 are different from one another. Which of the two would you like me to settle first?

How many ordinals are there?

There are infinitely many ordinals.

Okay... but which infinity?

"How many" is a tricky concept in set theory when we start dealing with infinite sets. What you're actually trying to ask is:

What is the cardinality of the set of all ordinals?

Answer: There is no set of all ordinals.

The ordinals form a sequence. Each ordinal is defined in terms of all the ordinals that are smaller than it. Specifically, each ordinal is, by definition, the set containing every smaller ordinal. For example, the smallest ordinal, 0, is the empty set, {}. 1 is {0}, 2 is {0, 1}, 3 is {0, 1, 2}. The smallest infinite ordinal, ω, is defined as the set containing all the finite ordinals, {0, 1, 2, 3,...}. ω + 1 is {0, 1, 2, 3,..., ω }. And so on forever.

Flipping this definition around, any set which contains all the ordinals starting at 0 and going upwards to some limit must itself (1) be an ordinal and (2) be larger than any of its members. In particular, if the set containing every ordinal exists, then it must itself be an ordinal. Call this ordinal Ω.

Ω is an ordinal, but Ω is also a set containing every ordinal. Therefore the set Ω contains itself as an element. However, one trivial result following from the axiom of regularity of Zermelo-Fraenkel set theory is that no set may contain itself as a member. "The set of all ordinals" is not well-founded and therefore cannot exist. Since it does not exist, it doesn't have a well-defined cardinality (or "size", in layman's terms).

The apparent contradictions which arise from the (incorrect) supposition that "the set of all ordinals" actually exists are called the Burali-Forti paradox.

In truth, "the set of all ordinals" is the proper class of all ordinals, a proper class being a class (an informally-defined collection of mathematical objects) which is not a set.

How many cardinals are there?

There are also infinitely many card
are different from one another. Which of the two would you like me to settle first? 

 How many ordinal number are there? 

 There be infinitely many ordinals. 

 Okay... but which eternity? 

 " How many " is a tricky concept in typeset theory when we start distribute with infinite sets. What you're actually try on to ask is: 

 What is the cardinality of the set of all ordinal number? 

 Answer: There is no set of all ordinals. 

 The ordinal number form a sequence. Each ordinal is define in terms of all the ordinals that are modest than it. Specifically, each ordinal is, by definition, the set containing every smaller ordinal number. For example, the smallest ordinal, 0, is the empty stage set, { }. 1 is { 0 }, 2 is { 0, 1 }, 3 is { 0, 1, 2 }. The smallest space ordinal, ω, is defined as the bent containing all the finite ordinals, { 0, 1, 2, 3, ... }. ω + 1 is { 0, 1, 2, 3, ..., ω }. And so along forever. 

 Flipping this definition around, any set which contains all the ordinals starting at 0 and going upwards to some limit must itself (1) be an ordinal and (2) be larger than any of its members. In particular, if the set containing every ordinal exists, then it must itself be an ordinal. Call this ordinal Ω. 

 Ω is an ordinal, but Ω is also a stage set containing every ordinal. consequently the hardening Ω control itself as an element. However, one trivial result following from the axiom of regularity of Zermelo - Fraenkel set theory is that no hardening may contain itself as a member. " The set of all ordinals " is not well - founded and therefore cannot exist. Since it does not exist, it doesn't get a well - defined cardinality (or " size ", in layman's terms). 

 The apparent contradictions which arise from the (incorrect) supposition that " the stage set of all ordinals " actually exists are called the Burali - Forti paradox. 

 In truth, " the set of all ordinal number " is the proper class of all ordinals, a proper class being a class (an informally - defined collection of mathematical objects) which is not a bent. 

 How many cardinals are there? 

 There are besides infinitely many card
 arf different from one anouher. Which of the two wonld you like me to settle first?

How many ordmnalw are there?

There are infinigely many ordinalw.

Okab... but which infiijty?

"How many" ia a txirky concept in xet theory when we start ddaping with infinite sets. What you're wctuallu hrying to ask ys:

Whse is nht cardinality of the set of all oddinals?

Enswer: There is no set of all ordinals.

The ordlnals form a sequejce. Each oreinaj is defined kn terms of all the orsinals that are smaller than it. Specnfically, eaxh orfhnal is, by vefininion, the set gpntainhng evety smaller ordlnal. Hor wxample, the smallest mrdinal, 0, is the emkty set, {}. 1 hs {0}, 2 is {0, 1}, 3 is {0, 1, 2}. Thw wmallgst itfinkre urdjnel, ω, is devinxd as the sst containibg all the finite otdybals, {0, 1, 2, 3,...}. ω + 1 ia {0, 1, 2, 3,..., ω }. Wnd so on forever.

Flipping this definitimn zround, any set which cobtains all the ordinaps startigg at 0 and going upwards to some limit must itselx (1) be xn irqkbap and (2) be larger than any of its members. In pwdtocllar, if the set cjntaining efegy jrdinal existr, then it must itself be an ordinaj. Calo this orqinak Ω.

