hcju/proofwikips huggingface dataset:
Double Angle Formulas
=== Double Angle Formula for Sine === {{:Double Angle Formulas/Sine}} === Double Angle Formula for Cosine === {{:Double Angle Formulas/Cosine}} === Double Angle Formula for Tangent === {{:Double Angle Formulas/Tangent}}
Half Angle Formulas
=== Half Angle Formula for Sine === {{:Half Angle Formulas/Sine}} === Half Angle Formula for Cosine === {{:Half Angle Formulas/Cosine}} === Half Angle Formula for Tangent === {{:Half Angle Formulas/Tangent}} === Half Angle Formula for Tangent: Corollary 1 === {{:Half Angle Formulas/Tangent/Corollary 1}} === Half Angle Formula for Tangent: Corollary 2 === {{:Half Angle Formulas/Tangent/Corollary 2}} === Half Angle Formula for Tangent: Corollary 3 === {{:Half Angle Formulas/Tangent/Corollary 3}} === One Plus Tangent Half Angle over One Minus Tangent Half Angle === {{:One Plus Tangent Half Angle over One Minus Tangent Half Angle}}
Primitive Recursive Function is URM Computable
Every primitive recursive function is URM computable.
Constant Function is Primitive Recursive
The constant function $f_c: \N \to \N$, defined as: :$\map {f_c} n = c$ where $c \in \N$ is primitive recursive.
Heron's Formula
Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively. Then the area $A$ of $\triangle ABC$ is given by: :$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c}}$ where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.
Addition is Primitive Recursive
The function $\Add: \N^2 \to \N$, defined as: :$\map \Add {n, m} = n + m$ is primitive recursive.
Exact Form of Prime-Counting Function
Let:  :$\displaystyle \map \Pi x = \map \Li x - \sum_\rho \map \Li {x^\rho} - \map \ln 2 + \int_x^\infty \frac {\d t} {t \paren {t^2 - 1} \, \map \ln t}$  where: :$\map \Li x$ is the offset logarithmic integral :the sum $\displaystyle \sum_\rho$ is taken over all $0 < \rho \in \R$ such that the zeta function $\map \zeta {\alpha + i \rho} = 0$ for some $\alpha \in \R$. Then the prime-counting function is precisely: :$\displaystyle \map \pi x = \sum_{n \mathop = 1}^\infty \paren {\frac {\map \mu n} n \map \Pi {x^{1/n} } }$ where $\map \mu n$ denotes the Möbius function.
Multiplication is Primitive Recursive
The function $\operatorname{mult}: \N^2 \to \N$, defined as: :$\map \Mult {n, m} = n \times m$ is primitive recursive.
Exponentiation is Primitive Recursive
The function $\exp: \N^2 \to \N$, defined as: :$\exp \left({n, m}\right) = n^m$ is primitive recursive.
Bézout's Lemma
Let $a, b \in \Z$ such that $a$ and $b$ are not both zero. Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\exists x, y \in \Z: a x + b y = \gcd \set {a, b}$ That is, $\gcd \set {a, b}$ is an integer combination (or linear combination) of $a$ and $b$. Furthermore, $\gcd \set {a, b}$ is the smallest positive integer combination of $a$ and $b$.
Predecessor Function is Primitive Recursive
The '''predecessor function''' $\operatorname{pred}: \N \to \N$ defined as: :$\operatorname{pred} \left({n}\right) = \begin{cases} 0 & : n = 0 \\ n-1 & : n > 0 \end{cases}$ is primitive recursive.
Cut-Off Subtraction is Primitive Recursive
The '''cut-off subtraction''' function, defined as: :$\forall \tuple {n, m} \in \N^2: n \mathop {\dot -} m = \begin{cases} 0 & : n < m \\ n - m & : n \ge m \end{cases}$ is primitive recursive.
Maximum Function is Primitive Recursive
The maximum function $\max: \N^2 \to \N$, defined as: :$\max \left({n, m}\right) = \begin{cases} m: & n \le m \\ n: & m \le n \end{cases}$ is primitive recursive.
Sum of Maximum and Minimum
For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$: :$a + b = \max \left({a, b}\right) + \min \left({a, b}\right)$
Absolute Difference Function is Primitive Recursive
The '''absolute difference''' function $\operatorname {adf}: \N^2 \to \N$, defined as: :$\map {\operatorname {adf} } {n, m} = \size {n - m}$ where $\size a$ is defined as the absolute value of $a$, is primitive recursive.
