α+1 be the space spanned by Vα and the element aVα. If λ is a limit ordinal, let Vλ be the union of all earlier Vα. It is easy to see that { aVβ | β < α} is a basis for Vα. Thus, the dimension of each Vα is exactly the cardinality of α. In particular, if ω1 is the first uncountable ordinal, then Vω1 will have uncountable dimension, yet be the union of all Vα for α < ω1, which all have countable dimension, as desired.
If you forbid one to use the Axiom of Choice, then it is no longer true that every vector space has a basis (since it is consistent with ZF that some vector spaces have no basis), and the concept of dimension suffers in this case. But some interesting things happen. For example, it is consistent with the failure of AC that the reals are a countable union of countable sets. R = U An, where each An is countable. (The irritating difficulty is that although each An is countable, one cannot choose the functions witnessing this uniformly, since of course R is uncountable.) But in any case, we may regard R as a vector space over Q, and if we let Vn be the space spanned by A1 U... U An, then we can still in each case make finitely many choices to witness the countability and conclude that each Vn is countable dimensional, but the union of all Vn is all of R, which is not countable dimensional.

 Q:

Questions regarding Falcon Heavy's crossfeed system

My understanding of the Falcon Heavy cross feed system is that the outer cores feed six of the nine engines directly on the center core.  Thus, the cross feeds don't pump propellent and oxidizer into the center core tanks but rather feed six of the nine center core engines directly.
This is very similar to how the Space Shuttle worked.  The Shuttle's cross feed between the External Tank and the Orbiter was 17" in diameter -- one for propellent, one for oxidizer.  A project to reduce the size of the cross feed to 14" was cancelled.  A video of the disconnect being tested can be seen here: 

.
Some questions:
How large are the cross feeds between the Falcon Heavy's outer cores and the inner core?
Are there four (2 for propellent, 2 for oxid
α+1 be the space spanned by Vα and the element aVα. If λ is a limit ordinal, lease Vλ be the coupling of all earlier Vα. It is easy to see that { aVβ | β < α } is a footing for Vα. Thus, the dimension of each Vα is exactly the cardinality of α. In especial, if ω1 is the first uncountable ordinal, then Vω1 will suffer uncountable dimension, yet be the marriage of all Vα for α < ω1, which all have countable dimension, as hope. 
 If you prevent one to use the Axiom of Choice, then it is no longer true that every vector quad has a basis (since it is reproducible with ZF that some vector spaces have no basis), and the concept of proportion suffers in this case. But some interesting things happen. For example, it is reproducible with the failure of AC that the reals are a countable union of countable sets. R = U An, where each An is countable. (The irritating trouble is that although each An is countable, one cannot choose the functions witnessing this uniformly, since of course R is uncountable .) But in any case, we may regard radius as a vector space over Q, and if we let Vn be the space spanned by A1 U... U An, then we can still in each case make finitely many choices to witness the countability and conclude that each Vn is countable dimensional, but the union of all Vn is all of roentgen, which is not countable dimensional. 

  Q: 

 Questions regarding Falcon Heavy's crossfeed organization 

 My reason of the Falcon Heavy cross feed system is that the outer effect feed six of the nine engines directly on the center core.   therefore, the cross feeds don't pump propellent and oxidizer into the center core tanks but rather feed six of the nine center core engines immediately. 
 This is very similar to how the Space Shuttle cultivate.   The Shuttle's cross feed between the External Tank and the Orbiter was 17 " in diameter -- one for propellent, one for oxidizer.   A project to reduce the size of the hybridization feed to 14 " was cancelled.   A video of the disconnect being quiz can be seen here: 

. 
 Some questions: 
 How large are the cross feed between the Falcon Heavy's outer cores and the inner core? 
 Are there four (2 for propellent, 2 for oxid
α+1 be the space spanned by Vα xnd the element aVα. If λ is a mimit orainal, let Vλ be the union of ell warlitg Vα. It is easy to see that { aVβ | β < α} is q bawus for Vα. Tihs, the dimensjln oy xach Vα is exactky the carginality of α. It oaxticular, if ω1 is the first uncountablq ordinsl, then Vω1 will hwve lnsounfable dimension, yet be the union or all Vα for α < ω1, which all have countable dimenslon, ws desired.
If you flrbid one ti usq the Axiom ow Choice, then it is no longer true that every vector rpace has a basus (sijwe it is coisistegt with ZF tmst soma vectot spaces have ko bavis), and the concept of dmmension suffers in jhis case. Tuc some interesting thunts hakpen. Xor dzamole, iv ia conslstxnt with ths failure od AC that the reals awv a countable union os countable sets. R = U An, where each An ps ckuntable. (The irritating difficulty is that apthough ewch An is countable, one cannot choose the functiots wivndssnkn thkw kniformly, since of course R is uncountable.) Bue im sny case, we man regard R as a vevtlr fpace over Q, xnd if we let Vn be the spafe spanged bt A1 U... U Ag, thrn we can still in each casw make finitvly nany choices to wicness the cobntabikity snd conclude that each Rn is dountable dlmensionam, but the union ow akl Vn is all of R, which is njt countaule dnmensionxl.

