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In a locally small category, the morphisms from $A$ to $B$ form the set $\mathcal{C}(A,B)$. Postcomposition and precomposition give the corresponding covariant and contravariant representable functors.

A family of countable sets indexed by a linear order is increasing if $i\le j$ implies $X_i\subseteq X_j$. Its union has cardinality at most ${\aleph }_1$, by the cardinal bound for an increasing chain of countable sets. The index order need not be a well-order.

For an infinite subset $A\subseteq \mathbb{N}$, repeatedly selecting the least element not yet used gives a bijection $f:\mathbb{N}\to A$. The well-order ensures each minimum exists, infinitude ensures the remaining set is nonempty, and strict increase gives injectivity. If $a\in A$ were omitted forever, all selected values would be at most $a$, impossible inside the finite set $\{0,\dots ,a\}$.

An infinite Dedekind-finite set is an infinite set with no subset that is a countably infinite set. The name Dedekind set is sometimes used for this combination of properties. Under the axiom of choice no such set exists; results about these sets must avoid obtaining arbitrary enumerations of finite subsets by unstated choices.

An infinite set is a set with infinitely many elements. Equivalently, assuming the axiom of choice, it contains a countably infinite set.

An inner model $W$ of ZF containing all ambient subsets of $\omega$ computes the same countable ordinals as the ambient universe. Every ambient countably infinite ordinal has a well-order code coded by a subset of $\omega$, hence that code belongs to $W$; finite ordinals are already shared. The Mostowski collapse theorem inside $W$ recovers the actual ordinal and its countability witness. Conversely any countability witness in $W$ remains one outside. If $W$ also contains the full power set of $\omega$ as a set, its Continuum hypothesis would give an ambient bijection from ${\omega }_1$ onto that same power set. Thus an ambient failure of this well-orderable formulation of the Continuum hypothesis is preserved in such an inner model. No internal axiom of choice is required for this argument.

The intersection of any family of equivalence relations on one set is an equivalence relation. Reflexivity and symmetry hold in every component. A pair of consecutive related points belongs to every component, so each component's transitivity supplies the required final pair. Its equivalence classes refine the component partitions. The empty intersection is the universal relation.

For a finite set $A\subseteq [n]$ and a finite list $L$ of allowed intersection sizes, the displayed polynomial satisfies $p_A({\chi }_B)={\prod }_{\ell \in L}(|A\cap B|-\ell )$ at characteristic vectors of sets. For an $r$-uniform set family whose pairwise intersection sizes lie in $L\subseteq \{0,\dots ,r-1\}$, these polynomials vanish at all other family members and have nonzero values at their own members. Their restrictions are therefore linearly independent. Multilinear reduction on the Boolean cube and homogenisation on a uniform layer then prove the Ray-Chaudhuri–Wilson theorem bound.

An isomorphism of categories is a functor with a strictly inverse functor. It is equivalently bijective on objects and on each hom-set. An equivalence of categories only requires inverse composites up to invertible natural transformations. For example, the category of partial functions and the category of pointed sets are equivalent but their actual object collections prevent an isomorphism: the former has one zero object, the empty set, while the latter has distinct singleton pointed set objects that are all zero objects.

For a set $A$ of size an uncountable regular cardinal $\kappa$, an increasing sequence $(A_{\alpha }{)}_{\alpha <\kappa }$ with union $A$, each $|A_{\alpha }|<\kappa$, and $A_{\delta }={\bigcup }_{\alpha <\delta }{A}_{\alpha }$ at every nonzero limit stage.

This representation of an ordered pair by sets satisfies $⟨a,b⟩=⟨c,d⟩$ exactly when $a=c$ and $b=d$. Its intersection recovers $\{a\}$, and its union recovers $\{a,b\}$; if the latter is a singleton then $b=a$. Its rank of a set is $\max (\mathrm{rank}(a),\mathrm{rank}(b))+2$, which bounds the ranks of function graphs in absoluteness of cardinalhood in limit ranks.

A meromorphic continuation extends a holomorphic function from a connected open set to a larger connected domain as a meromorphic function, allowing isolated poles. Equality on the initial open set determines the extension uniquely by the identity theorem. Integral formulas whose integrands depend holomorphically on a parameter often supply such an extension after contour deformation or subtraction of singular terms.

A metric on a set $X$ is a function $d:X\times X\to [0,\infty )$ that is symmetric, satisfies the triangle inequality, and obeys $d(x,y)=0$ exactly when $x=y$.

A multiset records a nonnegative integer multiplicity for each element of a set. A finite multiset has finite support and finite total multiplicity. Its cardinality is the sum of its multiplicities; its disjoint union with another multiset adds multiplicities. The hook lengths of a Young diagram form a multiset, since different cells can have equal lengths.

If $H=G_0\times G_1$ is generic for product forcing over $M$, then $G_1$ is generic over $M[G_0]$. For a name $\dot{D}$ for a dense subset of the second factor, take $p_0\in G_0$ forcing density. Ground-model pairs with first coordinate incompatible with $p_0$, or with first coordinate below $p_0$ forcing the second into $\dot{D}$, form a dense product set. The existential clause of syntactic forcing and the membership clause for a canonical forcing name give the density witnesses. The product filter meets this set, cannot use the incompatible case, and hence its second projection meets ${\dot{D}}^{G_0}$.