## [Sh:430.1-3] Elementary Submodel Conventions

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…