hcju/proofwikips huggingface dataset:
Definition:Vector Sum/Component Definition
Let $\mathbf u$ and $\mathbf v$ be represented by their components considered to be embedded in a real $n$-space: {{begin-eqn}} {{eqn | l = \mathbf u       | r = \tuple {u_1, u_2, \ldots, u_n} }} {{eqn | l = \mathbf v       | r = \tuple {v_1, v_2, \ldots, v_n} }} {{end-eqn}} Then the '''(vector) sum''' of $\mathbf u$ and $\mathbf v$ is defined as: :$\mathbf u + \mathbf v := \tuple {u_1 + v_1, u_2 + v_2, \ldots, u_n + v_n}$ Note that the $+$ on the  is conventional addition of numbers, while the $+$ on the  takes on a different meaning. The distinction is implied by which operands are involved.
Definition:Vector Length/Arrow Representation
Let $\mathbf v$ be a vector quantity represented as an arrow in a real vector space $\R^n$. The '''length''' of $\mathbf v$ is the length of the line segment representing $\mathbf v$ in $\R^n$.
Definition:Deceleration
'''Deceleration''' is the term used in natural language to mean '''negative acceleration'''. Thus, while '''acceleration''' is used to mean '''speeding up''', '''deceleration''' is used to mean '''slowing down'''. As in the applied mathematical context, '''acceleration''' is used to mean '''any''' change in velocity, including a change in direction while moving at a constant speed, the term '''deceleration''' is generally not used in mathematics.
Definition:Rectilinear Coordinate System
A coordinate system whose coordinate axes are straight lines is called a system of '''rectilinear coordinate system'''.
Definition:Curvilinear Coordinate System
A coordinate system such that at least one of the coordinate axes is a curved line is called a system of '''curvilinear coordinates'''.
Definition:Curved Line
A '''curved line''' is a curve which is specifically not a straight line.
Definition:Line/Curve/Space
A '''space curve''' is a curve which cannot be embedded in the plane, but can be embedded in $3$-space.
Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)
Let $S$ be a finite set. Let $\mathscr C$ be a non-empty set of subsets of $S$. === Definition 1 === {{:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1}} === Definition 2 === {{:Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2}}
Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1
{{begin-axiom}} {{axiom | n = C1         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \implies C_1 \not \subseteq C_2 }} {{axiom | n = C2         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z }} {{end-axiom}}
Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 2
{{begin-axiom}} {{axiom | n = C1         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \implies C_1 \not \subseteq C_2 }} {{axiom | n = C2'         | q = \forall C_1, C_2 \in \mathscr C         | mr = C_1 \neq C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : y \in C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z }} {{end-axiom}}
Definition:Curvilinear Coordinate System/Cartesian Representation
The relation between curvilinear coordinates and Cartesian coordinates can be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \map z {q_1, q_2, q_3} }} {{end-eqn}} where: :$\tuple {x, y, z}$ denotes the Cartesian coordinates :$\tuple {q_1, q_2, q_3}$ denotes their curvilinear equivalents.
Definition:Orthogonal Curvilinear Coordinates
Let $\tuple {q_1, q_2, q_3}$ denote a set of curvilinear coordinates. Let the relation between those curvilinear coordinates and Cartesian coordinates be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \map z {q_1, q_2, q_3} }} {{end-eqn}} where $\tuple {x, y, z}$ denotes the Cartesian coordinates. Let these equations have the property that: :$\dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j} = 0$ wherever $i \ne j$. Then $\tuple {q_1, q_2, q_3}$ are '''orthogonal curvilinear coordinates'''.
Definition:Functional
=== Real Functional === {{:Definition:Real Functional}}
Definition:Permutation Theory
'''Permutation theory''' is a branch of abstract algebra which studies the properties of permutations.
Definition:Landau Symbols
The '''Landau symbols''' are the $\OO$ and $\mathcal o$ that are used in the definition of big-$\OO$ notation and little-$\mathcal o$ notation: === Big-$\OO$ Notation === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Little-$\mathcal o$ Notation === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} {{NamedforDef|Edmund Georg Hermann Landau|cat = Landau}}
Definition:Vector Subtraction/Arrow Representation
Let $\mathbf u$ and $\mathbf v$ be represented by arrows embedded in the plane such that: :$\mathbf u$ is represented by $\vec {AB}$ :$\mathbf v$ is represented by $\vec {AC}$ that is, so that the initial point of $\mathbf v$ is identified with the initial point of $\mathbf u$. :300px Then their '''(vector) difference''' $\mathbf u - \mathbf v$ is represented by the arrow $\vec {CB}$.
