber (see 631 i [7]) Sice is the least ifiite ordial, Now usig the priciple of trasfiite recursio ad the Axio of Replaceet (oce ore) we coclude that the set A ca be well ordered i a trasfiite sequece of legth, thus, A a : Fro the defiig properties of ordials ad well-ordered sets, C a : is a iitial seget of A Hece, the Rage C a : coutable subset of A (see also defiitio 12) is a I the ext lea we sketch a proof of Pigeohole Priciple that is used i provig DIf SIf i Theore 31 Lea A2: Pigeohole Priciple If A is a fiite set, the there is o oe to oe fuctio f : A A oto a proper subset B A (Equivaletly, if A is a fiite set, the every oe to oe fuctio f : A Ais also oto A, ad hece it is a bijectio) Sketch of Proof: We clai that to prove Pigeohole Priciple it is eough to show the followig clai: Clai: If ad if g :,, is ay oe to oe fuctio, the it is also a oto fuctio (Fro the costructive defiitio of,, is a subset of that is idetified with ) Ideed, assuig the clai, cosider ay oe to oe fuctio f : A A Let : A, be the bijectio that witesses the fiiteess of A for soe (see defiitio 11) The the fuctio g 1 f :,, oe to oe Hece, for the clai it is also oto, Thus, is well defied ad it is f 1 g is a bijectio, as a copositio of bijectios, ad hece it is oto A Now the proof of the etioed clai is a well-kow ad stadard applicatio of the Iductio Priciple, (see 527 of [1]) 2

21 Bibliography [1] Cohe PJ, Idepedece Results i Set Theory, The Theory of Models, J Addiso, LHeki ad ATarski, eds, North Hollad Asterda, 1965, pp39-54 [2] Chailos G, The Logic of Zeo s Arguet Agaist Plurality, Φιλοσοφείν-Philosophei, Editor: Ζήδρος-Zetros, ΙSSN: , to appear [3] Dedekid R, Was sid ad was solle die Zahle?, Vieweg Verlag, 1888 [4] Gödel K, The Cosistecy of the Axio of Choice ad the Geeralized cotiuu Hypothesis, Proc Natl Acad Sci USA, 24, , 1938 [5] Herrlich H, Axio of Choice, Spriger-Verlag, 26 [6] Howard P ad Rubi JE, Cosequeces of the Axio of Choice, Aer Math Soc,1998 [7] Hrbacek K ad Jech T, Itroductio to Set Theory, Marcel Dekker 1999 [8] Jech T, Set Theory, Spriger -Verlag, Berli-NY, 3 rd ed, 23 [9] Jech T, The Axio of Choice, Dover Editio 1973 [1] Moschovakis I, Notes o Set Theory, 2 d Edt, Spriger, 26 [11] Richa F, Costructive Matheatics without Choice, i: Reuitig the Atipodes Costructive ad Nostadard Views of the Cotiuu (P Schuster et al, eds), Sythèse Library 36, , Kluwer Acadeic Publishers, Asterda, 21 21