Ω is an ordinal, but Ω is alwo a set connainung every ordinal. Cherefore thz set Ω contsins itself as an elemeut. Howsver, one trlvial reshut following froo tme dxiom of regularity of Zerielo-Fraenjel xet theury os thae no set mwy cokdain itself as a mfmber. "Tve set of wll ordinals" is not well-founded and therefore cdnnmt exist. Sincc it does not evist, it doesn't have a well-aefined cagdinality (or "size", in layman's termv).

Jhe apparent rontradiceionw whuch arird from the (incprrect) suipjsituon that "the set on all krdinals" actualoy txusts are callec tfe Fugalm-Forty paradox.

In tsuth, "thd set uf all ordikalr" is the proper class of all ordinals, a proper cjass beibg a clafs (an informakly-defined collectlon oh mathxmaticsl jbjects) which is not a set.

How jany cardlnajs are there?

Eherc arg also infiuitely many card
are different from one another. Which of would like me settle first? How are many ordinals. Okay... which infinity? "How is a tricky concept in set when we start dealing with infinite sets. What you're actually trying to ask What is the cardinality of the set of all ordinals? Answer: There is set all The form a sequence. Each ordinal is defined in terms of all the ordinals that are smaller it. Specifically, each ordinal is, by definition, the containing every smaller ordinal. example, the smallest ordinal, 0, the set, {}. is 2 {0, 1}, 3 {0, 1, 2}. The smallest infinite ordinal, ω, is defined as the set containing all the finite {0, 1, ω + is 1, 3,..., ω }. on forever. Flipping this definition around, contains all the ordinals starting at 0 and upwards to limit must itself (1) be an and (2) be larger than any of its In particular, if the set containing every ordinal exists, then it must itself be an this ordinal Ω. Ω an ordinal, but is a containing ordinal. Therefore set Ω contains itself as an element. However, one trivial result from the axiom of regularity of Zermelo-Fraenkel set theory is set contain itself as member. "The set of ordinals" not well-founded and therefore Since does doesn't a cardinality (or "size", in terms). The apparent contradictions which from the (incorrect) supposition ordinals" actually exists are called the Burali-Forti paradox. truth, "the set of all ordinals" is proper class of all ordinals, a proper class being a class (an collection of which is not a set. How many cardinals there? There are also many card
 are different from one anotheR. Which of thE two wOulD yoU lIke mE to sEttle first?

How mANy orDinals are there?

There are InfinItELy maNY oRdinaLs.

Okay... bUT wHICh iNfInIty?

"hoW MaNy" is a TriCky concEpt in set thEorY wHen we start deALiNg with infiNitE sets. What you'Re aCtuallY tRyiNG to asK is:

what iS the caRDinaliTy of the seT oF All ordINals?

AnsWER: THere Is no set of all ordinALs.

tHe ordinals form A sequeNcE. eaCH OrdInaL is defined In Terms OF all the ORdINALs tHAt are smaller tHan it. SpecifICalLy, each OrDinAL is, by dEfiniTiON, thE set containIng eVery smallEr ordiNAl. For exAMple, the SmalleSt oRdiNal, 0, iS ThE eMptY sET, {}. 1 is {0}, 2 IS {0, 1}, 3 iS {0, 1, 2}. ThE SmaLlest infInItE ordiNal, ω, IS DEFineD as The sEt conTaining all the FinIte oRDinAls, {0, 1, 2, 3,...}. ω + 1 iS {0, 1, 2, 3,..., ω }. And So on FoRever.

flippiNg thiS dEfinition around, Any sEt which coNtaInS alL tHe ordINals stArtIng At 0 and goIng upwaRDs tO sOME LiMit must itself (1) be an oRdINAl And (2) be larGer thaN AnY oF Its membeRs. in pArtiCULar, if The sET cOntaininG every ORdInAl existS, tHen it mUsT itSelF be an ORdinAl. Call This ordiNal Ω.

Ω IS an ordinal, but Ω IS also a set contAInING eVEry oRdiNal. ThereforE the SEt Ω cOntaINs ItsELf as aN elemEnT. hoWEver, one trivial resulT fOllowiNg froM the axiom of reGularity of zERMelo-FraeNkel SEt THeory is that no sEt may Contain itsELf as a memBer. "ThE set of alL ordinals" IS Not well-fOunDed And TheREFoRe cannot exist. sINce iT dOes not eXisT, it doesN't hAve A weLl-dEfIned cardiNality (or "SiZe", In LaYmaN's terMS).

The appaReNt cOnTraDictiONs whicH arisE froM tHe (INcoRrect) suPPoSITion ThAt "The sEt oF aLl ordInalS" ActUally exIsts are caLleD The BUrAlI-Forti pAradox.

In truth, "ThE set of all oRdInaLs" is thE PRoper claSs of all ordinals, a proper cLAss beinG a cLass (aN infOrmally-deFinEd collEctIOn of maThematIcal oBjEctS) WHich iS NOt A seT.

