Minimal Walks, Oscillation, and L- and S-Spaces: Examples and Counterexamples

The displayed $C_{\omega }$ is cofinal in $\omega$, contains $0$, and has finite intersection with every $m<\omega$. Together with $C_{\eta +1}=\{0,\eta \}$, it meets every local requirement of [F1] used in this calculation.

Aiming at $5$, the least points at or above the target are $\omega +1$ in $C_{\omega +2}$, $\omega$ in $C_{\omega +1}$, $6$ in $C_{\omega }$, and $5$ in $C_6$. Aiming at $\omega$, the first two choices are $\omega +1$ and $\omega$; aiming at $5$ from $\omega$, the choices are $6$ and $5$. This proves the three displayed walks and, after deleting their terminal points, the three asserted upper traces.

For target $5$, the successive intersections along the long walk are $C_{\omega +2}\cap 5=\{0\}$, $C_{\omega +1}\cap 5=\{0\}$, $C_{\omega }\cap 5=\{0,2,4\}$, and $C_6\cap 5=\{0\}$. Their cumulative maxima are $(0,0,4,4)$. For target $\omega$, the two intersections are both $\{0\}$, giving $(0,0)$; for target $5$ from $\omega$, the intersections are $\{0,2,4\}$ and $\{0\}$, giving $(4,4)$.

Passing from the running-max lists to their trace sets gives $L(\omega ,\omega +2)=\{0\}<\{4\}=L(5,\omega )$ and $L(5,\omega +2)=\{0,4\}$. Hence [F2] applies and its upper- and lower-trace unions agree exactly with the direct computations. No maximum of an empty set occurs, all three targets are strict lower endpoints, and multiplicities were retained until the final set calculation.

The maps $\xi \mapsto 3\xi$, $3\xi +1$, and $3\xi +2$ divide ${\omega }_1$ into successive three-point blocks: each displayed ordinal is countable, $3\xi <3\xi +1<3\xi +2<3(\xi +1)$, and different blocks are disjoint. Thus $A$ and $B$ are uncountable pairwise-disjoint families of two- and one-element subsets, respectively.

Use [F1] with $\pi (0)=\pi (1)=0$ and $\chi =(0,1)$. For the resulting $a<b$, its parity conclusion gives $c(a(0),b(0))=1-0=1$ and $c(a(1),b(0))=1-1=0$.

Because $a<b$, both $a(0)$ and $a(1)$ are strictly below $b(0)$. The defining endpoint clause in [F2] therefore turns the two equalities of step 2.1 into $b(0)\in W_{a(0)}$ and $b(0)\mathrm{\notin }{W}_{a(1)}$.

By [F2], $U=W_{a(0)}\cap ({\omega }_1\setminus W_{a(1)})$ is clopen and basic, and step 3.1 puts $b(0)$ in it. Hence $U$ is nonempty and is a neighborhood of the stated point. Both bits, the shared column, the strict order, and the complement bit are explicit; the invocation uses positive $k=2,l=1$ and makes no assertion beyond the functional graph allowed by [F1].

The first half of the proposed parallel is correct: [F1] constructs an L-space in ZFC. But [F1] obtains an S-space only in the separate CH branch and obtains the incompatible conclusion that no S-space exists in the PFA branch.

Under the source-consistency assumption in Given, [F1] proves that ZFC does not prove the existence of an S-space. Hence the assertion that both existence statements are ZFC theorems is false relative to that same standard large-cardinal consistency assumption.

Nor do the actual arguments arise by a formal interchange of two words: the L-space branch uses Moore's minimal-walk colouring in ZFC, the positive S-space branch uses CH, and the negative S-space branch uses PFA. These hypotheses and conclusions cannot be conjoined, and empty or singleton spaces supply neither kind of witness. Thus both the claimed parallel ZFC status and the claimed dual proof are refuted, with no converse consistency implication asserted.