hcju/proofwikips huggingface dataset:
Definition:Minimal (Model Theory)
Let $\mathcal M$ be an $\mathcal L$-structure. Let $M$ be the universe of $\mathcal M$. Let $A$ be a subset of $M$. Let $D \subseteq M^n$ be an infinite $A$-definable set. Let $\phi \left({\bar x, \bar a}\right)$ be an $\mathcal L$-formula with parameters $\bar a$ from $A\subseteq M$ and  free variables $\bar x$ which defines $D$. {{Disambiguate|Definition:Logical Formula}} $D$ is '''minimal''' in $\mathcal M$  every definable subset of $D$ is either finite or cofinite. $\phi$ is '''minimal''' in $\mathcal M$ if $D$ is minimal in $\mathcal M$. $D$ and $\phi$ are '''strongly minimal''' in $\mathcal M$  $\phi$ is minimal in any elementary extension $\mathcal N$ of $\mathcal M$. An $\mathcal L$-theory $T$ is '''strongly minimal''' if for every model $\mathcal N$ of $T$ with universe $N$, the set $N$ is strongly minimal.
Definition:Cofinite Set
Let $S$ be a set. Let $A$ be a subset of $S$. $A$ is '''cofinite in $S$'''  the complement of $A$ in $S$ is finite.
Definition:Fortissimo Space
Let $S$ be an uncountably infinite set. Let $p \in S$ be a particular point of $S$. Let $\tau_p \subseteq \powerset S$ be a subset of the power set of $S$ defined as: :$\tau_p = \leftset {U \subseteq S: p \in \relcomp S U}$ or $\set {U \subseteq S: \relcomp S U}$ is countable (either finitely or infinitely)$\rightset {}$ That is, $\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is countable. Then $\tau_p$ is a '''Fortissimo topology''' on $S$, and the topological space $T = \struct {S, \tau_p}$ is a '''Fortissimo space'''.
Definition:Divide (Model Theory)
Let $T$ be a complete $\mathcal L$-theory. Let $\mathfrak C$ be a monster model for $T$. Let $A$ be a subset of the universe of $\mathfrak C$. Let $\phi \left({\bar x, \bar b}\right)$ be an $\mathcal L$-formula with free variables $\bar x$ and parameters $\bar b$ from the universe of $\mathfrak C$. {{Disambiguate|Definition:Logical Formula}} Let $\operatorname{tp} \left({\bar b / A}\right)$ denote the type of $\bar b$ over $A$. Then $\phi \left({\bar x, \bar b}\right)$ '''$k$-divides in $\mathfrak C$ over $A$''' : :there exists a sequence $\left\langle{\bar b_i}\right\rangle_{i \mathop \in \N}$ such that $\operatorname{tp} \left({\bar b_i / A}\right) = \operatorname{tp} \left({\bar b / A}\right)$ for each $i \in \N$ and: :for any distinct $k$-many terms $\bar b_{i_1}, \ldots, \bar b_{i_k}$ of the sequence, the set $\left\{{\phi \left({\bar x, \bar b_{i_1} }\right), \ldots, \phi \left({\bar x, \bar b_{i_k} }\right)}\right\}$ is not satisfiable in $\mathfrak C$. Let $\pi \left({\bar x, \bar b}\right)$ be a set of formulas with parameters $\bar b$. $\pi \left({\bar x, \bar b}\right)$ '''$k$-divides''' over $A$ if it implies a formula $\phi \left({x, \bar c}\right)$ which $k$-divides over $A$. Formulas and sets of formulas are said to '''divide''' if they $k$-divide for some $k$.