 Q:

Auestijns regardlng Falcon Heavy's crossffed sistem

Mf understajding of the Falcon Heavy cross feed system ix dhan the outzr corcs feed six of ehe nine engings directjy on the centeg core.  Tius, the crosf feeds don't [omp propellenv and oxiqizee inro the zdnter core tanls but ranhtr feed siz of the nine centcr cote engines directoy.
Tyis is very simolaf tj jox the Vpace Shuttla wofkea.  The Shuttlt'f grors fred between the Extesnal Tank and the Orbiyev was 17" in diametew -- one for prppellent, one for odidizxr.  A 'rojecy tj reduce the size of the cross feed to 14" waf cancelled.  A vldeo of the dixconnect being tested can be seen here: 

.
Wome questions:
How oarge are the cross needs betweei the Salcon Hedvy's outer cores and the inner core?
Art there four (2 for propemlent, 2 for lxid
α+1 be the space spanned by Vα element If λ a limit ordinal, of earlier Vα. It easy to see { aVβ | β < α} a basis for Vα. Thus, the dimension of each Vα is exactly the of α. In particular, if ω1 is the first uncountable ordinal, then Vω1 have dimension, be union of all Vα for α < ω1, which all have countable dimension, as desired. If forbid one to use the Axiom of Choice, it is no longer that every vector space has basis it is with that vector spaces have basis), and the concept of dimension suffers in this case. But some interesting things happen. For example, is consistent failure of that reals a countable union sets. R = U An, where countable. (The irritating difficulty is that although each is countable, cannot choose the functions witnessing this since of course R is uncountable.) But in case, we may regard R as a vector space over Q, and if we let the space spanned by U... U An, we still each make finitely choices to witness the countability and conclude that each Vn is dimensional, but the union of all Vn is all of is countable dimensional. Q: regarding Falcon Heavy's crossfeed My of the Falcon Heavy system that feed of nine engines directly on center core. Thus, the cross don't pump propellent and tanks but rather feed six of the nine core engines directly. This is very similar how the Space Shuttle worked. The Shuttle's cross feed between the External and the 17" in diameter -- one for propellent, one oxidizer. A project to the size of the cross feed to 14" was A of the being tested can seen here: . questions: How large cross feeds the Heavy's the inner core? Are there four for 2 for oxid
α+1 be the space spanned by Vα and The element AVα. If Λ is A liMiT ordInal, Let Vλ be the unioN Of alL earlier Vα. It is easy to seE that { AVΒ | Β < α} is A BaSis foR Vα. Thus, THe DIMenSiOn Of eAcH vα Is exaCtlY the carDinality of Α. In PaRticular, if ω1 iS ThE first uncoUntAble ordinal, tHen vω1 will HaVe uNCountAblE dimeNsion, yET be the Union of alL VΑ For α < ω1, wHIch all hAVE cOuntAble dimension, as deSIrED.
If you forbid onE to use ThE axIOM of choIce, then it iS nO longER true thAT eVERY veCTor space has a bAsis (since it IS coNsisteNt WitH zF that Some vEcTOr sPaces have no BasiS), and the coNcept oF DimensiON sufferS in thiS caSe. BUt soME iNtEreStINg tHInGs hAPpeN. For examPlE, iT is coNsisTENT With The FailUre of aC that the realS arE a coUNtaBle unIon of CounTaBle seTs. R = U An, Where EaCh An is countable. (the iRritating DifFiCulTy Is thaT AlthouGh eAch an is couNtable, oNE caNnOT CHoOse the functions witNeSSInG this uniFormly, SInCe OF course R Is UncOuntABLe.) But In anY CaSe, we may rEgard R AS a VeCtor spaCe Over Q, aNd If wE leT Vn be THe spAce spaNned by A1 U... u An, thEN we can still in eACh case make finITeLY MaNY choIceS to witness tHe coUNtabIlitY AnD coNClude That eAcH vn IS countable dimensionAl, But the Union Of all Vn is all oF R, which is nOT COuntable DimeNSiONal.

 Q:

Questions rEgardIng Falcon HEAvy's crosSfeed System

My UnderstanDINg of the FAlcOn HEavY crOSS fEed system is thAT The oUtEr cores FeeD six of tHe nIne EngIneS dIrectly on The centeR cOrE.  THuS, thE crosS Feeds don'T pUmp PrOpeLlent ANd oxidIzer iNto tHe CeNTer Core tanKS bUT RathEr FeEd siX of ThE nine CentER coRe enginEs directlY.
ThIS is vErY sImilar tO how the Space SHuTtle worked.  thE ShUttle's CROss feed bEtween the External Tank anD The OrbiTer Was 17" in DiamEter -- one foR prOpelleNt, oNE for oxIdizer.  a projEcT to REDuce tHE SiZe oF tHe cross feeD TO 14" waS cancElLed.  A Video of The disconnect being TEstEd can be seen heRe: 

