Over a model of Generalized continuum hypothesis, add ${\aleph }_3$ Cohen reals with finite binary partial functions. The Delta-system lemma proves the countable chain condition for forcing. The generic reals give the continuum lower bound ${\aleph }_3$, while nice forcing names for subsets of ${\omega }_1$ number at most $({\aleph }_3{)}^{{\aleph }_1}={\aleph }_3$ in the ground model. Cardinal preservation then gives $2^{{\aleph }_0}=2^{{\aleph }_1}={\aleph }_3$.

An infinite cardinal number $\kappa$ is singular when $\mathrm{cf}(\kappa )<\kappa$. Thus a short cofinal function reaches arbitrarily high ordinals below $\kappa$. For example, ${\aleph }_{\omega }$ has cofinality $\omega$. The complementary case is a regular cardinal.

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 map $f:\delta \to \alpha$ whose range is cofinal: every $\xi <\alpha$ lies below or equals some $f(\eta )$. It need not be increasing.

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 $g(t)$ is the conditional giant fraction and $w(t)$ a second evolved-state fraction at common age $t$, independent component masses give the fraction of giant-containing systems with another evolved component as $\mathbb{E}[g^2+2gw]/\mathbb{E}[2g-g^2]$. Averaging must follow conditioning on age, rather than multiplying marginal state probabilities.

In an instantaneous burst all binary components have a fixed common age. If the initial component masses are independent, the probability of two red giants is the square of the individual giant probability at that age. This need not equal the age-averaged pair fraction in a continuously forming population.

On an oriented boundaryless Riemannian manifold, apply Stokes theorem to the compactly supported form $f\ast \alpha$. The graded Leibniz rule gives $\int df\wedge \ast \alpha =-\int fd\ast \alpha$. The Hodge star operator identifies these top forms with the scalar integrands displayed above. This proves that the codifferential on one-forms is $\delta =-\ast d\ast$, the formal adjoint of the exterior derivative. Compact support of $f$ suffices.

On the whole Euclidean space, an $L$-Lipschitz gradient of a convex function is $1/L$-cocoercive. Subtract the supporting affine function at one point, apply the descent lemma to a gradient step at the other point, then interchange the points and add. The conclusion needs convexity but no second derivative.

The notation $\Delta h=\sum h_{(1)}\otimes h_{(2)}$ suppresses the summation index of a comultiplication; iterated indices use coassociativity. For a comodule, write $\rho (u)=\sum u_{(0)}\otimes u_{(1)}$.

For a coalgebra $C$ and a unital associative algebra over a commutative ring $A$, maps $f,g:C\to A$ have convolution $f\ast g=m_A(f\otimes g){\Delta }_C$, with unit $j_A{\epsilon }_C$. The two-sided convolution inverse of $1_H$ is the antipode.

A right comodule over a coalgebra $(C,\Delta ,\epsilon )$ over a commutative ring $k$ is a $k$-module $M$ with a coaction $\rho :M\to M{\otimes }_{k}C$ satisfyingusing the canonical tensor identifications. In Sweedler notation, $\rho (m)=\sum m_{(0)}\otimes m_{(1)}$. The equations are the reversed-arrow counterparts of associativity and the unit law for an algebra module.^[ref]

For the group algebra coalgebra $kG$ with $\Delta (g)=g\otimes g$, a direct sum $M={⨁}_{g\in G}{M}_g$ gives a right comodule by $\rho (m_g)=m_g\otimes g$. Conversely, expanding a coaction in the basis of group elements recovers this direct sum decomposition. Thus comodules over this coalgebra encode a decomposition indexed by $G$, whereas modules over the same group algebra encode a group representation. More generally, the definition extends to comonoids in a monoidal category whenever the relevant tensor action is specified.

The principal CNO cycle follows ${}^{12}\mathrm{C}(p,\gamma ){}^{13}\mathrm{N}$, beta-plus decay to carbon-13, proton capture to nitrogen-14, proton capture to oxygen-15, beta-plus decay to nitrogen-15, and ${}^{15}\mathrm{N}(p,\alpha ){}^{12}\mathrm{C}$. The carbon seed is regenerated. The ordinary bottleneck is ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$, so nitrogen-14 accumulates in cycle equilibrium.

A branch of the CNO cycle proceeds through proton captures on nitrogen-15 and oxygen-16, beta-plus decay of fluorine-17, and ${}^{17}\mathrm{O}(p,\alpha ){}^{14}\mathrm{N}$. Returning through oxygen-15 and nitrogen-15 closes the cycle. It also catalyzes the conversion of four protons into helium-4, with branch-dependent neutrino loss.

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.

A subset of a regular uncountable cardinal number $\kappa$ is stationary if it meets every club set in $\kappa$. Intersecting a stationary set with a club set preserves stationarity. Every member of the club filter is stationary. At a regular infinite $\theta <\kappa$, the stationarity of ordinals of prescribed cofinality supplies important examples.

A sequence $⟨C_{\delta }⟩$ in which $C_{\delta }$ is closed and unbounded in $\delta$. At a successor ordinal its predecessor is a club singleton. Such sequences define minimal walks along a club sequence.

For $⟨C_{\alpha }:\alpha <\delta ⟩$, its diagonal intersection consists of $\beta <\delta$ belonging to every $C_{\alpha }$ with $\alpha <\beta$. For club sets on a regular uncountable cardinal, this is a club.

For a bounded self-adjoint operator $L$, every bounded sequence whose images converge has a norm-convergent subsequence exactly when its kernel is finite-dimensional and its range is closed, or equivalently $0\notin {\Sigma }_{\mathrm{e}}(L)$ for the essential spectrum of a bounded self-adjoint operator. Split the sequence into kernel and kernel complement: finite dimensionality gives a subsequence on the first part and the closed-range bound on the kernel complement makes the second part a Cauchy sequence. Infinite kernel or approximate null unit vectors in its complement obstruct the property.

The resolvent formalism studies the spectrum through the resolvent of an operator and its dependence on a complex spectral parameter. This connects operator theory with complex analysis.