C-sequences and the upper and lower traces of minimal walks on omega-one

Work in ZFC. A locally finite $C$-sequence on ${\omega }_1$ is a sequence $⟨C_{\gamma }:\gamma <{\omega }_1⟩$ such that

• $C_0=\mathrm{\varnothing }$;
• if $\gamma >0$, then $C_{\gamma }\subseteq \gamma$ is cofinal in $\gamma$;
• $C_{\gamma }\cap \eta$ is finite whenever $\eta <\gamma$; and
• $0\in C_{\gamma }$ whenever $\gamma >0$.

At a successor one may and shall take $C_{\eta +1}=\{0,\eta \}$, read as $\{0\}$ when $\eta =0$. At a nonzero countable limit $\gamma$, choose a strictly increasing cofinal $\omega$-sequence and adjoin $0$. The choice, simultaneously for all limit $\gamma <{\omega }_1$, is the use of The Axiom of Choice in this definition; the definitions made from a fixed $C$-sequence use no further choice.

Fix such a sequence. For $\alpha <\beta <{\omega }_1$, the minimal walk from $\beta$ down to $\alpha$ is the finite decreasing sequence

$\beta ={\beta }_0>{\beta }_1>\cdots >{\beta }_{n-1}>{\beta }_n=\alpha ,$

where, as long as ${\beta }_i>\alpha$,

${\beta }_{i+1}=\min (C_{{\beta }_i}\setminus \alpha )=\min \{\xi \in C_{{\beta }_i}:\alpha \le \xi \}\mathrm{.}$

The displayed set is nonempty: cofinality supplies an element at a limit stage, and the predecessor belongs to the chosen successor set. Its minimum is below ${\beta }_i$. If the recursion never reached $\alpha$, it would give an infinite strictly decreasing sequence of ordinals, contrary to the well-ordering in Ordinal (von Neumann). Thus $n<\omega$ and the walk is well defined.

Its upper trace is

$\mathrm{Tr}(\alpha ,\beta )=\{{\beta }_i:i<n\},$

with $\mathrm{Tr}(\alpha ,\alpha )=\mathrm{\varnothing }$. For $i<n$ put

$m_i(\alpha ,\beta )=\{\begin{array}{ll}\max \text{\hspace{0.17em}⁣}\left({\displaystyle \bigcup _{j\le i}(C_{{\beta }_j}\cap \alpha )}\right),& \alpha >0,\\ 0,& \alpha =0.\end{array}$

The union in the first line is a nonempty finite set: it contains $0$, and it is a finite union of finite initial intersections. The lower trace is

$L(\alpha ,\beta )=\{m_i(\alpha ,\beta ):i<n\},L(\alpha ,\alpha )=\mathrm{\varnothing },$

listed in its inherited nondecreasing order when multiplicities along the walk matter. Equivalently, with ${\beta }^{\mathrm{\prime }}=\min (C_{\beta }\setminus \alpha )$ and $m=\max (C_{\beta }\cap \alpha )$ for $\alpha >0$,

$L(\alpha ,\beta )=(L(\alpha ,{\beta }^{\mathrm{\prime }})\cup \{m\})\setminus m;$

here an ordinal $m$ is the set of its predecessors, so subtraction discards earlier values below the new running maximum. The explicit $\alpha =0$ convention avoids the undefined expression $\max \mathrm{\varnothing }$ and gives $L(0,\beta )=\{0\}$ for $\beta >0$.

For finite sets of ordinals, $a<b$ means that every member of $a$ is below every member of $b$. This convention will be used in the concatenation statements below; it is not the comparison of their cardinalities.