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A binary relation on a set $X$ is a subset of $X\times X$.

A set is a collection of distinct objects regarded as its set elements. Membership, written $a\in A$, is the basic question: is $a$ one of the objects in $A$? For example, $\{1,2,2\}=\{1,2\}=\{2,1\}$: repeated entries and their written order do not change a set. A multiset retains multiplicities, while a tuple retains positions as well. Choosing the right object therefore determines which distinctions the mathematics remembers.

A set can collect possible outcomes of a probability space, vertices of a graph, or allowed inputs of a function. The empty set contains none; a subset selects members of a previously given set; the power set collects every such selection. These constructions are useful because they can be stated once and reused regardless of what the members represent. In an axiomatic account, rules such as the axiom schema of separation specify which collections exist as sets.

A relation is extensional when distinct elements have distinct sets of predecessors.

This is the predecessor-set instance of the identity principle extensionality.

A relation $r$ on $X$ is well-founded when every nonempty subset of $X$ contains an element with no $r$-predecessor in that subset.

A set $X$ is transitive when every element of an element of $X$ is itself an element of $X$.

Forcing extends a model of set theory by adjoining a filter generic for a partially ordered set of finite or otherwise controlled approximations.

The definable power set $\mathcal{D}(X)=\mathrm{Def}(X)$ consists of the subsets of $X$ definable over the first-order structure $(X,\in )$ by a first-order formula with parameters from $X$. Satisfaction for a set structure makes this a definable operation in ZF. Iterating it produces the constructible hierarchy and the relative constructible hierarchy; their unions are the constructible universe and the relative constructible universe.

The von Neumann hierarchy isEvery level is transitive, the levels are increasing, and every set belongs to some level.

A null Schwarzschild geodesic with $b\gg r_s$ is deflected through $2r_s/b=4GM/(b{c}^2)$ to first order.

This is the leading spherical-mass deflection underlying a gravitational lens.

A static observer at Schwarzschild radius $r>2M$ must maintain the outward proper accelerationIt approaches the Newtonian value $M/r^2$ at large radius and diverges at the horizon.

For static observers at radii $r_A$ and $r_B$, equal Schwarzschild-coordinate time intervals correspond to proper times $d\tau =\sqrt{1-2M/r}dt$. Hence

This static-observer relation is a special case of gravitational redshift, not a definition of every gravitational frequency shift.

The surface $r=2M$ is the future event horizon of Schwarzschild spacetime. Its singularity in Schwarzschild coordinates is removable, whereas $r=0$ is a curvature singularity.

The tortoise coordinate satisfies

The Schwarzschild radius is $r_s=2GM/c^2$, or $2M$ in geometrized units.

For inverse radius $y=1/r$ and azimuth $\varphi$, a Schwarzschild null geodesic obeyswhere $b=L/E$ is the impact parameter.

With the curvature conventionthe vacuum Schwarzschild metric satisfiesThe opposite Riemann-sign convention reverses this sign.

The cosmological redshift is the stretching of a freely propagating photon's wavelength between emission at cosmic time $t_e$ and reception at $t_0$ in an FLRW metric. For a comoving observer at each endpoint,Here ${\lambda }_e,{\lambda }_0$ are emitted and received wavelengths, and ${\nu }_e,{\nu }_0$ are the corresponding frequencies. A positive cosmological redshift means that the scale factor grew while the photon travelled. The name describes a shift toward longer wavelengths; it applies to light beyond the visible range as well.

For a computed example, let the scale factor double from $a(t_e)=0.5$ to $a(t_0)=1$. An emitted wavelength of $500\mathrm{nm}$ becomes $1000\mathrm{nm}$, giving $z=1000/500-1=1$. The received frequency is half the emitted frequency. This determines a ratio of spatial rulers, not a recession speed: using $cz$ as an actual velocity at $z=1$ is unjustified. The wavelength panel in the illustration under Hubble's law shows this doubling.

A short derivation follows successive wave crests along a radial null geodesic in FLRW spacetime. Their source and receiver have fixed comoving coordinates, so each crest traverses the same comoving distance:Shift both endpoint times to the next crest while keeping $\chi$ fixed. To first order in the very short periods $\delta t_e,\delta t_0$, this gives $\delta t_0/a(t_0)=\delta t_e/a(t_e)$. Locally, wavelength equals the speed of light times the crest period, yielding the displayed cosmological redshift relation. These periods are much shorter than the timescale on which the scale factor changes.

The relation isolates the smooth expansion contribution. An observed redshift can also contain a Doppler effect from source or observer peculiar velocity and a gravitational redshift. See combined kinematic and cosmological redshift for the multiplicative combination of endpoint motion and expansion. Treating every observed redshift as pure cosmological redshift can therefore misassign a nearby source's distance.

The useful insight is that a received wavelength ratio records how much the scale factor changed during the photon's journey. Turning that ratio into a distance requires an expansion history, through the Hubble parameter, and a specified convention such as luminosity distance or proper distance in cosmology; the ratio alone supplies neither.

A comoving coordinate remains fixed for an observer following the homogeneous cosmological expansion.

A comoving coordinate provides labels for the comoving and proper distances comparison; proper separation additionally contains the cosmological scale factor.

The Hubble parameter is the instantaneous fractional rate of cosmic expansion:where the dot is a derivative with respect to cosmic time. Thus a short interval $\Delta t$ changes any fixed-comoving proper distance in cosmology by the fraction $\Delta d/d\simeq H\Delta t$. Its dimensions are inverse time; $H>0$ describes expansion and $H<0$ contraction. The Hubble parameter identity gives $\dot{H}+H^2=\ddot{a}/a$.

Knowledge of a valid public and private exponent gives $ed-1$, a nonzero multiple of $\lambda (N)$. Writing it as $2^{s}t$ with $t$ odd and testing random bases produces a nontrivial square root of one modulo $N$ with high probability; a greatest common divisor then factors $N$. Thus one customer's private key under a shared semiprime modulus compromises every exponent using that modulus.