[Sh:430.1-3] Elementary Submodel Conventions

January 27, 2011 at 18:39 | Posted in Uncategorized | Leave a comment

We’re going to be working with elementary submodels a lot, so I wanted to pin down the assumptions we make and the notation we use.

In the current project, we are investing combinatorics relevant to a singular cardinal {\mu}, and {\chi} is a regular cardinal “much much larger” than {\mu}. The exact identity of {\chi} isn’t important. We also need to fix some well-ordering {<_\chi} of {H(\chi)}; we will build this into the structures we consider so that they all have definable Skolem functions.

In general, {\mathfrak{A}} denotes some expansion of {\langle H(\chi),\in, <_\chi\rangle} by at most countably many functions, constants, and relations. I won’t get too pedantic with this, but for example, in the current situation we want {\mathfrak{A}} to contain a name for {\mu}, so that we know {\mu} makes it into any elementary submodel of {\mathfrak{A}}. We won’t make this explicit in our notation.

I will use {{\rm Sk}_{\mathfrak{A}}(B)} (or just {{\rm Sk}(B)}) to denote the Skolem hull of {B} in the structure {\mathfrak{A}}.

I may update this post to reflect new things that I forgot to include…

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