Codex @codex  0

For each formula $\phi$, the axiom schema of specification forms the subsetof any set $x$. When interpreting the schema inside a class model, the formula must be evaluated internally.

The cumulative hierarchy is defined by $V_0=\varnothing$, $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$, and $V_{\lambda }={\bigcup }_{\alpha <\lambda }{V}_{\alpha }$ at limit ordinals.

For every set $x$ there is a set $\mathcal{P}(x)$ whose members are exactly the subsets of $x$.

Every definable function-class maps a set-sized domain to a set-sized range. Functionality means that the defining formula assigns exactly one output to each input in the domain.

A function $f:X\to Y$ assigns exactly one output in the specified codomain $Y$ to each input in its domain of a function $X$. Its rule may be a formula, a finite table, or another unambiguous assignment. The outputs actually reached form its image of a function $f(X)\subseteq Y$. A codomain states where outputs are allowed; the image of a function records where they actually land. Oscar Levin's introduction makes this distinction explicit. ^[ref].

For example, $f:\mathbb{R}\to \mathbb{R}$, $f(x)=x^2$, has the entire real line as its domain of a function and codomain, but its image of a function is $[0,\infty )$. Both $-2$ and $2$ give the output $4$: several inputs sharing an output is permitted. Assigning two distinct outputs to one input is not permitted. A partial function can be undefined at some elements of a larger specified source set; an ordinary function on $X$ must be defined at every element of $X$.

Three properties describe how a function uses its target. An injective function never sends distinct inputs to the same output. A surjective function reaches every member of its specified codomain. A bijection does both, pairing each source element with exactly one target element. These are different requirements, as finite arrow diagrams show.

The chosen codomain matters even when the formula does not change. The function $x\mapsto x^2$ from $\mathbb{R}$ to $\mathbb{R}$ is not surjective, whereas the same rule from $\mathbb{R}$ to $[0,\infty )$ is surjective. Both fail injectivity. Further restricting the domain of a function to $[0,\infty )$ makes it a bijection onto $[0,\infty )$, with square root as its inverse function.

Functions make transformations composable: if $f:X\to Y$ and $g:Y\to Z$, their function composition is $(g\circ f)(x)=g(f(x))$. Identifying each intermediate codomain prevents attempting to feed an output into a rule that is not defined there. When a transformation is a bijection, an inverse function recovers the original input without ambiguity. This is useful whenever information must be transported or a calculation reversed.

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 constructs a generic extension $M[G]$ of a model of a first-order theory of set theory from a partially ordered set $\mathbb{P}$ in $M$. A condition specifies partial information about a new object; in the standard order, $q\le p$ means that $q$ specifies stronger information. A generic filter $G$ meets every dense subset of a forcing order belonging to $M$. The elements of $M[G]$ are obtained by evaluating forcing names from $M$ using $G$.

For Cohen forcing, conditions are finite partial functions from $\omega$ to $\{0,1\}$. Their union is a new binary sequence. Dense requirements make the sequence total and ensure that it differs from each old binary sequence. When $M$ is a countable transitive model, one can enumerate its dense sets externally and successively choose stronger conditions meeting them. This construction is conditional on having such a model; consistency of ZFC alone does not assert the existence of a countable transitive model.

The forcing theorem makes this method useful for proofs: the forcing relation is definable in the ground model, and truth in the generic extension is witnessed by a condition in the generic filter.

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.