Definition:Fork
Let $T$ be a complete $\mathcal{L}$-theory. Let $\mathfrak{C}$ be a monster model for $T$. Let $A$ be a subset of the universe of $\mathfrak{C}$. Let  $\pi (\bar{x}, \bar{b})$ be a set of $\mathcal{L}$-formulas with free variables $\bar{x}$ and parameters $\bar{b}$  from the universe of $\mathfrak{C}$. {{Disambiguate|Definition:Logical Formula}} $\pi(\bar{x}, \bar{b})$ '''forks''' over $A$ if it implies some disjunction $\phi_1 (\bar x, \bar c_1) \vee \cdots \vee \phi_n (\bar x, \bar c_n)$ where each $\phi_i (\bar x, \bar c_i)$ divides over $A$. An individual formula $\phi (\bar x, \bar b)$ is said to '''fork''' if the singleton $\{\phi (\bar x, \bar b)\}$ forks.
Definition:Forking Extension
Let $T$ be a complete $\mathcal{L}$-theory. Let $\mathfrak{C}$ be a monster model for $T$. Let $A\subseteq B$ be subsets of the universe of $\mathfrak{C}$. Let $p(\bar x)$ be a complete $n$-type over $B$. Denote by $p\restriction A$ the subset of $p$ consisting of those formulas which involve only parameters from $A$. $p$ is a '''non-forking extension''' of $p\restriction A$ if $p(\bar x)$ does not fork over $A$.  gdkrozd2nkd7dv0v00jxtvtxhtzt815
Definition:Monster Model
Let $T$ be a complete $\mathcal L$-theory. A '''monster model''' of $T$ whose cardinality $\kappa$ is infinite is a model of $T$ which is saturated and homogeneous.
Definition:Finite Set
A set $S$ is defined as '''finite''' : :$\exists n \in \N: S \sim \N_{<n}$ where $\sim$ denotes set equivalence. That is, if there exists an element $n$ of the set of natural numbers $\N$ such that the set of all elements of $\N$ less than $n$ is equivalent to $S$. Equivalently, a '''finite set''' is a set with a count.
Definition:Embedding (Model Theory)
Let $\MM$ and $\NN$ be $\LL$-structures with universes $M$ and $N$ respectively. $j: \MM \to \NN$ is an '''$\LL$-embedding'''  it is an injective map $M \to N$ which preserves interpretations of all symbols in $\LL$; that is, such that: :$\map j {\map {f^\MM} {a_1, \dots, a_{n_f} } } = \map {f^\NN} {\map j {a_1}, \ldots, \map j {a_{n_f} } }$ for all function symbols $f$ in $\LL$ and $a_1, \dots, a_{n_f}$ in $M$ :$\tuple {a_1, \ldots, a_{n_R} } \in R^\MM \iff \tuple {\map j {a_1}, \dots, \map j {a_{n_R} } } \in R^\NN$ for all relation symbols $R$ in $\LL$ and $a_1, \dots, a_{n_R}$ in $M$ :$\map j {c^\MM} = c^\NN$ for all constant symbols $c$ in $\LL$. === Partial Embedding === A common method of constructing isomorphisms and elementary embeddings in proofs is to recursively define them a finite number of elements at a time.  For this  purpose, it is useful to have a definition of embeddings using functions which are only defined on a subset of $M$: Let $A \subseteq M$ be a subset of $M$. $j: A \to \NN$ is a '''partial $\LL$-embedding'''  it is an injective map $A \to N$ which preserves interpretations of all symbols in $\LL$ applied to elements of $A$; that is, such that: :$\map j {\map {f^\MM} {a_1, \dots, a_{n_f} } } = \map {f^\NN} {\map j {a_1}, \ldots, \map j {a_{n_f} } }$ for  all function symbols $f$ in $\LL$ and $a_1, \dots, a_{n_f}$ in $A$ :$\tuple {a_1, \ldots, a_{n_R} } \in R^\MM \iff \tuple {\map j {a_1}, \dots, \map j {a_{n_R} } } \in R^\NN$ for all relation symbols $R$ in $\LL$ and $a_1, \dots, a_{n_R}$ in $A$ :$\map j {c^\MM} = c^\NN$ for all constant symbols $c$ in $\LL$. === Isomorphism === $j: \MM \to \NN$ is an '''$\LL$-isomorphism'''  it is a bijective $\LL$-embedding. === Automorphism === $j: \MM \to \NN$ is an '''$\LL$-automorphism'''  it is an $\LL$-isomorphism and $\MM = \NN$. It is often useful to talk about automorphisms which are constant on subsets of $M$.  So, there is a definition and a notation for doing so: Let $A \subseteq M$ be a subset of $M$, and let $b \in M$. An $\LL$-automorphism $j$ is an '''$A$-automorphism'''  $\map j a = a$ for all $a\in A$. An $\LL$-automorphism $j$ is an '''$A, b$-automorphism'''  it is an $\paren {A \cup \set b}$-automorphism; that is: $\map j a = a$ for all $a \in A$ and also $\map j b = b$.
