Minimal-walk weights, labelled lower traces, and the functions e-beta

Work in ZFC and retain the fixed $C$-sequence and traces from C-sequences and the upper and lower traces of minimal walks on omega-one. Write $2^{\omega }$ for Cantor space and $C(2^{\omega },\omega )$ for the continuous maps from Cantor space to discrete $\omega$. Such a map has finite image by compactness and is constant on the cells of a finite clopen partition. Every clopen subset of Cantor space is a finite union of basic cylinders, so there are only countably many such maps.

Use Solovay’s stationary partition theorem to partition ${\omega }_1$ into countably many stationary sets and fix a sequence $⟨w_{\xi }:\xi <{\omega }_1⟩$ in which every member of $C(2^{\omega },\omega )$ occurs on a stationary set. Cantor's theorem Cantor's theorem: $A\prec \mathcal{P}(A)$, together with AC's comparison of cardinals, gives an injection ${\omega }_1\to 2^{\omega }$; fix pairwise distinct $⟨z_{\alpha }:\alpha <{\omega }_1⟩$. These two simultaneous selections, and the stationary partition's ZFC proof, account for the The Axiom of Choice dependency.

Suppose $\alpha <\beta$ and write the walk and its running maxima as

$\beta ={\beta }_0>\cdots >{\beta }_n=\alpha ,m_0\le \cdots \le m_{n-1}\mathrm{.}$

For $\theta \in L(\alpha ,\beta )$ let $i(\theta )$ be the least $i<n$ with $m_i=\theta$. The labelled lower trace

$\mu (\alpha ,\beta ):L(\alpha ,\beta )⟶C(2^{\omega },\omega )$

is defined by $\mu (\alpha ,\beta )(\theta )=w_{{\beta }_{i(\theta )}}$. Thus the label at a repeated running maximum is the label from its first occurrence. For $\xi <{\omega }_1$, its evaluated form is the integer-valued function

$\mu (\alpha ,\beta ;\xi )(\theta )=\mu (\alpha ,\beta )(\theta )(z_{\xi })\mathrm{.}$

On the diagonal, $\mu (\alpha ,\alpha )$ is the empty function. In recursive language, the new minimum of the lower trace receives label $w_{\beta }$, and all strictly larger lower-trace points retain the labels from the next walk node. Consequently, whenever traces concatenate under the separation hypothesis, the labelled traces concatenate with the same restrictions; and for $0<\xi <\delta$,

$\mu (\xi ,\delta )(\min L(\xi ,\delta ))=w_{\delta }\mathrm{.}$

For $\alpha \le \beta$, the maximal weight is

${\rho }_1(\alpha ,\alpha )=0,{\rho }_1(\alpha ,\beta )=\max \{\mathrm{\mid }{C}_{\zeta }\cap \alpha \mathrm{\mid }:\zeta \in \mathrm{Tr}(\alpha ,\beta )\}\mathrm{.}$

This is a natural number because the trace and every displayed intersection are finite. Equivalently, if ${\beta }^{\mathrm{\prime }}=\min (C_{\beta }\setminus \alpha )$, then

${\rho }_1(\alpha ,\beta )=\max (\mathrm{\mid }{C}_{\beta }\cap \alpha \mathrm{\mid },{\rho }_1(\alpha ,{\beta }^{\mathrm{\prime }}))\mathrm{.}$

Finally define

$e_{\beta }:\beta ⟶\omega ,e_{\beta }(\alpha )={\rho }_1(\alpha ,\beta )\mathrm{.}$

The terms “coherent” and “finite-to-one” are conclusions of the next lemma, not assumptions smuggled into this definition.