Global singularization and the failure of SCH

Abstract

We say that κ is μ -hypermeasurable (or μ -strong) for a cardinal μ ≥ κ + if there is an embedding j : V → M with critical point κ such that H ( μ ) V is included in M and j ( κ ) > μ . Such a j is called a witnessing embedding. Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V ∗ where F is realised on all V -regular cardinals and moreover, all F ( κ ) -hypermeasurable cardinals κ , where F ( κ ) > κ + , with a witnessing embedding j such that either j ( F ) ( κ ) = κ + or j ( F ) ( κ ) ≥ F ( κ ) , are turned into singular strong limit cardinals with cofinality ω . This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality. As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ , at least for continuum functions which are simple in that 2 α ∈ { α + , α + + } for every cardinal α below κ (in this case every κ + + -hypermeasurable cardinal in the ground model is witnessed by a j with either j ( F ) ( κ ) ≥ F ( κ ) or j ( F ) ( κ ) = κ + ).