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Topic: Notation
Replies: 14   Last Post: May 27, 2013 1:46 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Notation
Posted: May 25, 2013 10:48 PM
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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 eta-th 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}





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