Definition:Homogeneous (Model Theory)
Let $T$ be an $\mathcal{L}$-theory. Let $\kappa$ be an infinite cardinal. A model $\mathcal{M}$ of $T$ is $\kappa$-'''homogeneous''' if for every subset $A$ and element $b$ in the universe of $\mathcal{M}$ with the cardinality of $A$ strictly less than $\kappa$, if $f:A\to \mathcal{M}$ is partial elementary, then $f$ extends to an elementary map $f^*: A\cup\{b\} \to \mathcal{M}$. That  is, $\mathcal{M}$ is $\kappa$-homogeneous if for all $A  \subseteq \mathcal{M}$ with  $A<\kappa$ and all $b\in \mathcal{M}$, every elementary $f:A\to \mathcal{M}$ extends to an elementary $f^*:A\cup\{b\}\to\mathcal{M}$. We say $\mathcal{M}$ is  '''homogeneous''' if it is $\kappa$-homogeneous where $\kappa$ is the cardinality of the universe of $\mathcal{M}$.
Definition:Arens-Fort Space
Let $S$ be the set $\Z_{\ge 0} \times \Z_{\ge 0}$ be the Cartesian product of the set of positive integers $\Z_{\ge 0}$: :$S = \set {0, 1, 2, \ldots} \times \set {0, 1, 2, \ldots}$ Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that: :$(1): \quad \forall H \subseteq S: \tuple {0, 0} \notin H \implies H \in \tau$ :$(2): \quad H \subseteq S: \tuple {0, 0} \in H$ and, for all but a finite number of $m \in \Z_{\ge 0}$, the sets $S_m$ defined as: ::::$S_m = \set {n: \tuple {m, n} \notin H}$ :::are finite. That is, $H$ is allowed to be in $\tau$ if, considering $S = \Z_{\ge 0} \times \Z_{\ge 0}$ as the lattice points of the first quadrant of a Cartesian plane: Either: :$H$ does not contain $\tuple {0, 0}$ :$H$ contains $\tuple {0, 0}$, and only a finite number of the columns of $S$ are allowed to omit an infinite number of points in $H$. Then $\tau$ is the '''Arens-Fort topology''' on $S = \Z_{\ge 0} \times \Z_{\ge 0}$, and the topological space $T = \struct {S, \tau}$ is the '''Arens-Fort space'''.