Primitive Recursive Set is URM Computable
Every primitive recursive set is URM computable.
Set Containing Only Zero is Primitive Recursive
The subset $\left\{{0}\right\} \subset \N$ is primitive recursive.
Set of Non-Zero Natural Numbers is Primitive Recursive
Let $\N^*$ be defined as $\N^* = \N \setminus \left\{{0}\right\}$. The subset $\N^* \subset \N$ is primitive recursive.
Signum Function is Primitive Recursive
Let $\operatorname{sgn}: \N \to \N$ be defined as the signum function. Then: : $\operatorname{sgn}$ is primitive recursive. : $\overline {\operatorname{sgn}}$ is primitive recursive.
Complement of Primitive Recursive Set
Let $S \subseteq \N$ be primitive recursive. Then its relative complement $\N \setminus S$ of $S$ in $\N$ is primitive recursive.
Intersection of Primitive Recursive Sets
Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$. Let $A$ and $B$ both be primitive recursive. Then $A \cap B$, the intersection of $A$ and $B$, is primitive recursive.
Union of Primitive Recursive Sets
Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$. Let $A$ and $B$ both be primitive recursive. Then $A \cup B$, the union of $A$ and $B$, is primitive recursive.
Set of Even Numbers is Primitive Recursive
Let $E \subseteq \N$ be the set of all even natural numbers. Then $E$ is primitive recursive.
Primitive Recursive Relation is URM Computable
Every primitive recursive relation is URM computable.
Equality Relation is Primitive Recursive
The relation $\operatorname{eq} \subseteq \N^2$, defined as: :$\map {\operatorname {eq} } {n, m} \iff n = m$ is primitive recursive.
Set Operations on Primitive Recursive Relations
Let $\RR_1 \subseteq N^k$ and $\RR_2 \subseteq N^k$ be $k$-ary relations on $N^k$. Let $\RR_1$ and $\RR_2$ be primitive recursive. Then the following are all primitive recursive relations: :$\TT = \neg \RR_1$ :$\UU = \RR_1 \land \RR_2$ :$\VV = \RR_1 \lor \RR_2$
Ordering Relations are Primitive Recursive
The ordering relations on $\N^2$: * $n < m$ * $n \le m$ * $n \ge m$ * $n > m$ are all primitive recursive.
Permutation of Variables of Primitive Recursive Function
Let $f: \N^k \to \N$ be a primitive recursive function. Let $\sigma$ be a permutation of $\left({1, 2, \ldots, k}\right)$. Then the function $h: \N^k \to \N$ defined as: :$h \left({n_1, n_2, \ldots, n_k}\right) = f \left({n_{\sigma \left({1}\right)}, n_{\sigma \left({2}\right)}, \ldots, n_{\sigma \left({k}\right)}}\right)$ is also primitive recursive.
Definition by Cases is Primitive Recursive
Let $\mathcal R_1, \mathcal R_2, \ldots, \mathcal R_k$ be primitive recursive relations on $\N^l$ such that: :$\forall i, j \in \set{1, 2, \ldots, k}: \mathcal R_i \implies \lnot \mathcal R_j$, that is, all relations are mutually exclusive :$\forall \tuple {n_1, n_2, \ldots, n_l} \in \N^l: \exists i \in \set {1, 2, \ldots, k}: \map {\mathcal R_i} {n_1, n_2, \ldots, n_l}$, that is, the set of relations is exhaustive. Let $\forall i \in \set {1, 2, \ldots, k}: g_i: \N^l \to \N$  be primitive recursive functions. Then the function $f: \N^l \to \N$ defined as: :$\map f {n_1, n_2, \ldots, n_l} = \begin{cases} \map {g_1} {n_1, n_2, \ldots, n_l} & : \map {\mathcal R_1} {n_1, n_2, \ldots, n_l} \\ \map {g_2} {n_1, n_2, \ldots, n_l} & : \map {\mathcal R_2} {n_1, n_2, \ldots, n_l} \\ \quad \vdots & \quad \vdots \\ \map {g_k} {n_1, n_2, \ldots, n_l} & : \map {\mathcal R_k} {n_1, n_2, \ldots, n_l} \end{cases}$ is primitive recursive.
Domain of Injection to Countable Set is Countable
Let $X$ be a set, and let $Y$ be a countable set. Let $f: X \to Y$ be an injection. Then $X$ is also countable.
