fom
Posts:
1,968
Registered:
12/4/12


Re: Notation
Posted:
May 25, 2013 10:48 PM


On 5/25/2013 8:36 PM, William Elliot wrote: > On Sat, 25 May 2013, fom wrote: > >> On 5/24/2013 10:57 PM, William Elliot wrote: >>> V is the ZFC universe. >>> L is the constructible universe. >>> L_omega0 is the omega_0th level of the constructible universe. >>> >>> Correct or needing correcting? >> >> Your apparently simple question seems to have generated >> some interesting replies. >> > Seemingly off the mark because I'm asking about notation and not about > theory. When constructing the constructible universe we start with > > L_0 = emptyset = 0 > L_1 = { 0, {0} }. Let {0} = 1. > L_2 = { 0, 1, {1}, {0,1} } > > L_(n+1) = L_n \/ P(L_n) > L_omega0 = \/_n L_n. > > Is L_eta the correct notation for the constructible > universe to the etath level? >
L_omega0 is correct.
L_(n+1) = GC(L_n \/ {L_n}) /\ P(L_n)
Where GC is the Goedel closure.
The constructible hierarchy is a cumulative hierarchy satisfying the order relation,
L_n subset L_(n+1) subset P(L_n)
Goedel closure is closure under composition of the operations,
F_01(U,V) = {U,V}
F_02(U,V) = U x V
F_03(U,V) = {<x,y> : x in U, y in V, x in y}
F_04(U,V) = U  V
F_05(U,V) = U /\ V
F_06(U) = /\U
F_06(U) = /\U
F_07(U) = {x : Ey(<x,y> in U)}
F_08(U) = {<x,y> : <y,x> in U}
F_09(U) = {<x,y,z> : <x,z,y> in U}
F_10(U) = {<x,y,z> : <y,z,x> in U}