Definition:Elementary Embedding
Let $\mathcal{M}$ and $\mathcal{N}$ be $\mathcal{L}$-structures with universes $M$ and $N$ respectively. An $\mathcal{L}$-embedding $j:\mathcal{M}\to \mathcal{N}$ is an '''elementary embedding'''  it preserves truth; that is: :$\mathcal{M} \models \phi(a_1,\dots, a_n) \iff \mathcal{N} \models \phi(j(a_1),\dots, j(a_n))$ holds for all $n\in\mathbb N$, all $\mathcal{L}$-formulas $\phi$ with $n$ free variables, and for all $a_1,\dots,a_n \in M$. === Partial Elementary Embedding === {{:Definition:Elementary Embedding/Partial Elementary Embedding}}
Definition:Modified Fort Space
Let $N$ be an infinite set. Let $\set a$ and $\set b$ be singleton sets such that $a \ne b$ and $a, b \notin N$. Let $S = N \cup \set a \cup \set b$. Let $\tau_{a, b}$ be the set of subsets of $S$ defined as: :$\tau_{a, b} = \set {H \subseteq N} \cup \set {H \subseteq S: \paren {a \in H \lor b \in H} \land N \setminus H \text { is finite} }$ That is, a subset $H$ of $S$ is in $\tau_{a, b}$  either: :$(1): \quad H$ is any subset of $N$ or: :$(2): \quad$ if $a$ or $b$ or both are in $H$, then $H$ is in $S$ only if it is cofinite in $S$, that is, that it contains all but a finite number of points of $S$ (or $N$, equivalently). Then $\tau_{a, b}$ is a '''modified Fort topology''' on $a$ and $b$, and the topological space $T = \struct {S, \tau_{a, b} }$ is a '''modified Fort space'''.
Definition:Algebraic (Model Theory)
Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$. Let $A$ be a subset of $M$. and let $\bar b$ be an ordered $n$-tuple of elements from $M$. Let $\mathcal{L}_A$ be the language formed by adding constant symbols to $\mathcal{L}$ for each element of $A$. $\bar b$ is '''algebraic''' over $A$ if there is an $\mathcal{L}_A$-formula $\phi(\bar x)$ with $n$ free variables such that $\mathcal{M}\models \phi(\bar b)$ and the set $\{\bar m \in M^n : \mathcal{M}\models \phi(\bar m)\}$ has only finitely many elements. {{Disambiguate|Definition:Logical Formula}}
Definition:Open Point
Let $T = \left({S, \tau}\right)$ be a topological space. Let $p \in S$ such that $\left\{{p}\right\} \in \tau$. That is, let $\left\{{p}\right\}$ be an open set of $T$. Then $p$ is an open point of $T$.  rs9ypa7mfatx7czrg2prm3peck65rh5
Definition:Age (Model Theory)
Let $\MM$ be an $\LL$-structure. An '''age''' of $\MM$ is a class $K$ of $\LL$-structures such that: * if $\AA$ is a finitely generated $\LL$-structure such that there is an $\LL$-embedding $\AA \to \MM$, then $\AA$ is isomorphic to some structure in $K$, * no two structures in $K$ are isomorphic, and * $K$ does not contain any structures which are not finitely generated or do not embed into $\MM$. That is, $K$ is an age of $\MM$ if it contains exactly one representative from each isomorphism type of the finitely-generated structures that embed into $\MM$. {{MissingLinks|finitely generated structure}}
Definition:Isolated Point (Metric Space)
Let $M = \left({A, d}\right)$ be a metric space. === Isolated Point in Subset === {{:Definition:Isolated Point (Metric Space)/Subset}} === Isolated Point in Space === When $S = A$ this reduces to: {{:Definition:Isolated Point (Metric Space)/Space}}
Definition:Isolated Point
=== Topology === {{:Definition:Isolated Point (Topology)/Space}} === Metric Space === {{:Definition:Isolated Point (Metric Space)/Space}} === Complex Analysis === {{:Definition:Isolated Point (Complex Analysis)}} === Real Analysis === {{:Definition:Isolated Point (Real Analysis)}}
Definition:Isolated Point (Real Analysis)
Let $S \subseteq \R$ be a subset of the set of real numbers. Let $\alpha \in S$. Then $\alpha$ is an '''isolated point of $S$'''  there exists an open interval of $\R$ whose midpoint is $\alpha$ which contains no points of $S$ except $\alpha$: :$\exists \epsilon \in \R_{>0}: \openint {\alpha - \epsilon} {\alpha + \epsilon} \cap S = \set \alpha$
