 → R having the eigen-value r when sampling S ⊆ U. Thus the set-version of the Born Rule is not some weird quantum notion of probability on sets but the perfectly ordinary Laplace-Boole rule for the conditional probability Pr (r|S) = |f −1(r)∩S| |S|, given S ⊆ U, of a random variable f : U → R having the value r. The collecting-together of some eigen-elements {u} ⊆ U into a subset S ⊆ U to form an "indefinite element" S has the vector sum |S〉 = ∑ u∈U 〈{u} |US〉 |{u}〉 in the vector space ℘ (U) over Z2 giving the superposition version of the indefinite element. This cements the interpretation of collecting together in sets as superposition in vector spaces. The indefinite element S is being measured using the observable f where the probability Pr (r|S) of getting the eigenvalue r is ‖f−1(r)∩S‖2 U ‖S‖2U = |f−1(r)∩S| |S| and where the "damned quantum jump" (Schrödinger) goes from S by the projection operator f−1 (r) ∩ () to the projected resultant state f−1 (r) ∩ S which is in the eigenspace ℘ ( f−1 (r) ) for that eigenvalue r. The partition operation in QM/sets that describes measurement is the partition join of the partition {S, Sc} and f−1 = { f−1 (r) } so that the initial pure state S (as a mini-blob) is refined into the mixture { f−1 (r) ∩ S } of possible resultant states. The other states { f−1 (r) ∩ Sc } in the join f−1 ∨ {S, Sc} are not possible or "potential" states starting from S. The state resulting from the measurement represents a more-definite element f−1 (r)∩S that now has the definite f -value of r-so a second measurement would yield the same eigenvalue r with probability Pr ( r|f−1 (r) ∩ S ) = |f−1(r)∩[f−1(r)∩
→ R having the eigen - value r when sampling S ⊆ U. therefore the hardening - version of the Born Rule is not some weird quantum impression of probability on sets but the absolutely ordinary Laplace - Boole rule for the conditional probability Pr (r|S) = |f −1(r)∩S| |S|, give S ⊆ U, of a random variable f: uracil → R having the value r. The collection - together of some eigen - elements { u } ⊆ U into a subset S ⊆ U to imprint an " indefinite element " S has the vector sum |S 〉 = ∑ u∈U 〈 { u } |US 〉 |{u } 〉 in the vector space ℘ (uracil) over Z2 giving the superposition version of the indefinite chemical element. This cements the interpretation of collecting in concert in sets as principle of superposition in vector spaces. The indefinite element S is being measured using the discernible f where the probability Pr (r|S) of getting the eigenvalue r is ‖f−1(r)∩S‖2 U ‖S‖2U = |f−1(r)∩S| |S| and where the " damned quantum jump " (Schrödinger) goes from S by the projection operator f−1 (r) ∩ () to the projected resultant state f−1 (r) ∩ S which is in the eigenspace ℘ (f−1 (r) ) for that eigenvalue r. The partition operation in QM / sets that describe measurement is the partition join of the division { S, Sc } and f−1 = { f−1 (radius) } so that the initial saturated state S (as a mini - blob) is refined into the mix { f−1 (r) ∩ S } of possible resultant country. The other states { f−1 (r) ∩ Sc } in the join f−1 ∨ { S, Sc } are not possible or " potential " state starting from S. The state resulting from the measurement represents a more - definite chemical element f−1 (r)∩S that now has the definite f -value of roentgen - so a second measurement would yield the same eigenvalue r with probability Pr (r|f−1 (r) ∩ S) = |f−1(r)∩[f−1(r)∩