Definition:Orientation of Coordinate Axes
The '''orientation''' of a coordinate system is the disposition of the coordinate axes relative to each other.
Definition:Orientation of Coordinate Axes/Cartesian Plane
There are $2$ different orientations of a Cartesian plane: :Right-hand Cartesian plane320px $\qquad \qquad \qquad$ Left-hand Cartesian plane320px
Definition:Orientation of Coordinate Axes/Cartesian Plane/Right-Handed
A Cartesian plane is defined as being '''right-handed''' if it has the following property: Let a '''right hand''' be placed, with palm uppermost, such that the thumb points along the $x$-axis in the positive direction, such that the thumb and index finger are at right-angles to each other. Then the index finger is pointed along the $y$-axis in the positive direction. :320px
Definition:Orientation of Coordinate Axes/Cartesian Plane/Left-Handed
A Cartesian plane is defined as being '''left-handed''' if it has the following property: Let a '''left hand''' be placed, with palm uppermost, such that the thumb points along the $x$-axis in the positive direction, such that the thumb and index finger are at right-angles to each other. Then the index finger is pointed along the $y$-axis in the positive direction. :320px
Definition:Orientation of Coordinate Axes/Cartesian 3-Space
There are $2$ different orientations of a Cartesian $3$-space: :450px $\qquad \qquad \qquad$ 320px
Definition:Orientation of Coordinate Axes/Cartesian 3-Space/Right-Handed
A Cartesian $3$-Space is defined as being '''right-handed''' if it has the following property: Let a '''right hand''' be placed such that: :the thumb and index finger are at right-angles to each other :the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm :the thumb points along the $x$-axis in the positive direction :the index finger points along the $x$-axis in the positive direction. Then the $3$rd finger is pointed along the $z$-axis in the positive direction. :450px
Definition:Orientation of Coordinate Axes/Cartesian 3-Space/Left-Handed
A Cartesian $3$-Space is defined as being '''left-handed''' if it has the following property: Let a '''left hand''' be placed such that: :the thumb and index finger are at right-angles to each other :the $3$rd finger is at right-angles to the thumb and index finger, upwards from the palm :the thumb points along the $x$-axis in the positive direction :the index finger points along the $x$-axis in the positive direction. Then the $3$rd finger is pointed along the $z$-axis in the positive direction. :320px
Definition:Pairwise Perpendicular
Let $S = \set {L_1, L_2, \ldots, L_n}$ be a set of lines in a Euclidean space. Then the elements of $S$ are '''pairwise perpendicular'''  :$\forall i, j \in \set {1, 2, \ldots, n}, i \ne j:$ $L_i$ and $L_j$ are perpendicular to each other.
Definition:Orthogonal Coordinate System
An '''orthogonal coordinate system''' is a coordinate system in which the coordinate axes are pairwise perpendicular.
Definition:Smallest Natural Number
Let $S \subseteq \N$ be a subset of the natural numbers $\N$. The '''smallest''' element $m$ of $S$ is defined as: :$\forall n \in S: m \le n$ That is, it is the minimal element of $S$ under the usual ordering.
Definition:Energy/Unit
The unit of measurement of '''energy'''  is the joule $\mathrm J$: :$1 \ \mathrm J = 1 \ \mathrm N \ \mathrm m$ where: :$\mathrm N$ denotes the newton :$\mathrm m$ denotes the metre.
Definition:Joule (Unit)
The '''joule''' is the SI unit of energy. It is defined as being: :the '''energy''' transferred to (or work done on) a body when a force of $1$ newton acts on that body in the direction of the force's motion through a distance of one metre :the '''energy''' dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second.
Definition:Joule (Unit)/Base Units
The SI base units of the '''joule''' are: :$\mathrm J := \mathrm {kg} \, \mathrm m^2 \mathrm s^{-2}$ where: :$\mathrm {kg}$ denotes kilograms :$\mathrm m$ denotes metres :$\mathrm s$ denotes seconds (of time).
Definition:Power (Physics)
'''Power''' the amount of energy transferred or converted per unit time. '''Power''' is a scalar quantity.
Definition:Power (Physics)/Dimension
The dimension of measurement of '''power''' is $\mathsf {M L}^2 \mathsf T^{-3}$. This derives from its definition as: :$\dfrac {\text{Energy} } {\text {Time} }$
Definition:Power (Physics)/Unit
The unit of measurement of '''power'''  is the watt $\mathrm W$: :$1 \ \mathrm W = 1 \ \mathrm J \ \mathrm s^{-1}$ where: :$\mathrm J$ denotes the joule :$\mathrm s$ denotes the second.
