More elementary submodels (II)

May 23, 2011 at 16:55 | Posted in Uncategorized | Leave a comment

In this post, I want to give a more precise description of “Skolem hull arguments”, as such arguments are critical ingredients in many of the proofs we are looking at in this blog. I’ve done some searching, and I think I like the way Holst, Steffens, and Weitz present things in their book “Introduction to Cardinal Arithmetic” (referred to in the sequel as [HSW]), as they actually fill in several details that are usually glossed over and we’ll shortly be in need of this level of precision. I’m just going to sketch a little material from section 4.4 of their book, mainly to fix notation.

Our convention is that {\mathfrak{A}} denotes a structure of the form {\langle H(\chi), \in , <_\chi\rangle}, where {<_\chi} is a well-ordering of {H(\chi)}. We’ll denote the usual language of set theory (i.e., one binary relation symbol standing for {\in}) by {L}, and let {L^*} be {L} together with another binary relation symbol standing for {<_\chi}.

Given an {L^*} formula {\phi(x_0,\dots, x_n)} in which {x_0} occurs freely, let {f_\phi} be the function from {H(\chi)^n\rightarrow H(\chi)} which sends each {n}-tuple {\langle a_1,\dots,a_n\rangle} to the {<_\chi}-least element {a\in H(\chi)} for which

\displaystyle  H(\chi)\models\phi(a, a_1,\dots, a_n), \ \ \ \ \ (1)

if there is such an {a}, and which returns the value {0} if there is no such {a}. Thus, {f_\phi} is a Skolem function for {\phi}.

We usually think of the Skolem hull of {X\subseteq H(\chi)} as being the closure of {X} under all Skolem functions, but the fact that our Skolem functions are definable in {L^*} gives us the following:

Lemma 1 (Lemma 4.4.2 of [HSW]) If {X\subseteq H(\chi)}, then

\displaystyle  {\rm Sk}_{\mathfrak{A}}(X)=\bigcup\{ f_\phi[X]:\phi\text{ a formula in }L^*\}. \ \ \ \ \ (2)

The above will help make some of the arguments we present less technical than they might have been. Of course we will still be sloppy and refer to “elementary submodels of {H(\chi)}”. One can take this as either “officially” referring to elementary substructures of {\mathfrak{A}}, or referring to the {L}-reducts of such structures.

For an easy example of a Skolem hull argument, let’s look again at the preceding post. Back then, we were considering elementary submodels {M} and {N} of {H(\chi)}, where {\chi} was a regular cardinal much larger than some fixed singular cardinal {\mu}. The theorem proved there needed two assumptions:

\displaystyle  \mu\in M\cap N, \ \ \ \ \ (3)

and

\displaystyle   (\forall\alpha\in N\cap\mu)(\exists A\in M)(\alpha\in A\wedge A\subseteq N), \ \ \ \ \ (4)

and now we can prove the following proposition:

Proposition 2 The two assumptions above are satisfied if

\displaystyle  N={\rm Sk}_{\mathfrak{A}}(\eta\cup p) \ \ \ \ \ (5)

where

  1. {\mu\in p\subseteq M},
  2. {\eta} is a cardinal below {\mu}, and
  3. {\eta=\sup(M\cap\eta)}.

Proof: Clearly only (4) is an issue, so assume {\alpha\in N\cap \mu}. We know there is an {L^*} formula {\phi}, ordinals {\alpha_0,\dots,\alpha_{m-1}} in {\mu}, and {a_0,\dots, a_{n-1}} in {p} such that

\displaystyle  \alpha=f_\phi(\alpha_0,\dots,\alpha_{m-1},a_0,\dots,a_{n-1}). \ \ \ \ \ (6)

Fix {\zeta\in M\cap\eta} greater than {\max(\alpha_0,\dots,\alpha_{m-1})}, and define

\displaystyle  A:=\{\beta<\mu:(\exists\beta_0<\zeta)\dots(\exists\beta_{m-1}<\zeta)\phi(\beta,\beta_0,\dots,\beta_{m-1},a_0,\dots, a_{n-1})\}. \ \ \ \ \ (7)

The set {A} definable in {\mathfrak{A}} with parameters from {M}, hence {A\in M}. Notice that {A} lies in {N} as well, as {\zeta\in N} and {p\subseteq N}. Since {\zeta} (and hence {\zeta^n}) is a subset of {N}, it follows that {A\subseteq N}. Clearly {\alpha\in A}, too, and so we have what we want. \Box

******
Edit 5/27/2011 I left a piece of formula out of (7) —- there should be a “such that beta is the least such that…” in the definition of A. I’ll fix it when I get time.

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