# Do black holes have a transinfinite density 1

## Forcing

Are a partial order *P.* ∈ *M.* and a *P.*- generic amount *G* given, it is closed *P.* and *G* appropriate generic extension *M.*[*G*] the smallest extension of *M.* (i.e. *M.* ⊆ *M.*[*G*]) to a countable transitive model of Φ, which *G* contains. The partial order *P.* controls which additional axioms in *M.*[*G*] be valid. So the art of forcing is to stick to a given axiom *φ*whose consistency with ZFC is to be shown, a suitable partial order *P.* to find so that *φ* in *M.*[*G*] is satisfied.

The formal construction of *M.*[*G*] is quite complicated and should only be sketched here. One defines to a given *P.* a class **V.**_{P.} so-called *P.*-Names. That means a lot *τ**P.*-Name if and only if *τ* is a relation and for all (*σ*, *p*) ∈ *τ* holds that *σ* a *P.*-Name is and *p* ∈ *P.*. This definition of *P.*-Name is to be understood as transfinite recursion. For the amount of *P.*- Names in M are written *M.*_{P.} : = *M.* ∩ **V.**_{P.}.

After all, you define *M.*[*G*] := {*τ*_{G} : *τ* ∈ *M. _{P.}*}, where the quantities

*τ*

_{G}through transfinite recursion as

\ begin {eqnarray} \ tau_ {G}: = \ Biggl \ {\ sigma_ {G}: \ bigvee \ limits_ {p \ in G} (\ sigma, p) \ in \ tau \ Biggr \} \ end {eqnarray }

are defined.For some proofs of consistency it is necessary to iterate the described process for the construction of generic extensions. It becomes an ordinal number *α* a chain of models

\ begin {eqnarray} M = M_ {0} \ subseteq M_ {1} \ subseteq \ ldots M _ {\ xi} \ subseteq \ cdots \ subseteq M _ {\ alpha} \ end {eqnarray}

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