.
SOme qUEStIonS:
hoW LarGe ARe tHE Cross feeds betweEn the FalcoN HEAvY's outer corES anD tHe inner Core?
Are There FOur (2 for pRopellent, 2 For oxid
α+1 be the space spanned b y Vα and t he el eme ntaV α. I f λis a limit ord i nal,  let Vλ be the union o f all  e a rlie r  V α. It  is eas y  t o see  t ha t {  a V β| β <  α}  is a b asis for V α.Th us, the dime n si on of each  Vα  is exactlythe  cardi na lit y  of α . I n par ticula r , if ω 1 is thefi r st unc o untable o rd inal , then Vω1 will h a ve uncountable di mensio n, ye t bethe  union ofal l Vαf or α <ω 1, w h ich all have coun table dimen s ion , as d es ire d .
If y ou fo rb i d o ne to use t he A xiom of C hoice, then it is no l ongertru e t hate ve ry  ve ct o r s p ac e h a s a  basis ( si nc e itis c o n s i sten t w ithZF th at some vecto r s pace s  ha ve no  basi s),an d the  conce pt of  d imension suffer s in  this cas e.Bu t s om e int e restin g t hin gs happ en. For exa mp l e ,  i t is consistent wi th t he  failure  of AC th at the real sare  a c o u ntabl e un i on  of coun tables et s.  R = UAn , wher eeac h A n isc ount able.(The irr itati n g difficulty i s  that althoug h  e a c hA n is  co untable, on e ca n notchoo s ethe funct ionswi t ne s sing this uniformly ,sinceof co urse R is unc ountable.) B u t in any  cas e ,w e may regard R  as a  vector sp a ce overQ, an d if welet Vn be t he space  sp ann edbyA 1  U ... U An, the n we c an  stillineach ca semak e f ini te ly many c hoices t owi tn es s t he co u ntabilit yand  c onc ludet hat ea ch Vn  isco un t abl e dimen s io n a l, b ut  t he u nio nof al l Vn isall ofR, whichisn ot c ou nt able di mensional.

Q: 

Question sreg arding F alcon He avy's crossfeed systemMy unde rst andin g of  the Falc onHeavycro s s feed  syste m isth att h e out e r  c ore sfeed six o f the  nine  e ngin es dire ctly on the center cor e.  Thus, the  cr ossf e ed s d o n' t  pu mp pro p e llent and oxidi zer into t he ce nter coret ank sbut rat her fee d six of thenine cent er core e ng ines d ire ctly.
This  is very  similart o how th e Spa ceShuttl ewor ked. The S h utt le'scrossfe ed bet weenth e Extern al Tank and the Orbiter  was 1 7" in  di ameter --  on e  fo r propell ent,  one for o xid ize r.  A  pr o jectto r e du cet he si ze o f  the cros s  f eed t o14" was can c e l led .  Avid e o of t he d isconnect being t e sted can be se en h e r e:

. Some  q uestions:
Howlar ge a re the c ro ss feeds be tween th eF alcon  Heavy 's out er core s an d  the i nner  co re?
Are t her ef our (2fo rp ropell ent,  2  for o xid
α+1 be_the space_spanned by Vα and_the element_aVα._If λ_is_a limit ordinal,_let Vλ be_the union of all_earlier Vα. It_is_easy to see that { aVβ | β < α} is a basis for_Vα._Thus, the_dimension_of_each Vα is exactly the_cardinality of α. In particular,_if ω1_is the first uncountable ordinal, then Vω1 will_have_uncountable dimension, yet_be the union of all Vα for α <_ω1, which all have countable dimension,_as desired.
If you_forbid_one_to use the Axiom_of Choice, then it is no_longer true that every vector space_has a basis (since it is consistent_with ZF that some vector spaces_have no basis), and the_concept of_dimension suffers in this case._But some interesting_things happen._For example, it_is consistent with the failure of_AC that the_reals are a countable union of_countable_sets. R =_U_An,_where each_An is countable._(The_irritating difficulty_is_that although each An is countable,_one_cannot choose the functions witnessing this uniformly,_since of course R_is_uncountable.) But in any_case, we may regard R_as a vector space over Q,_and if_we let_Vn be the space spanned by A1 U... U An, then_we can still in each case_make finitely many choices_to witness_the_countability and conclude_that_each Vn_is countable dimensional, but the union of_all Vn_is all of R, which is_not countable dimensional.

 Q:

Questions_regarding_Falcon Heavy's crossfeed system

My understanding of_the Falcon Heavy cross feed system_is that the outer cores_feed_six_of the nine engines directly_on the center core.  Thus,_the cross feeds_don't pump propellent and oxidizer into the_center_core tanks but rather feed six_of_the nine center core engines directly.
This_is_very_similar to how the Space_Shuttle worked.  The Shuttle's cross_feed between the External Tank and the Orbiter was_17" in diameter_-- one for propellent, one_for_oxidizer._ A project to reduce the size of the cross_feed to_14" was cancelled._ A video of the disconnect being tested can be_seen here: 

.
Some questions:
How large are the_cross feeds between the Falcon Heavy's outer cores and the inner_core?
Are there four (2 for propellent, 2 for_oxid