Unique Code for URM Instruction
Each basic instruction $I$ in a URM Program can be identified with a unique '''code number''' $\beta \left({I}\right)$. We also define the following sets: * $\operatorname{Zinstr}$ is the set of codes of all the <tt>Zero</tt> instructions * $\operatorname{Sinstr}$ is the set of codes of all the <tt>Successor</tt> instructions * $\operatorname{Cinstr}$ is the set of codes of all the <tt>Copy</tt> instructions * $\operatorname{Jinstr}$ is the set of codes of all the <tt>Jump</tt> instructions. Then we define $\operatorname{Instr}$ to be the set of codes of all basic URM instructions. That is: :$\operatorname{Instr} = \operatorname{Zinstr} \cup \operatorname{Sinstr} \cup \operatorname{Cinstr} \cup \operatorname{Jinstr}$.
URM Instructions are Countably Infinite
The set $\Bbb I$ of all basic URM instructions is countably infinite.
URM Programs are Countably Infinite
The set $\mathbf P$ of all URM programs is countably infinite.
Unique Code for URM Program
Any URM program can be assigned a unique '''code number'''.
URM Computable Functions of One Variable is Countably Infinite
The set $\mathbf U$ of all URM computable functions of $1$ variable is countably infinite.
Natural Number Functions are Uncountable
The set of all natural number one-variable functions $\set {f: \N \to \N}$ is uncountably infinite.
Not All Natural Number Functions are Primitive Recursive
Not all functions $f: \N \to \N$ are primitive recursive.
Bounded Summation is Primitive Recursive
Let the function $f: \N^{k+1} \to \N$ be primitive recursive. Then so is the function $g: \N^{k+1} \to \N$ defined as: :$\displaystyle g \left({n_1, n_2, \ldots, n_k, z}\right) = \begin{cases} 0 & : z = 0 \\ \sum_{y \mathop = 1}^z f \left({n_1, n_2, \ldots, n_k, y}\right) & : z > 0 \end{cases}$
Bounded Product is Primitive Recursive
Let the function $f: \N^{k+1} \to \N$ be primitive recursive. Then so is the function $g: \N^{k+1} \to \N$ defined as: :$\displaystyle g \left({n_1, n_2, \ldots, n_k, z}\right) = \begin{cases} 1 & : z = 0 \\ \prod_{y=1}^z f \left({n_1, n_2, \ldots, n_k, y}\right) & : z > 0 \end{cases}$
Divisor Relation is Primitive Recursive
The divisor relation $m \divides n$ in $\N^2$ is primitive recursive.
Divisor Counting Function is Primitive Recursive
The divisor counting ($\tau$) function is primitive recursive.
Set of Prime Numbers is Primitive Recursive
The set $\Bbb P$ of prime numbers is primitive recursive.
Prime Enumeration Function is Primitive Recursive
Let the function $p: \N \to \N$ be the prime enumeration function, defined as: :$p \left({n}\right) = \begin{cases} 1 & : n = 0 \\ \mbox{the } n \mbox{th prime number} & : n > 0 \end{cases}$ Then $p$ is primitive recursive.
Length Function is Primitive Recursive
Let $n \in \N$. Let $\map \len n$ denote the length of $n$. Then the function $\len: \N \to \N$ is primitive recursive.
Prime Exponent Function is Primitive Recursive
Let $n \in \N$ be a natural number. Let $\left({n, j}\right): \N^2 \to \N$ be defined as: :$\left({n, j}\right) = \left({n}\right)_j$ where $\left({n}\right)_j$ is the prime exponent function. Then $\left({n, j}\right)$ is primitive recursive.
Set of Codes for URM Instructions is Primitive Recursive
The set $\operatorname{Instr}$ of codes of all basic URM instructions is primitive recursive.
Set of Sequence Codes is Primitive Recursive
Let $\operatorname{Seq}$ be the set of all code numbers of finite sequences in $\N$. Then $\operatorname{Seq}$ is primitive recursive.
Set of Codes for URM Programs is Primitive Recursive
Let $\operatorname{Prog}$ be the set of all code numbers of URM programs. Then $\operatorname{Prog}$ is a primitive recursive set.
Upper Bounds for Prime Numbers
Let $p: \N \to \N$ be the prime enumeration function. Then $\forall n \in \N$, the value of $\map p n$ is bounded above. In particular:
Minimization on Relation Equivalent to Minimization on Function
Let $\mathcal R$ be a $k+1$-ary relation on $\N^{k+1}$. Then the function $g: \N^{k+1} \to \N$ defined as: :$g \left({n_1, n_2, \ldots, n_k, z}\right) = \mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ where $\mu y \ \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ is the minimization operation on $\mathcal R$ is equivalent to minimization on a total function.
