Sh:E114

• Shelah, S. Strong Covering Lemma and CH in V[r]. Preprint.
  Ch. 7 of [Sh:g]
• Abstract:
  For an inner model \mathbf{W} of \mathbf{V}, the (\mathbf{W},\mathbf{V})-covering lemma states that for cardinals \lambda, \kappa with \lambda > \kappa = cf(\kappa) (usually \kappa \geq \aleph_1), the set \big([\lambda]^{< \kappa} \big)^{\!\mathbf{W}} = [\lambda]^{< \kappa} \cap \mathbf{W} is cofinal in [\lambda]^{< \kappa} (where [\lambda]^{< \kappa} = \big\{ A \subseteq \lambda : |A| < \kappa\big\}, ordered by inclusion).

  The strong (\mathbf{W},\mathbf{V})-covering lemma for (\lambda,\kappa) states that \big([\lambda]^{< \kappa} \big)^{\!\mathbf{W}} is a stationary subset of [\lambda]^{< \kappa}, which means that for every model M \in \mathbf{V} with universe \lambda and vocabulary of cardinality < \kappa, there is N \prec M with universe \in \big([\lambda]^{< \kappa} \big)^{\!\mathbf{W}}.

  We give sufficient conditions for the strong (\mathbf{W},\mathbf{V})-covering lemma to hold, which are satisfied in the classical cases where the original lemma holds (i.e. covering, squares, and reals). In fact, we place stronger conditions on M. The proof does not use fine structure theory, but only some well-known combinatorial consequences thereof.

  We use this to solve problems about the aspects of adding a real to a universe \mathbf{V}.

  Earlier versions appeared as [Sh:b][XIII,§1-4] in the author’s book Proper Forcing (Springer-Verlag 940, 1982), and later versions as Chapter VII of Cardinal Arithmetic (Oxford University Press, Clerendon Press, Vol. 24).

• Version 2024-12-02 (36p)

@article{Sh:E114,
 author = {Shelah, Saharon},
 title = {{Strong Covering Lemma and  CH in  $V[r]$}},
 note = {Ch. 7 of [Sh:g]},
 refers_to_entry = {Ch. 7 of [Sh:g]}
}