Do black holes have a transinfinite density 1


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 φwhose 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 τP.-Name if and only if τ is a relation and for all (σ, p) ∈ τ holds that σ a P.-Name is and pP.. 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] := {τG : τM.P.}, where the quantities τG 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}