Function Obtained by Minimization from URM Computable Functions
Let the function $f: \N^{k+1} \to \N$ be a URM computable function. Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus: :$g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \left({f \left({n_1, n_2, \ldots, n_k}\right) = 0}\right)$. Then $g$ is also URM computable.
Function Obtained by Minimization from URM Computable Relations
Let $\mathcal R$ be a URM computable $k+1$-ary relation on $\N^{k+1}$. Let the function $f: \N^{k+1} \to \N$ be a URM computable function. Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus: :$g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$ Then $g$ is also URM computable.
Recursive Function is URM Computable
Every recursive function is URM computable.
Unique Code for State of URM Program
Every state of a URM program can be assigned a unique '''code number'''. This code number is called the '''state code''' (or '''situation code''').
State Code Function is Primitive Recursive
Let $k \in \N^*$. Let $e = \gamma \left({P}\right)$ be the code number of a URM program $P$. Let $\left({n_1, n_2, \ldots, n_k}\right)$ be the input of $P$. Let $S_k: \N^{k+2} \to \N$ be the function defined as: :$S_k \left({e, n_1, n_2, \ldots, n_k, t}\right)$ is the state code for $P$ at stage $t$ of computation of $P$. If $e$ does not code a URM Program then $S_k = 0$. Also, if $P$ terminates at stage $t_0$, then we put: :$\forall t > t_0: S_k \left({e, n_1, n_2, \ldots, n_k, t}\right) = S_k \left({e, n_1, n_2, \ldots, n_k, t_0}\right)$. Then for all $k \ge 1$, the function $S_k$ is primitive recursive.
URM Computable Function is Recursive
Every URM computable function is recursive.
Kleene's Normal Form Theorem
For each integer $k \ge 1$, there exists: * a primitive recursive $k+1$-ary relation $T_k$; * a primitive recursive function $U: \N \to \N$ such that a partial function $f: \N^k \to \N$ is recursive iff, for some $e \in \N$ and all $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$: :$f \left({n_1, n_2, \ldots, n_k}\right) \approx U \left({\mu z \ T_k \left({e, n_1, n_2, \ldots, n_k, z}\right)}\right)$
Recursive Function uses One Minimization
Every recursive function can be obtained from the basic primitive recursive functions using: * substitution * primitive recursion * at most one minimization on a function.
Universal URM Computable Functions
For each integer $k \ge 1$, there exists a URM computable function: :$\Phi_k: \N^{k+1} \to \N$ such that for each URM computable function $f: \N^k \to \N$ there exists a natural number $e$ such that: :$\forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: f \left({n_1, n_2, \ldots, n_k}\right) \approx \Phi_k \left({e, n_1, n_2, \ldots, n_k}\right)$. This function $\Phi_k$ is '''universal for URM computable functions of $k$ variables'''.
Universal URM Programs
For each integer $k \ge 1$, there exists a URM program $P_k$ such that: For each URM program $P$ there exists a natural number $e$ such that: For all $\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$, the computation using the program $P_k$ with input $\left({e, n_1, n_2, \ldots, n_k}\right)$ has the same output as the computation using the program $P$ with input $\left({n_1, n_2, \ldots, n_k}\right)$. This function $P_k$ is a '''universal program for URM computations with $k$ inputs'''.
Combination of Recursive Functions
Let $f: \N^k \to \N$ and $g: \N^k \to \N$ be recursive functions (not necessarily total), where $k \ge 1$. Let $\mathcal R$ be a $k$-ary relation such that: : if $\mathcal R \left({n_1, n_2, \ldots, n_k}\right)$ holds, then $f \left({n_1, n_2, \ldots, n_k}\right)$ is defined : if $\mathcal R \left({n_1, n_2, \ldots, n_k}\right)$ does not hold, then $g \left({n_1, n_2, \ldots, n_k}\right)$ is defined. Let $h: \N^k \to \N$ be the function defined as: :$h \left({n_1, n_2, \ldots, n_k}\right) = \begin{cases} f \left({n_1, n_2, \ldots, n_k}\right) & : \text{if } \mathcal R \left({n_1, n_2, \ldots, n_k}\right) \text { holds} \\ g \left({n_1, n_2, \ldots, n_k}\right) & : \text{otherwise} \end{cases}$ so that $h$ is total. Then $h$ is recursive.
