An aleph omega plus one scale on an infinite set of successor alephs

We first give the cofinal-suborder transfer used below. Suppose an eventual-comparison product $P$ has a strict cofinal regular $\lambda$-chain $(p_{\xi }{)}_{\xi <\lambda }$, and $j:Q\to P$ preserves and reflects weak and strict comparisons and has a cofinal image. Every family of fewer than $\lambda$ elements in $P$ has a strict bound: choose for each member a weakly dominating chain term, bound these indices below $\lambda$ by F4 and take a later term. To build a chain in $j[Q]$, recurse through $\eta <\lambda$, strictly bounding all earlier chosen image elements together with $p_{\eta }$ by this rule, then choosing an image element weakly above that strict bound. A1 fixes a choice function on the nonempty witness subsets before the F6 recursion. The new image element strictly exceeds every predecessor and $p_{\eta }$; hence the image chain is strict and cofinal in $P$, and reflection makes its preimage a strict cofinal chain in $Q$. No cofinal family of size less than $\lambda$ exists in either order: in $P$ the strict bound just constructed contradicts cofinality, and a cofinal family in $Q$ would map to one in $P$. Thus this procedure transfers precisely the regular true cofinality $\lambda$. It uses no assertion that a ceiling map preserves strict inequalities.

Take the chain $f$ from F1. F2 gives all its uncountable regular projection properties below $\mu$, in particular at ${\mathrm{\aleph }}_1$. By F5, $\lambda$ is regular and greater than ${\mathrm{\aleph }}_1$, so F3 gives an exact least bound $v$ with positive limit values. The function $a(n)={\mathrm{\aleph }}_n$ is a pointwise bound; leastness gives $v{\le }^{\ast }a$. Set $h(n)=\min \{v(n),a(n)\}$, so $h{=}^{\ast }v$ and $h$ remains positive and limit-valued everywhere. Exactness and all eventual coordinate-cofinality conclusions are unchanged. Apply the ${\mathrm{\aleph }}_2$ conclusion and discard its finite exceptional set, leaving infinite $N^{\mathrm{\prime }}\subseteq N$, with $c(n)=\mathrm{cf}(h(n))\ge {\mathrm{\aleph }}_2>{\mathrm{\aleph }}_1$ for every $n\in N^{\mathrm{\prime }}$. For all these coordinates $c(n)\le {\mathrm{\aleph }}_n<\mu$: a cofinal subset of $h(n)\le {\mathrm{\aleph }}_n$ has size at most ${\mathrm{\aleph }}_n$. Further, for each finite $k$, the ${\mathrm{\aleph }}_{k+1}$ cofinality bound holds outside a finite set (using ${\mathrm{\aleph }}_1$ when $k=0$). Hence $c(n)>{\mathrm{\aleph }}_k$ eventually. In particular $c$ tends to $\mu$ in the sense of eventually exceeding every smaller cardinal. Restriction to $N^{\mathrm{\prime }}$ preserves exactness by F3.

Every $f_{\xi }{<}^{\ast }h$, since $f_{\xi }{<}^{\ast }{f}_{\xi +1}{\le }^{\ast }h$. After restriction to $N^{\mathrm{\prime }}$, reset its finitely many failure coordinates to zero to obtain ${\tilde{f}}_{\xi }\in {\prod }_{n\in N^{\mathrm{\prime }}}h(n)$. These resets preserve strict comparisons. Given $g$ in that product, exactness gives $g{<}^{\ast }{f}_{\xi }{=}^{\ast }{\tilde{f}}_{\xi }$ for some $\xi$, so the reset chain is cofinal. By F4 and A1 choose increasing cofinal maps $e_n:c(n)\to h(n)$. The map $E(t)(n)=e_n(t(n))$ from $\prod c(n)$ into $\prod h(n)$ preserves and reflects pointwise comparisons at every coordinate, hence also eventual weak and strict comparisons. Its image is cofinal: for each $g$ take the least ${\gamma }_n<c(n)$ with $g(n)\le e_n({\gamma }_n)$, which exists by cofinality. Apply step 1.1 to obtain true cofinality $\lambda$ on $\prod c(n)$ modulo finite sets.

Put $D=\mathrm{ran}(c\upharpoonright N^{\mathrm{\prime }})$. Each $d\in D$ is a regular cardinal by F4, at least ${\mathrm{\aleph }}_2$ and less than $\mu$. Each fiber $\{n:c(n)=d\}$ is finite: choose finite $k$ with $d\le {\mathrm{\aleph }}_k$; step 1.2 gives $c(n)>{\mathrm{\aleph }}_k$ outside a finite set. Consequently the preimage of a finite subset of $D$ is finite. Conversely if $E\subseteq D$ is infinite, its preimage is infinite, because the image of a finite set cannot be infinite and $c$ is onto $D$. For $u\in {\prod }_{d\in D}d$, define $R(u)(n)=u(c(n))$. Its comparison failure sets are exactly the preimages of those on $D$; thus $R$ preserves and reflects eventual weak and strict comparisons. Its image is cofinal: for $t\in \prod c(n)$ put

Apply the transfer of step 1.1 to the cofinal embedding $R$ of step 2.2 and the strict cofinal chain of step 2.1. This gives a scale of length $\lambda$ on $\prod D$ modulo finite sets, with no shorter cofinal family. Each $d\in D$ is a unique ${\mathrm{\aleph }}_k$ with finite $k\ge 2$, by F5 and the bound ${\mathrm{\aleph }}_2\le d<\mu$. Set $B=\{k\ge 2:{\mathrm{\aleph }}_k\in D\}$. The bijection $k\mapsto {\mathrm{\aleph }}_k$ from infinite $B$ to $D$ carries finite sets to finite sets in both directions and carries product functions to product functions. Reindexing the scale therefore preserves its strict comparisons and cofinality, giving the stated sequence $(s_{\alpha }{)}_{\alpha <{\mathrm{\aleph }}_{\omega +1}}$ on $B$. This proves the assertion, including its true cofinality claim. QED.