Definition:Limit Point/Topology/Set
Let $A \subseteq S$. ==== Definition from Open Neighborhood ==== {{:Definition:Limit Point/Topology/Set/Definition 1}} ==== Definition from Closure ==== {{:Definition:Limit Point/Topology/Set/Definition 2}} ==== Definition from Adherent Point ==== {{:Definition:Limit Point/Topology/Set/Definition 3}} ==== Definition from Relative Complement ==== {{:Definition:Limit Point/Topology/Set/Definition 4}}
Definition:Limit Point/Complex Analysis
Let $S \subseteq \C$ be a subset of the set of complex numbers. Let $z_0 \in \C$.  Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$ for a given $\epsilon \in \R$ such that $\epsilon > 0$. Then $z_0$ is a '''limit point of $S$'''  ''every'' deleted $\epsilon$-neighborhood $\map {N_\epsilon} {z_0} \setminus \set {z_0}$ of $z_0$ contains a point in $S$: :$\forall \epsilon \in \R_{>0}: \paren {\map {N_\epsilon} {z_0} \setminus \set {z_0} } \cap S \ne \O$ that is: :$\forall \epsilon \in \R_{>0}: \set {z \in S: 0 < \cmod {z - z_0} < \epsilon} \ne \O$
Definition:Limit Point/Metric Space
Let $M = \struct {S, d}$ be a metric space. Let $A \subseteq S$ be a subset of $S$. Let $\alpha \in S$.  Then $\alpha$ is a '''limit point of $A$'''  ''every'' deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$: :$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap A \ne \O$ that is: :$\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$ Note that $\alpha$ does not have to be an element of $A$ to be a '''limit point'''.
Definition:Deleted Neighborhood/Topology
Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Let $V \subseteq S$ be a neighborhood of $x$. Then $V \setminus \left\{{x}\right\}$ is called a '''deleted neighborhood of $x$'''. That is, it is a neighborhood of $x$ with $x$ itself ''removed''.
Definition:Deleted Neighborhood/Metric Space
Let $M = \left({A, d}\right)$ be a metric space. Let $x \in A$. Let $B_\epsilon \left({x}\right)$ be the open $\epsilon$-ball neighborhood of $x$. Then the '''deleted $\epsilon$-neighborhood''' of $x$ is defined as $B_\epsilon \left({x}\right) \setminus \left\{{x}\right\}$. That is, it is the open $\epsilon$-ball neighborhood of $x$ with $x$ itself ''removed''. It can also be defined as: : $\left\{{y \in A: 0 < d \left({x, y}\right) < \epsilon}\right\}$
Definition:Deleted Neighborhood/Complex Analysis
Let $z_0 \in \C$ be a point in the complex plane. Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$. Then the '''deleted $\epsilon$-neighborhood''' of $z_0$ is defined as $\map {N_\epsilon} {z_0} \setminus \set {z_0}$. That is, it is the $\epsilon$-neighborhood of $z_0$ with $z_0$ itself ''removed''. It can also be defined as: :$\map {N_\epsilon} {z_0} \setminus \set {z_0} : = \set {z \in A: 0 < \cmod {z_0 - z} < \epsilon}$ from the definition of $\epsilon$-neighborhood.
Definition:Deleted Neighborhood/Real Analysis
Let $\alpha \in \R$ be a real number. Let $N_\epsilon \left({\alpha}\right)$ be the $\epsilon$-neighborhood of $\alpha$: :$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon \,.\,.\, \alpha + \epsilon}\right)$ Then the '''deleted $\epsilon$-neighborhood''' of $\alpha$ is defined as $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\}$. That is, it is the $\epsilon$-neighborhood of $\alpha$ with $\alpha$ itself ''removed''. It can also be defined as: : $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} : = \left\{{x \in \R: 0 < \left \vert{\alpha - x}\right \vert < \epsilon}\right\}$ or : $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} : = \left({\alpha - \epsilon \,.\,.\, \alpha}\right) \cup \left({\alpha \,.\,.\, \alpha + \epsilon}\right)$ from the definition of $\epsilon$-neighborhood.
