116c- Lecture 8

April 24, 2008

We defined infinite sums and products and showed that if $\max{2,\kappa_i}\le\lambda_i$ for all $i\in I$, then $\sum_{i\in I}\kappa_i\le\prod_{i\in I}\lambda_i$.

We also showed that if $(\kappa_i:i<\mbox{cf}(\kappa))$ is an increasing sequence of cardinals cofinal in $\kappa$, then $\prod_{i\in\mbox{\small cf}(\kappa)}\kappa_i=\kappa^{\mbox{\small cf}(\kappa)}$. In particular, $\prod_n\aleph_n=\aleph_\omega^{\aleph_0}$.

We defined singular cardinals and showed that (with choice) all successor cardinals are regular and all limit cardinals $\aleph_\alpha$ are singular unless $\alpha=\aleph_\alpha$. We showed that, indeed, there are fixed points of the aleph function, as a particular case of a result about normal functions. We defined (weakly) inaccessible cardinals as the regular limit cardinals (thus, regular fixed points of the aleph function).

Correction. I believe during lecture I mixed two arguments by mistake, making one of the proofs come out unnecessarily confusing, so I will present the correct argument here, for clarity.

In lecture we showed that if $F$ is a normal function, then it has a proper class of fixed points. Thus, we can enumerate them in increasing order. Let $G$ be this enumeration.

Claim. $G$ was also normal.

Proof. We need to check that $G$ is continuous. Let $\gamma$ be a limit ordinal and suppose that $\tau=\sup_{\beta<\gamma}G(\beta)$. We need to show that $G(\gamma)=\tau$.

By definition, this means that:

1. $\tau$ is a fixed point of $F$, and
2. $\tau$ is the $\gamma$-th fixed point of $F$.

But, clearly, if $\tau$ is a fixed point, then it must be the $\gamma$-th one, since we have already enumerated $\gamma$ fixed points below $\tau$, and any fixed point below $\tau$ is below some $G(\beta)$ with $\beta<\gamma$, so it is not even the $\beta$-th one.

So we only need to check that $F(\tau)=\tau$. But $\tau=\sup_{\beta<\gamma}G(\beta)$ and each $G(\beta)$ is a fixed point of $F$ (again, by definition of $G$), so $\tau=\sup_{\beta<\gamma}F(G(\beta))=F(\sup_{\beta<\gamma}G(\beta))=F(\tau)$, where the previous to last equality is by continuity of $F$. ${\sf QED}$

It follows that $G$ itself has a proper class of fixed points. It is also the case that there is a proper class of fixed points of $F$ that are limits of fixed points of $F$: Simply notice that the argument above shows that any limit of fixed points of $F$ is itself a fixed point. Thus, we have:

Corollary. The function $H:{\sf ORD}\to{\sf ORD}$ enumerating the limit points of $G$ (i.e., the fixed points of $F$ that are themselves limit of fixed points) is normal.

I believe during lecture I mixed at some point $H$ and $G$ (although I never explicitly mentioned $H$). Hopefully the above clarifies the argument. For the particular case of $F(\alpha)=\aleph_\alpha$, we have that $G$ enumerates the ordinals $\alpha$ such that $\alpha=\aleph_\alpha$, so $G(0)$ is the first such cardinal. The function $H$ enumerates the limit points of $G$, so $H(0)=G(\omega)$. Notice that $\mbox{cf}(H(0))=\mbox{cf}(G(\omega))=\omega$. One can easily see that if $\kappa$ is a weakly inaccessible cardinal, then $\kappa$ is a fixed point of $F$, $G$ and $H$.

In fact, define $F_0(\alpha)=\aleph_\alpha$ for all $\alpha$, let $F_{\beta+1}$ be the enumeration of the fixed points of $F_\beta$, and let $F_\gamma$ (for $\gamma$ limit) enumerate the ordinals $\kappa$ that are simultaneously fixed points of all the $F_\beta$ for $\beta<\gamma$. Then, if $\kappa$ is weakly inaccessible, then $F_\alpha(\kappa)=\kappa$ for all $\alpha<\kappa$.

Remark.

1.  We did not prove that weakly inaccessible cardinals exist. The examples given in lecture of fixed points of the aleph function have cofinality $\omega$ and, similarly, we can produce fixed points of arbitrarily large cofinality, but the argument falls short of finding regular fixed points (in fact, we can show that each $F_\alpha$ as defined above is normal, but the argument does not show that we can “diagonalize” to obtain a $\kappa$ fixed for all $F_\alpha$ with $\alpha<\kappa$). In fact, it is consistent with ${\sf ZFC}$ that all limit cardinals are singular. However, it is the general consensus among set theorists that the existence of inaccessible cardinals is one of the axioms of set theory that the original list ${\sf ZFC}$ somehow missed.
2. We defined normal functions as proper classes; however, we can as well define for any ordinal $\alpha$ a function $F:\alpha\to{\sf ORD}$ to be normal iff it is strictly increasing and continuous. The same argument as in lecture (or above) then shows that if $\mbox{cf}(\kappa)>\omega$ and $F:\kappa\to\kappa$ is normal, then there is a closed and unbounded subset of $\kappa$ consisting of fixed points of $F$. It turns out that closed unbounded sets are very important in infinitary combinatorics, and we will study them in more detail in subsequent lectures.