Not All URM Computable Functions are Primitive Recursive
There exist URM computable functions which are not primitive recursive.
Cantor's Diagonal Argument
Let $S$ be a set such that $\card S > 1$, that is, such that $S$ is not a singleton. Let $\mathbb F$ be the set of all mappings from the natural numbers $\N$ to $S$: :$\mathbb F = \set {f: \N \to S}$ Then $\mathbb F$ is uncountably infinite.
Infinite if Injection from Natural Numbers
Let $S$ be a set. Let there exist an injection $\phi: \N \to S$ from the natural numbers to $S$. Then $S$ is infinite.
Halting Problem is Not Algorithmically Decidable
Let $H: \N^2 \to \N$ be the function given by: :$\map H {m, n} = 1$ if $m$ codes a URM program which halts with input $n$ :$\map H {m, n} = 0$ otherwise. Then $H$ is not recursive.
Set of Total Functions is Not Recursive
The set $\operatorname{Tot}$ of natural numbers which code URM programs which compute total functions of one variable is not recursive.
Infinitely Many Programs for URM Computable Function
Let $g: \N^k \to \N$ be a URM computable function. Then there is an infinite number of URM programs which compute $g$.
Bounds of GCD for Sum and Difference Congruent Squares
Let $x, y, n$ be integers. Let: :$x \not \equiv \pm y \pmod n$ and: :$x^2 \equiv y^2 \pmod n$ where $a \equiv b \pmod n$ denotes that $a$ is congruent to $b$ modulo $n$. Then: :$1 < \gcd \set {x - y, n} < n$ and: :$1 < \gcd \set {x + y, n} < n$ where $\gcd \set {a, b}$ is the GCD of $a$ and $b$.
Triangle Angle-Side-Angle Equality
If two triangles have: :two angles equal to two angles, respectively :the sides between the two angles equal then the remaining angles are equal, and the remaining sides equal the respective sides. That is to say, if two pairs of angles and the included sides are equal, then the triangles are congruent.
Triangle Side-Angle-Angle Equality
If two triangles have: :two angles equal to two angles, respectively :the sides opposite one pair of equal angles equal then the remaining angles are equal, and the remaining sides equal the respective sides. That is to say, if two pairs of angles and a pair of opposite sides are equal, then the triangles are congruent.
Equal Alternate Angles implies Parallel Lines
Given two infinite straight lines which are cut by a transversal, if the alternate angles are equal, then the lines are parallel. {{:Euclid:Proposition/I/27}}
Parallelism is Transitive Relation
Parallelism between straight lines is a transitive relation. {{:Euclid:Proposition/I/30}}
Sum of Sequence of Products of Consecutive Integers
{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n j \paren {j + 1}       | r = 1 \times 2 + 2 \times 3 + \dotsb + n \paren {n + 1}       | c =  }} {{eqn | r = \frac {n \paren {n + 1} \paren {n + 2} } 3       | c =  }} {{end-eqn}}
Sum of Sequence of Products of Consecutive Reciprocals
:$\displaystyle \sum_{j \mathop = 1}^n \frac 1 {j \paren {j +1} } = \frac n {n + 1}$
Sum of Sequence of Fibonacci Numbers
:$\displaystyle \forall n \in \Z_{\ge 0}: \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$
Sum of Sequence of Even Index Fibonacci Numbers
{{begin-eqn}} {{eqn | lo= \forall n \ge 1:       | l = \sum_{j \mathop = 1}^n F_{2 j}       | r = F_2 + F_4 + F_6 + \cdots + F_{2 n}       | c =  }} {{eqn | r = F_{2 n + 1} - 1       | c =  }} {{end-eqn}}
Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers
:$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$
Lucas Number as Sum of Fibonacci Numbers
Let $L_k$ be the $k$th Lucas number, defined as: :$L_n = \begin{cases} 2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$ Then: :$L_n = F_{n - 1} + F_{n + 1}$
Sum of Sequence of Product of Lucas Numbers with Powers of 2
Let $L_k$ be the $k$th Lucas number. Let $F_k$ be the $k$th Fibonacci number. Then: :$\displaystyle \forall n \in \N_{>0}: \sum_{j \mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$ That is: :$2^0 L_1 + 2^1 L_2 + 2^2 L_3 + \cdots + 2^{n - 1} L^n = 2^n F_{n + 1} - 1$
Sum of Odd Positive Powers
Let $n \in \N$ be an odd positive integer. Let $x, y \in \Z_{>0}$ be (strictly) positive integers. Then $x + y$ is a divisor of $x^n + y^n$.
