116c- Lecture 8

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.
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