Definition:Isolated Point (Complex Analysis)
Let $S \subseteq \C$ be a subset of the set of real numbers. Let $z \in S$. Then $z$ is an '''isolated point of $S$'''  there exists a neighborhood of $z$ in $\C$ which contains no points of $S$ except $z$: :$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} z \cap S = \set z$
Definition:Lipschitz Quaternion
A '''Lipschitz quaternion''' is a quaternion whose components are all integers. The set $\Bbb L$ of all '''Lipschitz quaternions''' can therefore be defined as: :$\Bbb L = \left\{{a + b \mathbf i + c \mathbf j + d \mathbf k \in \Bbb H: a, b, c, d \in \Z}\right\}$ {{NamedforDef|Rudolf Lipschitz|cat=Lipschitz}}
Definition:Hurwitz Quaternion
A '''Hurwitz quaternion''' is a quaternion whose components are all either: :integers or: :integers plus a half, that is, halves of odd integers. The set $H$ of all '''Hurwitz quaternions''' can therefore be defined as: :$H = \left\{{a + b \mathbf i + c \mathbf j + d \mathbf k \in \Bbb H: \left({a, b, c, d \in \Z}\right) \text { or } \left({a, b, c, d \in \Z + \dfrac 1 2}\right)}\right\}$
Definition:Multiple Pointed Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $A$ be a finite set whose cardinality is greater than $1$. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on $A$. Let $T \times D$ be the product space of $T$ and $D$. Then $T \times D$ is known as the '''multiple pointed topology''' on $T$. It is seen that $T \times D$ is conceptually equivalent to taking the space $T$ and replacing each point with a finite set of topologically indistinguishable points.
Definition:Box Topology
Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Define: :$\displaystyle \BB := \set {\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}$ Then $\BB$ is a synthetic basis on $X$, as shown on Basis for Box Topology. The '''box topology''' on $X$ is defined as the topology $\tau$ generated by the synthetic basis $\BB$.
Definition:Big Model
Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$. Let $\kappa$ be a cardinal. $\mathcal{M}$ is '''$\kappa$-big''' if for every subset $A\subset M$ with cardinality $|A| < \kappa$, the following holds: :if $\mathcal{L}_A$ is the language obtained from $\mathcal{L}$ by adding new constant symbols for each $a\in A$, then :for every language $\mathcal{L}_A^*$ obtained by adding a new relation symbol $R$ to $\mathcal{L}_A$, and :for every $\mathcal{L}_A^*$-structure $\mathcal{N}$ such that $\mathcal{M}$ and $\mathcal{N}$ are elementary equivalent as $\mathcal{L}_A$-structures, :there is a relation $R^\mathcal{M}$ on $M$ such that $(\mathcal{M},R^\mathcal{M})$ is elementary equivalent to $\mathcal{N}$ as an $\mathcal{L}_A^*$-structure.
Definition:Word Metric
Let $\struct {G, \circ}$ be a group. Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x \in S$). The '''word metric''' on $G$ with respect to $S$ is the metric $d_S$ defined as follows: :For any $g, h \in G$, let $\map {d_S} {g, h}$ be the minimum length among the finite sequences $\tuple {x_1, \dots, x_n}$ with each $x_i \in S$ such that $g \circ x_1 \circ \cdots \circ x_n = h$. Informally, $\map {d_S} {g, h}$ is the smallest number of elements from $S$ that one needs to multiply by to get from $g$ to $h$.
