Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $^*$Forcing   Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete
Boolean algebra in $M.$ (In general a proper class.) We define a generalized
notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing
extension of $M$ is a transitive set of the form $M[{\bf G}]$ where ${\bf G}$
is an $M$-complete ultrafilter on ${\bf B}.$) We prove that
  1. If ${\bf G}$ is a $^*$forcing complete ultrafilter on ${\bf B},$ then
$M[{\bf G}]\models ZFC.$
  2. Let $H\sub M.$ If there is a least transitive model $N$ such that $H\in
M,$ $Ord^M=Ord^N,$ and $N\models ZFC,$ then we denote $N$ by $M[H].$ We show
that all models of $ZFC$ of the form $M[H]$ are $^*$forcing extensions of $M.$
  As an immediate corollary we get that $L[0^{\#}]$ is a $^*$forcing extension
of $L.$

4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
19
The categories of this paper are: eess