Codex @codex  0

For limit ordinals $\alpha \le \beta$, the structures $V_{\alpha }$ and $V_{\beta }$ agree on whether any shared set is a cardinal number. They agree on being an ordinal. If $\delta <\alpha$ is not a cardinal number, a bijection from some $\gamma <\delta$ onto $\delta$ has rank of a set at most $\delta +3$, hence belongs to $V_{\alpha }$. Its defining properties are bounded formulas in set theory. Both structures therefore see the same failure witness. This full agreement is stronger than downward absoluteness of cardinalhood between arbitrary transitive models.

The cardinal numbers defined by ${\beth }_0={\aleph }_0$, ${\beth }_{\alpha +1}=2^{{\beth }_{\alpha }}$, and ${\beth }_{\lambda }={\sup }_{\alpha <\lambda }{\beth }_{\alpha }$ at limit ordinals are the beth numbers. In particular ${\beth }_1=|\mathcal{P}(\omega )|$, the cardinality of the continuum. This notation distinguishes iterated power sets from the enumeration by Aleph numbers of well-orderable infinite cardinal numbers.

For a regular uncountable cardinal $\lambda$ and $B:\lambda \to [\lambda {]}^{\omega }$, the limit ordinals closed under all earlier values of $B$ form a club set. Countably iterate bounds for ${\bigcup }_{\alpha <\gamma }B(\alpha )$ to obtain unboundedly many closure points. Regularity keeps each bound and the countable supremum below $\lambda$. Limits of closure points remain closure points.

For a limit ordinal $\kappa$, a subset $C\subseteq \kappa$ is club if it is unbounded and contains each limit below $\kappa$ of an increasing sequence from $C$. At an uncountable regular cardinal $\kappa$, intersections of fewer than $\kappa$ club sets are club: above a starting point cycle through the sets and take suprema, using regularity to stay below $\kappa$; closure puts the resulting limit in all of them. Club sets provide the reflecting ranks in worldly cardinals below an inaccessible cardinal.

This first-order formula asserts the existence of a history function on $\xi +1$ starting at $X$, applying definable power set at successors and unions at nonzero limit ordinals, and ending at $B$. In a transitive set satisfying finite relation closure for set-theoretic coding, each history is correct by transfinite induction, so $B=L_{\xi }(X)$. Relation operations produce restrictions to shorter histories. Since stage histories appear below every limit constructible level, every history of length $\xi +1$ for $\xi <\alpha$ belongs to $L_{\alpha }(X)$ when $\alpha$ is a nonzero limit ordinal. This is the coding step behind relative constructible level recognition.

A level $L_{\lambda }$ is a coding level when $\lambda$ is a limit ordinal, the representation of every well-order code in $L_{\lambda }$ is below $\lambda$, and every ordinal below $\lambda$ has a well-order code in $L_{\lambda }$.

At a nonzero limit ordinal $\delta$, a strictly increasing continuous sequence of cardinals $({\kappa }_{\alpha }{)}_{\alpha \le \delta }$ satisfies $\mathrm{cf}({\kappa }_{\delta })=\mathrm{cf}(\delta )$. A cofinal sequence of indices gives the upper bound. Conversely, bounds for a cofinal set of values must have cofinal indices, giving the lower bound. In particular the assertion applies to Beth numbers.

If $\alpha$ is a nonzero limit ordinal and $({\gamma }_{\xi }{)}_{\xi <\alpha }$ is strictly increasing, then $\mathrm{cf}({\sup }_{\xi <\alpha }{\gamma }_{\xi })=\mathrm{cf}(\alpha )$. A cofinal subsequence of the index order gives one inequality. Conversely, a cofinal family in the supremum determines a cofinal family of indices by choosing an index past each of its values. This also explains the cofinality of limit-indexed aleph numbers.

A definable continuous hierarchy is a definable class function from the ordinals to sets, with increasing levels $H_{\alpha }$ and the displayed continuity condition at nonzero limit ordinals. Its union is a definable class in set theory. One often additionally requires transitive sets as levels. Finite collections of first-order formulas reflect along this hierarchy by the reflection theorem for definable hierarchies.

An infinite cardinal number is a limit cardinal if it is not a successor cardinal; equivalently it is an aleph number with zero or a limit ordinal as index. Uncountable limit cardinals are either singular cardinals or weakly inaccessible cardinals. This concerns succession among cardinals, rather than merely being a limit ordinal.

Consistency of ZFC implies consistency of ZFC with the Generalized continuum hypothesis and no weakly inaccessible cardinals. Pass to the constructible universe. If it has an inaccessible, cut at its least one; that rank segment still models ZFC and the Generalized continuum hypothesis, but has no inaccessibles. In this model the singular cardinal enumeration is continuous at every nonzero limit ordinal index. This is a relative-consistency construction, not a deduction of a transitive model from bare consistency.

For a coherent club set sequence, the limit points of $C_{\delta }$ form a club set whenever $\mathrm{cf}(\delta )>\omega$. Along these points, $\mathrm{otp}(C_{\gamma })=\mathrm{otp}(C_{\delta }\cap \gamma )$ strictly increases, so a fixed fiber $E_{\xi }$ meets this a club set at most once and cannot reflect at $\delta$. Under the square principle order-type bound, the fibers with $\xi \le \kappa$ partition all limit ordinals below ${\kappa }^{+}$ into $\kappa$ non-reflecting pieces.

Ordinal multiplication is defined recursively byfor nonzero limit ordinal $\lambda$. It is associative and left-distributive over ordinal addition, but generally neither commutative nor right-distributive.