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 small categories $\mathcal{C},\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ and $Q:\mathcal{C}\to \mathbf{Set}$, choose an infinite cardinal number $\kappa$ bounding the number of morphisms of $\mathcal{D}$ and the total number of elements of $Q$. Given a natural transformation $\eta :Q\to GF$, generate a subfunctor $G^{\prime }\subseteq G$ from all ${\eta }_c(x)$ by applying every outgoing $\mathcal{D}$-morphism. Each $G^{\prime }(d)$ has at most $\kappa$ elements. Label bounded values by subsets of $\kappa$; only a set of such functors and transformations exists. These factorizations give the solution-set condition for precomposition, whose left adjoint is the left Kan extension.

There is an uncountable cardinal number $\kappa$ with ${\sigma }_{\kappa }=\kappa$. Iterate ${\kappa }_{n+1}={\sigma }_{{\kappa }_n}$ starting with ${\aleph }_0$. If there is no earlier fixed point, the increasing supremum $\kappa$ has countable cofinality, hence is a singular cardinal. The singular cardinal enumeration is cofinal in $\kappa$ below index $\kappa$, and continuity at this singular supremum gives equality.

An ordinal $\kappa$ is a cardinal number precisely when no smaller ordinal has a bijection onto it. Ordinalhood and the graph predicate for a bijection are bounded formulas in set theory, so $\mathrm{Ord}(\kappa )\wedge \forall f\forall \alpha \in \kappa \neg \mathrm{Bij}(f,\alpha ,\kappa )$ is a Pi-one formula modulo ZF. This does not assert that every set can be assigned an ordinal cardinal without the axiom of choice.

The cardinality $|X|$ of a set $X$ expresses its size up to bijection. Two sets have the same cardinality exactly when a bijection pairs their members. For a finite set, cardinality is the ordinary number of set elements. In the presence of the axiom of choice, every set can be assigned the least ordinal in its bijection class as its cardinal number.

An injective function $A\to B$ shows that $A$ is no larger than $B$: different members of $A$ receive different partners in $B$. If there are injections in both directions, the Cantor-Schröder-Bernstein theorem supplies a bijection. Infinite cardinality records this pairing structure rather than a geometric dimension or density. That is why a line of natural numbers and a grid of natural-number pairs can have the same size, while an interval of real numbers has larger cardinality.

For an infinite well-orderable cardinal number $\kappa$, each strict total order is a subset of ${\kappa }^2$, giving at most $2^{\kappa }$ possibilities by the square of an infinite cardinal. For a lower bound, order $\kappa$ two-element blocks by the ordinal index and independently reverse each block according to a subset of $\kappa$. Transport along $\kappa \times 2\cong \kappa$. Distinct subsets give distinct orders. The usual axiom of choice development supplies these product identities; for well-orderable cardinals they are already provable in ZF.

Let $M$ be a transitive set of cardinality $\kappa$ that models enough set theory and contains $\kappa$. The internal successor cardinal $\alpha =({\kappa }^{+}{)}^M$ is an ordinal in $M$, so transitivity gives $\alpha \subseteq M$ and hence $|\alpha |\le \kappa$ externally. Although $M$ regards $\alpha$ as a cardinal number, an ambient rank containing a bijection between $\kappa$ and $\alpha$ does not. Cardinalhood can therefore fail to be absolute even between transitive models.

If $\kappa$ is a regular cardinal and $\mathbb{P}$ has the $\kappa$-chain condition, then forcing with $\mathbb{P}$ preserves every cardinal number and cofinality at least $\kappa$. Indeed, a proposed surjection $f:\mu \to \lambda$ with $\mu <\lambda$ has its range contained in ${\bigcup }_{\xi <\mu }{B}_{\xi }$ from the possible-values lemma for chain-condition forcing. The regularity of $\kappa$ and infinite cardinal arithmetic make this union have cardinality below the ground-model cardinal $\lambda$, a contradiction.

A $\lambda$-closed forcing order preserves a ground-model cardinal number $\lambda$, even when it is singular. A proposed surjection from an ordinal below $\lambda$ to $\lambda$ would be an old function because closed forcing adds no short ordinal sequences, contradicting old cardinalhood. This concerns $\lambda$ and smaller cardinals; closure alone does not supply preservation of all larger cardinals.

A forcing order preserves cardinals over a ground model if every ground-model cardinal number remains a cardinal in every generic extension. This assertion and any sufficient chain condition for forcing are evaluated inside the ground model. External countability of a countable transitive model does not make all of its forcing orders internally ccc.

On a regular uncountable cardinal number $\kappa$, the club filter consists of all subsets containing a club set. It is a kappa-complete filter by club filter completeness. Its members are stationary sets, although a stationary set need not be a filter member, and a filter member need not itself be closed.

The club filter on a regular uncountable cardinal number $\kappa$ is $\kappa$-complete. Intersections of fewer than $\kappa$ club sets are closed. For unboundedness, repeatedly step past a point of each club and take a countable supremum; regularity keeps all suprema below $\kappa$, and closure puts the final supremum in every club.

For an infinite cardinal number $\kappa$, use cardinal arithmetic with the axiom of choice. If $\lambda =2^{\kappa }$ had cofinality $\mu \le \kappa$, choose cofinal ordinals ${\alpha }_i<\lambda$, $i<\mu$. Their cardinalities $a_i$ satisfy ${\sum }_i{a}_i=\lambda$. The König theorem for cardinal numbers gives $\lambda <{\prod }_i{a}_i^{+}\le {\lambda }^{\mu }\le (2^{\kappa }{)}^{\kappa }=\lambda$, a contradiction.

If $\kappa$ is a regular uncountable cardinal number, every family of $\kappa$ finite sets has a subfamily of cardinality $\kappa$ whose distinct members have one common pairwise intersection.

If a larger transitive model regards an ordinal $\kappa$ as a cardinal number, then a smaller transitive model does too: any bijection in the smaller model witnessing otherwise would remain in the larger one. The converse can fail when the larger model contains a new bijection between $\kappa$ and a smaller ordinal.

For an infinite ground-model cardinal number $\kappa$, use finite partial functions from $\omega$ to $\kappa$, ordered by reverse inclusion. The union of a generic filter is a surjection $\omega \to \kappa$. The order has size $\kappa$, hence the ${\kappa }^{+}$-chain condition, and preserves cardinals at least ${\kappa }^{+}$.

If $V_{\lambda }$, for an infinite cardinal number $\lambda$, satisfies Axiom schema of replacement for every external functional class, then an external cofinal function from an ordinal below $\lambda$ cannot exist: its range would have rank of a set $\lambda$ and would have to belong to $V_{\lambda }$. Similarly, a surjection $\mathcal{P}(\theta )\to \lambda$ with $\theta <\lambda$ is impossible. Infinity therefore makes $\lambda$ an uncountable regular cardinal and a strong limit cardinal. Full semantics for second-order logic, allowing arbitrary external functional relations, is essential.

The generalized continuum hypothesis states that $2^{\kappa }={\kappa }^{+}$ for every infinite cardinal number $\kappa$.

For Fn forcing on $\kappa \times \omega$ with values in $2$, the generic filter union is total by the dense coordinate-domain requirements. For each pair of distinct rows, the set of conditions giving them opposite bits at some column is dense: choose a column untouched in both finite rows. Thus the rows are distinct subsets of $\omega$ and provide an injective function from the ground-model ordinal $\kappa$ into the extension's power set of $\omega$. This construction alone does not prove preservation of its ground-model cardinal number status.