Thursday, June 30, 2016

On Weak Compactness II [edited]


\(
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\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
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\)

Lingering a little on the material from the last post just for curiosity's sake:

Theorem 

The following are equivalent for an inaccessible cardinal $\kappa$:

1.  $\kappa$ is weakly compact.

2.  For every sequence of functions $\langle f_\alpha:\alpha<\kappa\rangle$ with $f_\alpha:\alpha\rightarrow\alpha$, there is an $f:\kappa\rightarrow\kappa$ such that for every $\alpha<\kappa$ there is a $\beta\geq\alpha$ with $f_\beta\restr\alpha= f\restr\alpha$.

3. For every sequence of sets $\langle A_\alpha:\alpha<\kappa\rangle$ with $A_\alpha\subseteq\alpha$, there is a set $A\subseteq\kappa$ such that for every $\alpha<\kappa$ there is a $\beta\geq\alpha$ with $A_\beta\cap\alpha = A\cap\alpha$.

4. For every $C$-sequence $\langle C_\alpha:\alpha<\kappa\rangle$ there is a closed unbounded $C\subseteq\kappa$ such that for all $\alpha<\kappa$ there is a $\beta\geq\alpha$ with $C_\beta\cap\alpha= C\cap\alpha$.


We'll work this out next time to make sure, but (1) implies (2) by Shelah's result used last time,  and (4) implies (1) by a result of Todorcevic  (Theorem 6.3.5 of his book "Walks on Ordinals and their Characteristics").


One thing to note is that subtle cardinals, weakly ineffable cardinals, and ineffable cardinals all are defined using something that looks like condition (3).  It might be worthwhile to see if the corresponding modifications of (2) and (4) give equivalent characterizations.

See  Cantor's Attic for more information.


--------------

Edit:  (2) follows from (3) immediately, as we just set $f_\alpha$ to be the characteristic function of $A_\alpha$, and then $f$ is the characteristic function of the $A$ we need.

(4) follows from (3) immediately, as it is easily seen that if we apply (3) to a $C$-sequence $\langle C_\alpha:\alpha<\kappa\rangle$ then the resulting $A$ must be closed.










On weak compactness


\(
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\DeclareMathOperator{\tcf}{tcf}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\newcommand{\restr}{\upharpoonright}
\DeclareMathOperator{\ch}{Ch}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

We're firing the blog up again after a long hiatus.  Our immediate plan is to give a simplified proof of one of the main results from [GiSh:1013]  (Gitik-Shelah:  Applications of pcf for mild large cardinals to elementary embeddings).

We start with a lemma on weakly compact cardinals. I haven't seen this exact formulation before but certainly there aren't any new ideas involved.

Theorem

The following two conditions are equivalent for an uncountable cardinal $\kappa$:

1.  $\kappa$ is weakly compact.

2. If $c:[\kappa]^2\rightarrow\sigma$ with $\sigma<\kappa$, there is a function $f:\kappa\rightarrow\sigma$ and a set $H\subseteq\kappa$ of cardinality $\kappa$ such that for each $\alpha<\kappa$ we have $c(\alpha,\beta)=f(\alpha)$ for all sufficiently large $\beta\in H$.

Proof:

(2) implies (1)

Suppose $c:[\kappa]^2\rightarrow \{0, 1\}$, and let $f$ and $H$ be as in (2).   Define a function $g:\kappa\rightarrow\kappa$ by letting $g(\alpha)$ equal the least $\gamma<\kappa$ such that
\begin{equation}
\beta\in H \wedge \gamma\leq\beta\Longrightarrow c(\alpha,\beta)= f(\alpha).
\end{equation}

Thin out $H$ to $H_0$ of cardinality $\kappa$ so that
\begin{equation}
\alpha<\beta\text{ in }H\Longrightarrow g(\alpha)\leq\beta.
\end{equation}

This means that for $\alpha<\beta$ in $H_0$, $c(\alpha,\beta)=f(\alpha)$.  Now choose $\epsilon\in\{0,1\}$ so that $\{\alpha\in H_0:f(\alpha)=\epsilon\}$ has cardinality $\kappa$.  This set is homogeneous for the coloring $c$, and we conclude that $\kappa$ is weakly compact.



(1) implies (2)

We use a result due to Shelah (Theorem 1 of  [Sh:94] "Weakly compact cardinals: a combinatorial proof") which states that an inaccessible cardinal $\kappa$ is weakly compact if and only if for every family of functions $f_\alpha:\alpha\rightarrow\alpha$ (for $\alpha<\kappa$) there is a function $f:\kappa\rightarrow\kappa$ such that
$$(*) \qquad\qquad (\forall\alpha<\kappa)(\exists\beta<\kappa)[\alpha\leq\beta\wedge f_\beta\upharpoonright\alpha= f\restr\alpha].$$

Suppose $c:[\kappa]^2\rightarrow\sigma$ for some $\sigma<\kappa$.

 For $\alpha<\sigma$ we let $f_\alpha:\alpha\rightarrow\alpha$ be identically 0 (these $\alpha$ will be irrelevant), and for $\sigma\leq\alpha<\kappa$ we define $f_\alpha:\alpha\rightarrow\alpha$ by

$$f_\alpha(\gamma)=c(\gamma,\alpha).$$

Now fix a function $f:\kappa\rightarrow\kappa$ as in Shelah's theorem. We define an increasing sequence $\langle \beta_\xi:\xi<\kappa\rangle$ by the following recursion:


First, we set $\beta_0 =\sigma$.

Given $\langle \beta_\zeta:\zeta<\xi\rangle$, we define

$$\alpha_\xi=\sup\{\beta_\zeta:\zeta<\xi\}+1$$

and then use (*) to choose $\beta_\xi$ such that

$$\alpha_\xi\leq\beta_\xi\text{ and }f_{\beta_\xi}\restr\alpha_\xi = f\restr\alpha_\xi.$$


The preceding generates an increasing sequence $\langle\beta_\xi:\xi<\kappa\rangle$, and we note our construction guarantees that $c(\alpha,\beta_\xi)=f(\alpha)$ whenever $\alpha\leq\beta_\zeta$ and $\zeta<\xi<\kappa$.



In particular, if $\alpha<\kappa$ then $c(\alpha,\beta_\xi)= f(\alpha)$ for all sufficiently large $\xi<\kappa$, and $H:=\{\beta_\xi:\xi<\kappa\}$ is as required.