Continuity Conditions and Exact Upper Bounds

I want to use this post to work through some material which solidifies the relationship between square-like principles and scales. We begin with a theorem due to Shelah.

Theorem: Let \lambda, \kappa be regular cardinals with \kappa^{++}<\lambda. Then there is a stationary set S\subseteq S^{\lambda}_{\kappa}, and a sequence \bar C=\langle C_\alpha : \alpha<\lambda\rangle such that:

  1. Each C_\alpha is a closed subset of \alpha;
  2. If \beta\in nacc(C_\alpha), then C_\beta=C_\alpha\cap\beta;
  3. If \delta\in S, then C_\delta is a club subset of \delta with order-type cf(\delta)=\kappa.

The above sequence \bar C is referred to as a continuity condition for S by Shelah in Cardinal Arithmetic. We can think of these continuity conditions as weak fragments of square, which always hold in ZFC. The interesting thing is that continuity conditions allow us to produce exact upper bounds.

Definition: Let \bar C=\langle C_\alpha : \alpha<\lambda be a continuity condition for some stationary set S\subseteq S^\lambda_\kappa as above, and let \vec f=\langle f_\alpha :\alpha<\lambda \rangle be a sequence of functions from \kappa to the ordinals. We say \vec f weakly obeys \bar C if:

If \alpha<\lambda, then for each \beta\in nacc(C_\alpha), we have f_\beta(i)<f_\alpha(i) for each i<\kappa.

One thing to note is that despite having been published in 1994, Cardinal Arithmetic has no new material past 1989. In particular, the existence of continuity conditions over stationary sets was not known in the case that \lambda=\mu^+ for \mu singular when cardinal arithmetic was sealed. However, Todd Eisworth’s chapter in the handbook has a nice exposition about the above theorem. On the other hand, the following is included in Cardinal Arithmetic (appearing as Claim 2.6A on page 16):

Theorem: Let I be an ideal on a regular cardinal \kappa. Assume that:

  1. \lambda>\kappa^+ is regular such that there is some stationary S\subseteq S^\lambda_{\kappa^+} which has a continuity condition \bar C;
  2. \vec f=\langle f_\alpha : \alpha<\lambda\rangle is a sequence of functions from \kappa to the ordinals;
  3. \vec f obeys \bar C.

Then \vec f has a \leq_I-exact upper bound.

Proof: We first produce a \leq_I-least upper bound, and then show that this bound must be exact. In order to produce the desired lub, we inductively produce better and better upper bounds g_\xi which are better and better approximations to a lub as follows:

Stage \xi=0: Let g_0 be defined by g_0(i)=\sup_{\alpha<\lambda} f_\alpha(i) + 1.

Stage \xi=\eta+1: Suppose that g_\eta has been defined, and is a \leq_I-upper bound for \vec f. If g_\eta is a least-upper bound, then we can terminate the induction. Otherwise, there is some g_{\eta+1}\leq_I g_\eta which is an upper bound for \vec f such that g_{\eta+1}\neq_I g_\eta.

Stage \xi limit: Suppose that g_\eta has been defined for each \eta<\xi. For each i<\kappa, define the set S_\xi(i)=\{g_\eta(i) : \eta<\xi\}, and for each \alpha<\lambda, define f^\xi_\alpha ,the projection of f_\alpha to S_\xi=\langle S_\xi(i) : i<\kappa, by setting f^\xi_\alpha(i)=\min(S_\xi(i)\setminus f_\alpha(i)).

Note that each f^\xi_\alpha is defined I-almost everywhere, and that the sequence \vec{f}^\xi=\langle f^\xi_\alpha: \alpha<\lambda\rangle is \leq_I-increasing. We claim that there exists some \alpha_\xi<\lambda such that for each \alpha\in[\alpha_\xi, \lambda), we have f^\xi_\alpha=_I f^\xi_{\alpha_\xi}, and we set g_\xi=f^\xi_\alpha. To see this, assume otherwise.

As the sequence \vec f^\xi never stabilizes we can find a club set E\subseteq \lambda such that

\alpha,\beta\in E and \alpha<\beta implies f^\xi_\alpha\neq_I f^\xi_\beta.

Let \delta\in acc(E)\cap S, then C_\delta\subseteq \delta is club in \delta of order-type \kappa^+. We then inductively define an increasing sequence \{\beta_\epsilon : \epsilon<\kappa^+\} of ordinals in nacc( C_\delta) as follows:

We let \beta_0\in nacc(C_\delta). If \beta_\epsilon has been defined, then we can find \beta_{\epsilon+1}\in nacc(C_\delta) be such that f_{\beta_\epsilon}\neq_I f_{\beta_{\epsilon+1}}. Just pick \beta<\delta such that \beta\in E and \beta>\beta_\epsilon (which we can do since \delta is an accumulation point of E), and let \beta_{\epsilon+1}>\beta be in nacc(C_\delta). At limit stages, let \beta_{\epsilon}'=\sup_{\gamma<\epsilon}\beta_\gamma\in C_\delta, and let \beta_\epsilon>\beta_\epsilon' be in nacc(C_\delta). Then our collection has the following properties:

  1. \beta_\epsilon\cap C_\delta=C_{\beta_\epsilon};
  2. f^\xi_{\beta_\epsilon}\neq_I f^\xi_{\beta_{\epsilon+1}}.