Condition for Well-Foundedness
Let $\struct {S, \preceq}$ be an ordered set. Then $\struct {S, \preceq}$ is well-founded  there is no infinite sequence $\sequence {a_n}$ of elements of $S$ such that: :$\forall n \in \N: a_{n + 1} \prec a_n$ That is,  there is no infinite sequence $\sequence {a_n}$ such that $a_0 \succ a_1 \succ a_2 \succ \cdots$.
Subset Relation on Power Set is Partial Ordering
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the relation $\subseteq$. Then $\struct {\powerset S, \subseteq}$ is an ordered set. The ordering $\subseteq$ is partial  $S$ is neither empty nor a singleton; otherwise it is total.
Ordering is Equivalent to Subset Relation
Let $\struct {S, \preceq}$ be an ordered set. Then there exists a set $\mathbb S$ of subsets of $S$ such that: :$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$ where: :$\struct {\mathbb S, \subseteq}$ is the relational structure consisting of $\mathbb S$ and the subset relation :$\cong$ denotes order isomorphism. Hence any ordering on a set can be modelled uniquely by a set of subsets of that set under the subset relation.
Subset Relation is Ordering
Let $\mathbb S$ be a set of sets. Then $\subseteq$ is an ordering on $\mathbb S$. In other words, let $\struct {\mathbb S, \subseteq}$ be the relational structure defined on $\mathbb S$ by the relation $\subseteq$. Then $\struct {\mathbb S, \subseteq}$ is an ordered set.
Principle of Mathematical Induction/Well-Ordered Set
Let $\struct {S, \preceq}$ be a well-ordered set. Let $T \subseteq S$ be a subset of $S$ such that: :$\forall s \in S: \paren {\forall t \in S: t \prec s \implies t \in T} \implies s \in T$ Then $T = S$.
Order Isomorphism from Woset onto Subset
Let $\struct {S, \preceq}$ be a woset. Let $T \subseteq S$. Let $f: S \to T$ be an order isomorphism. Then $\forall x \in S: x \preceq \map f x$.
Order Isomorphism between Wosets is Unique
Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be wosets. Let $\struct {S_1, \preceq_1} \cong \struct {S_2, \preceq_2}$, that is, let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be order isomorphic. Then there is exactly one mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism.
No Isomorphism from Woset to Initial Segment
Let $\struct {S, \preceq}$ be a woset. Let $a \in S$, and let $S_a$ be the initial segment of $S$ determined by $a$. Then there is no order isomorphism between $S$ and $S_a$.
Woset is Isomorphic to Set of its Initial Segments
Let $\struct {S, \preceq}$ be a well-ordered set. Let: :$A = \set {a^\prec: a \in S}$ where $a^\prec$ is the strict lower closure of $S$ determined by $a$. Then: :$\struct {S, \preceq} \cong \struct {A, \subseteq}$ where $\cong$ denotes order isomorphism.
Ordering on Ordinal is Subset Relation
Let $\struct {S, \prec}$ be an ordinal. Then $\forall x, y \in S:$ :$x \in y \iff x \prec y \iff S_x \subsetneqq S_y \iff x \subsetneqq y$ where $S_x$ and $S_y$ are the initial segments of $S$ determined by $x$ and $y$ respectively. Thus there is no need to specify what the ordering on an ordinal is -- it is always the subset relation.
Initial Segment of Ordinal is Ordinal
Let $S$ be an ordinal, and suppose that $a \in S$. Then the initial segment $S_a = a$ of $S$ determined by $a$ is also an ordinal. In other words, every element of an ordinal is also an ordinal.
Ordinal Subset of Ordinal is Initial Segment
Let $S$ be an ordinal. Let $T \subset S$ also be an ordinal. Then $\exists a \in S: T = S_a$, where $S_a$ is the initial segment of $S$ determined by $a$. That is, $T = S_a = a \in S$.
Intersection of Two Ordinals is Ordinal
Let $S, T$ be ordinals. Then $S \cap T$ is an ordinal.
Relation between Two Ordinals
Let $S$ and $T$ be ordinals. Then either $S \subseteq T$ or $T \subseteq S$.
Isomorphic Ordinals are Equal
Let $A$ and $B$ be ordinals that are order isomorphic. Then $A = B$.