Definition:Watt (Unit)
The '''watt''' is the SI unit of power. It is defined as being: :the rate at which work is done when the velocity of a body is held constant at $1$ metre per second against a constant opposing force of $1$ newton :the rate at which work is performed when an electric current of $1$ ampere flows across an electrical potential difference of $1$ volt.
Definition:Volt
The '''volt''' is the SI unit of electromotive force. It is defined as being the difference in '''electric potential between two points of an electrical conductor: :when an electric current of $1$ ampere dissipates $1$ watt of power between those points :that will impart $1$ joule of energy per coulomb of electric charge that passes through it.
Definition:Volt/Base Units
The SI base units of the '''volt''' are: :$\mathrm W := \mathrm {kg} \, \mathrm m^2 \mathrm s^{-3} \mathrm A^{-1}$ where: :$\mathrm {kg}$ denotes kilograms :$\mathrm m$ denotes metres :$\mathrm s$ denotes seconds (of time). :$\mathrm A$ denotes ampere.
Definition:Ohm (Unit)
The '''ohm''' is the SI unit of electrical resistance. It is defined as being the electrical resistance between two points of an electrical conductor when a constant electric potential of $1$ volt, applied to these points, produces in the conductor an electric current of $1$ ampere, the  conductor not being the seat of any electromotive force.
Definition:Ohm (Unit)/Base Units
The SI base units of the '''ohm''' are: :$\mathrm W := \mathrm {kg} \, \mathrm m^2 \mathrm s^{-3} \mathrm A^{-2}$ where: :$\mathrm {kg}$ denotes kilograms :$\mathrm m$ denotes metres :$\mathrm s$ denotes seconds (of time). :$\mathrm A$ denotes ampere.
Definition:Strong Nuclear Force
The '''strong nuclear force''' is a force which acts in the range of the order of $10^{-15} \ \mathrm m$. It is what holds protons and neutrons together in the nucleus of an atom. Beyond the range of approximately the diameter of a large atom its effect is diminished. This may be the reason why stable isotopes of large atoms are rare.
Definition:Elementary Charge/Symbol
The symbol used to denote the '''elementary charge''' is usually $\E$ or $e$. The preferred symbol on  is $\E$.
Definition:Zero Vector/Vector Quantity
A '''vector quantity''' whose magnitude is zero is referred to as a '''zero vector'''.
Definition:Scalar Multiplication/Vector Quantity
Let $\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. The operation of scalar multiplication by $m$ of $\mathbf a$ is denoted $m \mathbf a$ and defined such that: :the magnitude of $m \mathbf a$ is equal to $m$ times the magnitude of $\mathbf a$: ::$\size {m \mathbf a} = m \size {\mathbf a}$ :the direction of $m \mathbf a$ is the same as the direction of $\mathbf a$.
Definition:Vector Quantity/Component/Cartesian Plane
Let $\mathbf a$ be a vector quantity embedded in a Cartesian plane $P$. Let $\mathbf a$ be represented with its initial point at the origin of $P$. Let $\mathbf i$ and $\mathbf j$ be the unit vectors in the positive direction of the $x$-axis and $y$-axis. Then: :$\mathbf a = x \mathbf i + y \mathbf j$ where: :$x \mathbf i$ and $y \mathbf j$ are the '''component vectors''' of $\mathbf a$ in the $\mathbf i$ and $\mathbf j$ directions :$x$ and $y$ are the '''components''' of $\mathbf a$ in the $\mathbf i$ and $\mathbf j$ directions.
Definition:Vector Quantity/Component/Cartesian 3-Space
Let $\mathbf a$ be a vector quantity embedded in Cartesian $3$-space $S$. Let $\mathbf a$ be represented with its initial point at the origin of $S$. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then: :$\mathbf a = x \mathbf i + y \mathbf j + z \mathbf k$ where: :$x \mathbf i$, $y \mathbf j$ and $z \mathbf k$ are the '''component vectors''' of $\mathbf a$ in the $\mathbf i, \mathbf j, \mathbf k$ directions :$x$, $y$ and $z$ are the '''components''' of $\mathbf a$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions. It is usual to arrange that the coordinate axes form a right-handed Cartesian $3$-space.
Definition:Variational Principle
A '''variational principle''' is a statement defining a condition that is required to be fulfilled in order to solve a problem in the calculus of variations.