Definition:Continuous Extension
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $A, B \subseteq S_1$ be subsets of $S_1$ such that $A \subseteq B$. Let $f: A \to S_2$ and $g: B \to S_2$ be continuous mappings. Then $g$ is a '''continuous extension''' of $f$ : :$\forall s \in A: f \left({s}\right) = g \left({s}\right)$ That is, a '''continuous extension''' of $f$ is a continuous mapping on a superset which agrees with $f$ on the domain of $f$. Simply, it is a continuous mapping which is an extension. === Real Function === {{:Definition:Continuous Extension/Real Function}}
Definition:Diagonal Mapping
Let $S$ be a set. Let $S \times S$ be the Cartesian product of $S$ with itself. Then the '''diagonal mapping on $S$''' is defined as $\Delta: S \to S \times S$: :$\forall x \in S: \Delta \left({x}\right) = \left({x, x}\right)$ Clearly $\Delta$ is an injection, and is not a surjection unless $S$ is a singleton.  tgmqf7v43mpneequ5iqkxn1v1zkho6g
Definition:Minimal Arithmetic
'''Minimal arithmetic''' is the set $Q$ of theorems of the recursive set of sentences in the language of arithmetic containing exactly: {{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x         | m = \map s x \ne 0 }} {{axiom | n = \text M 2         | q = \forall x, y         | m = \map s x = \map s y \implies x = y }} {{axiom | n = \text M 3         | q = \forall x         | m = x + 0 = x }} {{axiom | n = \text M 4         | q = \forall x, y         | m = x + \map s y = \map s {x + y} }} {{axiom | n = \text M 5         | q = \forall x         | m = x \cdot 0 = 0 }} {{axiom | n = \text M 6         | q = \forall x, y         | m = x \cdot \map s y = \paren {x \cdot y} + x }} {{axiom | n = \text M 7         | q = \forall x         | m = \neg x < 0 }} {{axiom | n = \text M 8         | q = \forall x, y         | m = x < \map s y \iff \paren {x < y \lor x = y} }} {{axiom | n = \text M 9         | q = \forall x         | m = 0 < x \iff x \ne 0 }} {{axiom | n = \text M 10         | q = \forall x, y         | m = \map s x < y \iff \paren {x < y \land y \ne \map s x} }} {{end-axiom}}
Definition:Language of Arithmetic
A '''language of arithmetic''' is a signature for predicate logic consisting of: * the binary operation symbols: $+$ and $\cdot$ * the unary function symbol: $s$ * the binary relation symbol: $<$ * the constant symbol: $0$
Definition:Euclid's Definitions - Book I/2 - Line
A '''line''' is breadthless length.
Definition:Euclid's Definitions - Book I/4 - Straight Line
A '''straight line''' is a line which lies evenly with the points on itself.
Definition:Euclid's Definitions - Book I/3 - Line Extremities
The extremities of a line are points.
Definition:Euclid's Definitions - Book XI/2 - Extremity of Solid
An extremity of a solid is a surface.
Definition:Euclid's Definitions - Book I/7 - Plane Surface
A '''plane surface''' is a surface which lies evenly with the straight lines on itself.
Definition:Euclid's Definitions - Book I/6 - Surface Extremities
The extremities of a surface are lines.
Definition:Limit Point/Real Analysis
Let $S \subseteq \R$ be a subset of the real numbers. Let $\xi \in \R$ and let $S_\xi$ be the set defined as: :$S_\xi := \left\{{x: x \in S, x \ne \xi}\right\}$ Then $\xi$ is a '''limit point''' of $S$  $\xi$ is at zero distance from $S_\xi$.
Definition:Euclid's Definitions - Book I/1 - Point
A '''point''' is that which has no part.
Definition:Euclid's Definitions - Book I/5 - Surface
A '''surface''' is that which has length and breadth only.
Definition:Euclid's Definitions - Book I/8 - Plane Angle
A '''plane angle''' is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
Definition:Euclid's Definitions - Book I/9 - Rectilineal Angle
And when the lines containing the angle are straight, the angle is called '''rectilineal'''.
Definition:Euclid's Definitions - Book I/10 - Right Angle
When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is '''right''', and the straight line standing on the other is called a '''perpendicular''' to that on which it stands.
Definition:Euclid's Definitions - Book I/11 - Obtuse Angle
An '''obtuse angle''' is an angle greater than a right angle.