 → R having the eigen-value r when sampling S ⊆ U. Thns the aet-versiun of the Born Rule is not slmw weied quantum notion of pfobabilitj on sets but rhe perfecvmy ordikcry Lzilace-Yoile rule for tme conditiotal probabilitf Or (r|S) = |f −1(r)∩S| |S|, given S ⊆ U, of a random dariablr v : U → R having the dalus r. The collecting-together of some eigen-enements {u} ⊆ U onto a subset S ⊆ U to form an "lndefinite element" S has the cectje sum |S〉 = ∑ u∈U 〈{u} |US〉 |{u}〉 in the vector apace ℘ (U) over Z2 giving the supefposicion versiob if hve indefinive eleient. This cements tha interlretation of cpllxctibg together in sets av superposition in vector s[aees. The indefinite elwmwnt S is teine mexsudev uaing tje kbservable f where thw probability Pr (r|S) os getting the sigenvwlte r is ‖f−1(r)∩S‖2 U ‖S‖2U = |f−1(r)∩S| |S| and where the "ddmnsd quantum jump" (Schrödinter) goes from S by thg projectijn operator f−1 (r) ∩ () to the projected resultant stata f−1 (r) ∩ S cmlch kw ln the eigenspace ℘ ( f−1 (r) ) for that eigenvalue w. Tne partition opevation in QM/sets tnah cgscribes measutement na fhe partition join of the partution {S, Ss} anc f−1 = { f−1 (r) } so that the initual pure stane S (as a mini-blob) is xefined into the mixtire { f−1 (r) ∩ S } of possiblz resumtant stated. The othsf states { f−1 (r) ∩ Sz } pn tve join f−1 ∨ {S, Sc} are not pofsible or "potzntial" sgatex stareing from D. The state resulting frlm thg measgrement reoresents a more-definite element f−1 (r)∩S that now hds nhe definnte f -yalue of r-so a fecond measurekent wobld yidld the saje eigeivalue r witr probability Kr ( r|f−1 (r) ∩ S ) = |f−1(r)∩[f−1(r)∩
→ R having the eigen-value r when ⊆ Thus the of the Born quantum of probability on but the perfectly Laplace-Boole rule for the conditional probability (r|S) = |f −1(r)∩S| |S|, given S ⊆ U, of a random variable : U → R having the value r. The collecting-together of some eigen-elements ⊆ into subset ⊆ U to form an "indefinite element" S has the vector sum |S〉 = ∑ u∈U |US〉 |{u}〉 in the vector space ℘ (U) Z2 giving the superposition of the indefinite element. This the of collecting in as in vector spaces. indefinite element S is being measured using the observable f where the probability Pr (r|S) of getting eigenvalue r U ‖S‖2U |f−1(r)∩S| and the "damned quantum goes from S by the projection ∩ () to the projected resultant state f−1 ∩ S is in the eigenspace ℘ ( (r) ) for that eigenvalue r. The partition in QM/sets that describes measurement is the partition join of the partition {S, Sc} and { f−1 (r) } that the initial state (as mini-blob) refined into mixture { f−1 (r) ∩ S } of possible resultant states. other states { f−1 (r) ∩ Sc } in the ∨ Sc} are not or "potential" states starting S. state resulting from the a element now the f -value of r-so second measurement would yield the eigenvalue r with probability S ) = |f−1(r)∩[f−1(r)∩
 → R having the eigen-value r when Sampling S ⊆ U. thus tHe sEt-vErSion Of thE Born Rule is not SOme wEird quantum notion of proBabilItY On seTS bUt the PerfectLY oRDInaRy laPlaCe-bOoLe rulE foR the conDitional prObaBiLity Pr (r|S) = |f −1(r)∩S| |S|, GIvEn S ⊆ U, of a ranDom Variable f : U → R hAviNg the vAlUe r. tHe colLecTing-tOgetheR Of some Eigen-elemEnTS {u} ⊆ U intO A subset s ⊆ u To Form An "indefinite elemeNT" S HAs the vector sum |s〉 = ∑ u∈U 〈{u} |US〉 |{U}〉 iN ThE VEctOr sPace ℘ (U) over Z2 GiVing tHE superpOSiTION veRSion of the indeFinite elemeNT. ThIs cemeNtS thE InterpRetatIoN Of cOllecting toGethEr in sets aS superPOsition IN vector Spaces. the IndEfinITe ElEmeNt s Is bEInG meASurEd using tHe ObServaBle f WHERE the ProBabiLity PR (r|S) of getting tHe eIgenVAluE r is ‖f−1(R)∩S‖2 U ‖S‖2U = |F−1(r)∩S| |S| AnD wherE the "daMned qUaNtum jump" (SchrödiNger) Goes from S By tHe ProJeCtion OPeratoR f−1 (r) ∩ () To tHe projeCted resULtaNt STATe F−1 (r) ∩ S which is in the eigEnSPAcE ℘ ( f−1 (r) ) for thAt eigeNVaLuE R. The partItIon OperATIon in qM/seTS tHat descrIbes meASuReMent is tHe PartitIoN joIn oF the pARtitIon {S, Sc} And f−1 = { f−1 (r) } so That tHE initial pure stATe S (as a mini-bloB) Is REFiNEd inTo tHe mixture { f−1 (r) ∩ s } of pOSsibLe reSUlTanT StateS. The oThER sTAtes { f−1 (r) ∩ Sc } in the join f−1 ∨ {S, sc} Are not PossiBle or "potentiaL" states staRTINg from S. THe stATe REsulting from thE measUrement repREsents a mOre-deFinite elEment f−1 (r)∩S tHAT now has tHe dEfiNitE f -vALUe Of r-so a second mEASureMeNt would YieLd the saMe eIgeNvaLue R wIth probabIlity Pr ( r|F−1 (r) ∩ s ) = |f−1(R)∩[f−1(R)∩