Definition:Vectorial Mechanics
'''Vectorial mechanics''' is an approach to problems in mechanics which handles quantities as vector quantities and uses vector algebra and vector calculus for their solution.
Definition:Kinematics
'''Kinematics''' is the branch of the mechanics describing the motion of bodies without considering the forces that may affect this motion.
Definition:Like Vector Quantities
Let $\mathbf a$ and $\mathbf b$ be vector quantities. Then $\mathbf a$ and $\mathbf b$ are known as '''like vector quantities'''  they have the same direction.
Definition:Negative of Vector Quantity
Let $\mathbf a$ be a vector quantities. The '''negative''' $-\mathbf a$ of $\mathbf a$ is the vector quantity which has the same magnitude but the opposite direction.
Definition:Scalar Division/Vector Quantity
Let $\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. The operation of scalar division by $m$ of $\mathbf a$ is defined as: :$\dfrac {\mathbf a} m = \dfrac 1 m \size {\mathbf a}$ that is, scalar multiplication by $\dfrac 1 m$ of $\mathbf a$.
Definition:Scalar Division
=== Vector Quantity === {{:Definition:Scalar Division/Vector Quantity}}
Definition:Scalar Multiplication by Negative Scalar
Let $\mathbf a$ be a vector quantity. Let $-m$ be a negative scalar quantity. The operation of scalar multiplication by $-m$ of $\mathbf a$ is defined as: :$-m \mathbf a := \paren {-m} \mathbf a = m \paren {-\mathbf a}$ that is, the negative of the scalar product of $m$ with $\mathbf a$.
Definition:Direct Product Norm
Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces. Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\tuple {x, y} \in V$, $x \in X$ and $y \in Y$. Then the '''direct product norm''' on $V$ is defined as: :$\norm {\tuple {x, y} } := \map \max {\norm x, \norm y}$ where $\max$ denotes the max operation.
Definition:Coplanar Vectors
Three or more vectors are '''coplanar''' if they can be embedded in the same plane.
Definition:Coplanar Vectors/Non-Coplanar
Three or more vectors are '''non-coplanar''' if they are not '''coplanar'''. That is if they cannot be embedded in the same plane.
Definition:Direction Cosine
Let $\mathbf a$ be a vector quantity embedded in a Cartesian $3$-space. Let the angles which $\mathbf a$ makes with the $x$-axis, $y$-axis and $z$-axis be $\alpha$, $\beta$ and $\gamma$ respectively. Then the '''direction cosines''' of $\mathbf a$ are $\cos \alpha$, $\cos \beta$ and $\cos \gamma$, defined individually such that: :$\cos \alpha$ is the '''direction cosine of $\mathbf a$ with respect to the $x$-axis''' :$\cos \beta$ is the '''direction cosine of $\mathbf a$ with respect to the $y$-axis''' :$\cos \gamma$ is the '''direction cosine of $\mathbf a$ with respect to the $z$-axis'''.
Definition:Birkhoff-James Orthogonality
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed linear space. Let $x, y \in V$. Then $x$ and $y$ are '''Birkhoff-James orthogonal'''   :$\norm {x + \lambda y} \ge \norm y$ for all scalars $\lambda$. This is denoted: :$x \perp_B y$
Definition:Orthogonal Basis of Vector Space
Let $V$ be a vector space. Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $V$. Then $\BB$ is an '''orthogonal basis'''  $\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$ are pairwise perpendicular.
