## [Sh:430.1-2] Framework for proof

January 26, 2011 at 18:46 | Posted in Cardinal Arithmetic, [Sh:430] | Leave a commentLet us assume that is a singular cardinal with . We will prove that there is a family of cardinality such that every member of is a subset of some member of .

Fix a sufficiently large regular cardinal ; we will be working with elementary submodels of the structure where is some appropriate well-ordering of used to give us definable Skolem functions.

Let be a -approximating sequence, that is, is a -increasing and continuous sequence of elementary submodels of such that for each ,

- ,
- has cardinality ,
- is an initial segment of (so ), and
- .

Let denote . Then is an elementary submodel of of cardinality containing as both an element and subset. Define

Clearly and , so we need only verify that for any , there is a with .

In broad terms, this will be done via an “ argument” with , but we’ll fill in the details in further posts.

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