Fleissner's HYP covering interface

Work in $\mathrm{Z}\mathrm{F}\mathrm{C}$ (The Axiom of Choice). HYP is the assertion that there exist an infinite cardinal $\kappa$, an increasing sequence of cardinals $({\kappa }_n{)}_{n\in \omega }$ cofinal in $\kappa$, and a set $E$ such that

$(1\mathrm{a}){\sup }_n{\kappa }_n=\kappa ;(1\mathrm{b})2^{{\kappa }_n}<\kappa \text{ for every }n;(2)2^{\kappa }={\kappa }^{+};$ $(3\mathrm{a})E\subseteq \{\delta <{\kappa }^{+}:\mathrm{cf}(\delta )=\omega \}\text{ and }E\text{ is stationary in }{\kappa }^{+};(3\mathrm{b})E\cap \beta \text{ is not stationary in }\beta \text{ for every }\beta <{\kappa }^{+}\text{ with }\mathrm{cf}(\beta )>\omega \mathrm{.}$

Clause $(1\mathrm{b})$ together with $(1\mathrm{a})$ says that $\kappa$ is a strong limit cardinal of countable cofinality; $(2)$ is the $\kappa$-continuum hypothesis; $(3\mathrm{a})$ and $(3\mathrm{b})$ say that $E$ is a nonreflecting stationary subset of the ordinals below ${\kappa }^{+}$ of countable cofinality (Cardinal (initial ordinal) and cardinality, The successor cardinal ${\kappa }^{+}$, the alephs ${\mathrm{\aleph }}_{\alpha }$, the beths ${\mathrm{\beth }}_{\alpha }$, successor and limit cardinals, and the identifications ${\mathrm{\aleph }}_0=\omega$ and ${\mathrm{\aleph }}_1={\omega }_1$, Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals, For every ordinal $\alpha$ there is a least ordinal $\beta$ admitting a map $\beta \to \alpha$ with cofinal range, and that map may always be taken strictly increasing, Closed unbounded subsets of ordinals, The club filter and nonstationary ideal, Cofinality strata, trace, and reflection).

Ladders. In a structure satisfying HYP fix, for each $\delta \in E$, an increasing sequence $({\delta }_i{)}_{i\in \omega }$ of nonlimit ordinals cofinal in $\delta$. For each $\delta$, For every ordinal $\alpha$ there is a least ordinal $\beta$ admitting a map $\beta \to \alpha$ with cofinal range, and that map may always be taken strictly increasing gives a strictly increasing cofinal sequence $c_i<\delta$; replacing $c_i$ by $c_i+1$ gives successor ordinals still strictly below the limit $\delta$, strictly increasing and cofinal. The axiom of choice then fixes one such sequence for every $\delta \in E$ (The Axiom of Choice); nothing below depends on which ladders are chosen beyond the two properties just named.

• The CH case. Under $\mathrm{C}\mathrm{H}$, take $\kappa =\omega$, ${\kappa }_n=n$ and $E$ the nonzero limit ordinals below ${\omega }_1$. The finite cardinals have supremum $\omega$ and $2^n<\omega$, while CH gives $2^{\omega }={\omega }_1$. The set $E$ is stationary: given any club $C\subseteq {\omega }_1$, choose a strictly increasing sequence from $C$. Its supremum is a countable nonzero limit ordinal, belongs to $C$ by closure, and belongs to $E$. Every nonzero limit $\beta <{\omega }_1$ has cofinality $\omega$, so the successor-sequence club described above avoids $E\cap \beta$. In particular, both $\{\omega n:1\le n<\omega \}$ and $\{\omega n+1:n<\omega \}$ are clubs in ${\omega }^2$ and are disjoint. Containing a club at an ordinal of countable cofinality therefore does not imply meeting every club. This is precisely Fleissner's CH instance on printed p.367; ladder separation may also be proved directly in that case.

• The separation function is not recorded here. Fleissner's Lemma 1 derives, from $(3\mathrm{b})$ and the ladders, a function $m_{\beta }$ separating distinct ladders below $\beta$; that derivation is proved locally in Ladder separation from HYP, and nothing in this definition asserts it.

• The large-cardinal reading is an input, not a consequence. That HYP is implied by the nonexistence of an inner model with a measurable cardinal is a theorem about the Dodd-Jensen core model, recorded in Dodd-Jensen covering supplies Fleissner HYP data and not folded into this definition.