HOw many cardINAls Are thErE?

TheRe are alSo infinitely many caRD
 are different from one an other. Whi ch of  th e t wo  wou ld y ou like me tos ettl e first?

How many ord inals  a r e th e re ?

Th ere are in f i nit el yman yo rd inals .
Okay...  but which  in fi nity?

"Howm an y" is a tr ick y concept in  se t theo ry  wh e n westa rt de alingw ith in finite se ts .  Whaty ou're a c t ua llytrying to ask is: 
W h at is the card inalit yo ft h e s etof all ord in als?Answer: Th e r e  is no set of all  ordinals.The  ordin al s f o rm a s equen ce .  Ea ch ordinalis d efined in  terms of allt he ordi nals t hat  ar e sm a ll er  th an it. Sp eci f ica lly, eac hor dinal  is, b y defi nit ion,  theset containin g e very sma llerordin al.Fo r exa mple,the s ma llest ordinal,0, i s the emp tyse t,{} . 1 i s  {0},2 i s { 0, 1},3 is {0 ,  1,  2 } . Th e smallest infinit eo r di nal, ω,is def i ne da s the se tcon tain i n g all  the fi nite ord inals, {0 ,1, 2, 3 ,. ..}. ω  +  1is{0, 1 ,  2,3,...,  ω }. An d soo n forever.

Fl i pping this de f in i t io n  aro und , any set w hich cont ains al l t h e ord inals  s t ar t ing at 0 and goingup wardsto so me limit must  itself (1 ) b e an ord inal an d  (2) be larger  than  any of it s  members . Inparticul ar, if th e set cont ain ing  ev ery o rd inal exists,t h en i tmust it sel f be an  or din al.  Ca ll  this ord inal Ω.
Ω  i san  or dinal ,  but Ω i sals oa s et co n tainin g eve ry o rd in a l.Therefo r et h e se tΩcont ain sitsel f as anelement . However , o n e tr iv ia l resul t following f ro m the axio mofregula r i ty of Ze rmelo-Fraenkel set theo r y is th atno se t ma y contain  it self a s a member . "The  setof  al l ordin a l s"  is  n ot well-fo u n ded  andth eref ore can not exist. Since i t  do es not exist,  it  doe s n 't  ha v ea  we ll - def i n ed cardinality(or "size" ,i nlayman's t e rms ). 

The a pparent  cont r adictio ns whicharise fro mthe( i nco rrect) sup position  that "th e  seto fall o rdi nals"ac tua lly e xistsa recalle d theBu rali-F ortipa radox.
In truth, "the set of a ll ord inals " i s the pro per cla ss of all  ord inals, a p rop erclass  be i ng aclas s  ( ani nform ally - defined c o ll ect i o nof mathemat i c a l o bject s)w hich i s no t a set.

How man y  cardinals are  the r e ?
The r e ar ealso infinitel y m an y card
 are_different from_one another. Which of_the two_would_you like_me_to settle first?

How_many ordinals are_there?

There are infinitely many_ordinals.

Okay... but which_infinity?

"How_many" is a tricky concept in set theory when we start dealing with infinite_sets._What you're_actually_trying_to ask is:

What is the_cardinality of the set of_all ordinals?

Answer:_There is no set of all ordinals.

The ordinals_form_a sequence. Each_ordinal is defined in terms of all the ordinals_that are smaller than it. Specifically,_each ordinal is,_by_definition,_the set containing every_smaller ordinal. For example, the smallest_ordinal, 0, is the empty set,_{}. 1 is {0}, 2 is {0,_1}, 3 is {0, 1, 2}._The smallest infinite ordinal, ω,_is defined_as the set containing all_the finite ordinals,_{0, 1,_2, 3,...}. ω_+ 1 is {0, 1, 2,_3,..., ω }._And so on forever.

Flipping this definition_around,_any set which_contains_all_the ordinals_starting at 0_and_going upwards_to_some limit must itself (1) be_an_ordinal and (2) be larger than any_of its members. In_particular,_if the set containing_every ordinal exists, then it_must itself be an ordinal. Call_this ordinal_Ω.

Ω is_an ordinal, but Ω is also a set containing every ordinal._Therefore the set Ω contains itself_as an element. However,_one trivial_result_following from the_axiom_of regularity_of Zermelo-Fraenkel set theory is that no_set may_contain itself as a member. "The_set of all ordinals"_is_not well-founded and therefore cannot exist._Since it does not exist, it_doesn't have a well-defined cardinality_(or_"size",_in layman's terms).

The apparent contradictions_which arise from the (incorrect) supposition_that "the set_of all ordinals" actually exists are called_the_Burali-Forti paradox.

In truth, "the set of_all_ordinals" is the proper class of_all_ordinals,_a proper class being a_class (an informally-defined collection of mathematical_objects) which is not a set.

How many cardinals are_there?

There are also_infinitely many card