Now for each \epsilon<\kappa^+, define the set

t_\epsilon=\{i<\kappa : f^\xi_{\beta_\epsilon}(i)<f^\xi_{\beta_{\epsilon+1}}(i)\}.

By condition 2, each t_\epsilon\notin I, and so for each \epsilon<\kappa^+ we can pick i(\epsilon)\in t_\epsilon. Since we have \kappa^+-many of these sets, and i ranges through \kappa, there must be some unbounded subset of \kappa^+ for which i(\epsilon) is constant. Call this constant value i(*). Now suppose \epsilon(1)<\epsilon(2)<\kappa^+ are such that i(\epsilon(1))=i(\epsilon(2))=i(*), then \beta_{\epsilon(1)+1}\in C_\delta\cap\beta_{\epsilon(2)}=C_{\beta_(\epsilon(2))} (in fact, it’s in nacc(C_{\beta(\epsilon(2))}. Since \vec f obeys \bar C, we have that f_{\beta_{\epsilon(1)+1}}(i(*))<f_{\beta_{\epsilon(2)}}(i(*)), and so f^\xi_{\beta_{\epsilon(1)+1}}(i(*))\leq f^\xi_{\beta_{\epsilon(2)}}(i(*)). Finally, since i(\epsilon(1))=i(\epsilon(2)) are both in t_{\epsilon(1))} and t_{\epsilon(2))} respectively, we get the following string of inequalities:

f^\xi_{\beta_{\epsilon(1)}}(i(*))<f^\xi_{\beta_{\epsilon(1)+1}}(i(*))\leq f^\xi_{\beta_{\epsilon(2)}}(i(*))<f^\xi_{\beta_{\epsilon(2)+1}}(i(*)).

But then, the sequence \langle f^\xi_{\beta_{\epsilon +1}}: i(\epsilon)=i(*)\rangle is strictly increasing along S_\xi(i), which has size \kappa. Since i(\epsilon)=i(*) happens \kappa^+-many times, this is absurd. Thus, the induction can be carried through limit stages.

Next, we claim that this induction cannot have been carried through \kappa^+-many stages. Otherwise, assume that g_\xi has been defined for each \xi<\kappa^+, and let \alpha(*)=\sup_{\xi<\kappa^+} \alpha_{\xi}<\lambda. Next note since the sequence \langle S_\xi (i) : \xi<\kappa^+ is \subseteq-increasing for each i<\kappa, that the sequence \langle f^\xi_{\alpha(*)} : \xi<\kappa^+\rangle must be decreasing and hence stabilize. Thus, we see that \langle f^\xi_{\alpha(*)}: \xi<\kappa^+\rangle must also be eventually constant. On the other hand, f^\xi_{\alpha(*)}=_I g_\xi for each \xi<\kappa^+, contradicting the fact that g_\xi\neq_I g_\eta for \xi,\eta large enough.

Therefore, \vec f has a \leq_I-upper bound g. If we can show that g is exact, then we are done. So suppose otherwise and let h witness this. That is, h is a function from \kappa to the ordinals such that h<_I g, but there is no \alpha<\lambda such that h<_I f_\alpha. So for each \lambda define t_\alpha=\{ i<\kappa : h(i)\leq f_\alpha(i)\}, which is I-positive by assumption. Further, note that \langle t_\alpha : \alpha<\lambda\rangle cannot stabilize modulo I else if it does stabilize at some \alpha(*), then define F:\kappa\to ON by

f(i)=h(i) if i\in t_\alpha and f(i)=g(i) otherwise.

It is straightforward to check that F would then be an upper bound of \vec f with F\leq g and F\neq_I g, contradicting that g is a lub. So as before, we can find a club E\subseteq \lambda such that for all \alpha<\beta in E, we have that t_\alpha\neq t_\beta with t_\beta\subseteq_I t_\alpha. Letting \delta\in S\cap acc(E), we can again find \{\beta_\epsilon : \epsilon<\kappa^+\} such that

  1. \beta_\epsilon\cap C_\delta=C_{\beta_\epsilon};
  2. t_{\beta_\epsilon}\neq t_{\beta_{\epsilon +1}}.

Let i(\epsilon)\in t_{\beta_\epsilon}\setminus t_{\beta_{\epsilon+1}}, and again find an unbounded subset of \kappa^+ such that i(\epsilon)=i(*). If we let \epsilon(1)<\epsilon(2)<\kappa^+ be such that i(\epsilon(1))=i(\epsilon(2))=i(*), we then get that

h(i(*))\geq f_{\beta_{\epsilon(2)}}(i(*))>f_{\beta_{\epsilon(1)+1}} (i(*))> h(i(*)).

This is silly, and so g is an exact upper bound of \vec f.

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