Definition:Orthonormal Basis of Vector Space
Let $V$ be a vector space. Let $\BB = \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be a basis of $V$. Then $\BB$ is an '''orthonormal basis of $V$''' : :$(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an orthogonal basis of $V$ :$(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_1} = 1$
Definition:Vector Quantity/Component/Einstein Summation Convention
Let $\mathbf a$ be a vector quantity. $\mathbf a$ can be expressed in component form using the Einstein summation convention as: :$\mathbf a = a_i \mathbf e_i$
Definition:Dot Product/Einstein Summation Convention
Let $\mathbf a$ and $\mathbf b$ be vector quantities. The dot product of $\mathbf a$ and $\mathbf b$ can be expressed as: {{begin-eqn}} {{eqn | l = \mathbf a \cdot \mathbf b        | o = :=       | r = a_i b_j \delta_{i j}       | c =  }} {{eqn | r = a_i b_i       | c =  }} {{eqn | r = a_j b_j       | c =  }} {{end-eqn}}
Positive Integer Greater than 1 has Prime Divisor
== Lemma ==  Every positive integer greater than $1$ has at least one divisor which is prime.  == Proof 1 == {{:Positive Integer Greater than 1 has Prime Divisor/Proof 1}} == Proof 2 (not using FTA) == {{:Positive Integer Greater than 1 has Prime Divisor/Proof 2}} == Sources == * {{BookReference|Elements of Abstract Algebra|1971|Allan Clark|prev = Definition:Prime Number/Sequence|next = Euclid's Theorem}}: Chapter $1$: Properties of the Natural Numbers: $\S 22 \alpha$
Axiom:Axiom of Extension
== Axiom == The '''axiom of extension''' is the fundamental definition of the nature of a collection: it is completely determined by its elements.  === Set Theory === {{:Axiom:Axiom of Extension/Set Theory}} === Class Theory === The '''axiom of extension''' in the context of class theory has the same form: {{:Axiom:Axiom of Extension/Class Theory}} == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Set Equality * Definition:Equals == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} Extension Extension Extension nyvyz3hodadhs77097a0v1mjkjr01o3
Axiom:Axiom of Empty Set
== Axiom ==  The '''axiom of the empty set''' posits the existence of a set which has no elements. Depending on whether this axiom is declared in the context of set theory or class theory, it exists in different forms. === Set Theory === {{:Axiom:Axiom of Empty Set/Set Theory}} === Class Theory === In class theory, the existence of the empty class is not axiomatic, as it has been derived from previous axioms. Hence the '''axiom of the empty set''' takes this form: {{:Axiom:Axiom of Empty Set/Class Theory}} where $V$ denotes the basic universe.  == Also known as == {{:Axiom:Axiom of Empty Set/Also known as}} == Also see == * Definition:Zermelo-Fraenkel Axioms * Definition:Empty Set * Empty Set is Unique Empty Set Empty Set
Axiom:Axiom of Pairing
== Axiom ==  === Set Theory === {{:Axiom:Axiom of Pairing/Set Theory}} === Class Theory === {{:Axiom:Axiom of Pairing/Class Theory}} == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Definition:Doubleton * Definition:Ordered Pair {{LinkToCategory|Axiom of Pairing|the axiom of pairing}} Pairing Pairing Pairing cuzikepuwbqucmjwfidhnaql1cq6qna
Axiom:Axiom of Specification
== Axiom ==  For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. Because we cannot quantify over functions, we need an axiom for every condition we can express. Therefore, this axiom is sometimes called an axiom ''schema'', as we introduce a lot of similar axioms. This axiom schema can be formally stated as follows: === Set Theory === {{:Axiom:Axiom of Specification/Set Theory}} === Class Theory === The '''axiom of specification''' in the context of class theory has a similar form: {{:Axiom:Axiom of Specification/Class Theory}} == Also known as == {{:Axiom:Axiom of Specification/Also known as}} == Also see == * Axiom:Comprehension Principle -- do not confuse that with this {{refactor|Include the following in its own theorem page, and present that deduction formally}} The '''axiom of specification''' can be deduced from the Axiom of Replacement. == Historical Note == {{:Axiom:Axiom of Specification/Historical Note}} {{Languages|Axiom of specification}} {{language|German|Aussonderungsaxiom|lit = axiom of segregation}} {{end-languages}} == Sources == * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Definition:Class (Class Theory)|next = Definition:Finite Set}}: Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets Specification Specification Specification 2mhp95orxrykoudc2fopbmzs46tzk0a
Axiom:Axiom of Unions
== Axiom ==  === Set Theory === {{:Axiom:Axiom of Unions/Set Theory}} === Class Theory === {{:Axiom:Axiom of Unions/Class Theory}} == Also known as == {{:Axiom:Axiom of Unions/Also known as}} Unions Unions Unions 614u0u5kgomeahmakt1wp1tl5beot67
Axiom:Axiom of Powers
== Axiom ==  === Set Theory === {{:Axiom:Axiom of Powers/Set Theory}} === Class Theory === {{:Axiom:Axiom of Powers/Class Theory}} == Also known as == {{:Axiom:Axiom of Powers/Also known as}} == Also see == * Definition:Power Set Powers Powers Powers ehj2nolsk55rwqbwequ79o8o70oaewj
Axiom:Axiom of Infinity
== Axiom ==  === Set Theory === {{:Axiom:Axiom of Infinity/Set Theory}} === Class Theory === {{:Axiom:Axiom of Infinity/Class Theory}} Infinity Infinity Infinity 9gjyglkd40zx3kol5ox0syoks4l30ng