 → R having the eigen-valu e r when s ampli ngS ⊆  U . Th us t he set-version of t he Born Rule is not so me we ir d  qua n tu m not ion ofp ro b a bil it yonse t sbut t heperfect ly ordinar y L ap lace-Boole r u le  for the c ond itional prob abi lity P r(r| S ) = | f − 1(r)∩ S| |S| ,  given  S ⊆ U, o fa  rando m  variab l e  f  : U  → R having the v a lu e  r. The collec ting-t og e th e r  of  so me eigen-e le ments {u} ⊆ U in t o a s u bset S ⊆ U to  form an "i n def initeel eme n t" S h as th ev ect or sum |S〉= ∑u∈U 〈{u}|US〉 | { u}〉 int he vect or spa ce℘ ( U) o v er  Z 2 g iv i ngt he  su p erp ositionve rs ion o f th e i n defi nit e el ement . This cement s t he i n ter preta tionof c ol lecti ng tog ether  i n sets as super posi tion in v ect or  sp ac es. T h e inde fin ite  elemen t S isb ein gm e a su red using the obse rv a b le  f where  the p r ob ab i lity Pr(r |S)  ofg e tting  the ei genvalue  r is‖ f− 1( r)∩S‖2U‖S‖2U=|f− 1(r )∩S|| S| a nd whe re the " damne d  quantum jump" (Schrödinger) go e s  f r om S  by  the projec tion oper ator f− 1 ( r ) ∩ ( ) toth e  p r ojected resultant s ta te f−1  (r)∩ S which isin the eig e n s pace ℘ (  f−1 (r )  ) for that ei genva lue r. The partitio n ope ration i n QM/sets t hat desc rib esmea sur e m en t is the part i t ionjo in of t hepartiti on{S,  Sc } a nd  f−1 = {f−1 (r)}so  t ha t t he in i tial pur esta te  S(as a mini-b lob)is r ef in e d i nto the mi x t ure{f− 1 (r ) ∩  S  } of  pos s ibl e resul tant stat es. Theot he r state s { f−1 (r) ∩  S c } in the  j oin  f−1 ∨ { S, Sc} a re not possible or "pot e ntial"sta tes s tart ing fromS.The st ate result ing fr om th emea s u remen t re pre se nts a more - d efi niteel emen t f−1 ( r)∩S that now hast hedefinite f -v alu e of r -s o a se c ond  m e asu r e ment would yiel d the same  e i ge nvalue r w i thpr obabili ty Pr (  r|f− 1  (r) ∩S ) = |f− 1(r)∩[f−1 (r )∩
 →_R having_the eigen-value r when_sampling S_⊆_U. Thus_the_set-version of the_Born Rule is_not some weird quantum_notion of probability_on_sets but the perfectly ordinary Laplace-Boole rule for the conditional probability Pr (r|S) =_|f_−1(r)∩S| |S|,_given_S_⊆ U, of a random_variable f : U →_R having_the value r. The collecting-together of some eigen-elements_{u}_⊆ U into_a subset S ⊆ U to form an "indefinite_element" S has the vector sum_|S〉 = ∑_u∈U_〈{u}_|US〉 |{u}〉 in the_vector space ℘ (U) over Z2_giving the superposition version of the_indefinite element. This cements the interpretation of_collecting together in sets as superposition_in vector spaces. The indefinite_element S_is being measured using the_observable f where_the probability_Pr (r|S) of_getting the eigenvalue r is ‖f−1(r)∩S‖2_U ‖S‖2U =_|f−1(r)∩S| |S| and where the "damned_quantum_jump" (Schrödinger) goes_from_S_by the_projection operator f−1_(r)_∩ ()_to_the projected resultant state f−1 (r)_∩_S which is in the eigenspace ℘_( f−1 (r) )_for_that eigenvalue r. The_partition operation in QM/sets that_describes measurement is the partition join_of the_partition {S,_Sc} and f−1 = { f−1 (r) } so that the_initial pure state S (as a_mini-blob) is refined into_the mixture_{_f−1 (r) ∩_S_} of_possible resultant states. The other states {_f−1 (r)_∩ Sc } in the join_f−1 ∨ {S, Sc}_are_not possible or "potential" states starting_from S. The state resulting from_the measurement represents a more-definite_element_f−1_(r)∩S that now has the_definite f -value of r-so a_second measurement would_yield the same eigenvalue r with probability_Pr_( r|f−1 (r) ∩ S )_=_|f−1(r)∩[f−1(r)∩