Note on set theory - Real Analysis
Naive set theory (朴素集合论) is the informal theory of sets developed in the 19th century by B. Bolzano, P. du Bois-Reymond, R. Dedekind, etc. The fundamental paradoxes discovered by G. Cantor and B. Russell within naive set theory brought about a fundamental revision of the foundations of mathematical logic. Axiomatic set theory is a formal theory of sets formulated by methods of mathematical logic, which consists of: (1) a definition of the language for formulating propositions; and (2) the principles of naive set theory expressed in this language as axioms or axiom schemes. There are other types of formal theories of sets, such as descriptive set theory and combinatorial set theory. Descriptive set theory is the study of definable sets (analytic, Borel, or coanalytic sets) and mappings in polish spaces (spaces homeomorphic to a complete separable metric space), created in the early 20th century by E. Borel, R. Baire and H. Lebesgue.
Types of axiomatic set theories:
Z is suitable for developing arithmetic, analysis, and functional analysis and for studying cardinal numbers smaller than $\aleph_\omega$. ZF, the default axiomatic set theory, allows the formalization of all ordinary mathematical theorems. NBG is obtained from ZF by adding the class variables and a finite number of axioms for forming classes.
Strengths of axiomatic systems: ZF ≥ Z > T; ¬ NF < T; A statement about sets is provable in NBG iff it is provable in ZF.
The language of an axiomatic set theory consists of primitive symbols (variables, operators) and expressions. Variables are common names of sets, e.g. $x, y, z$. Operators can be grouped into: predicates, definite description operator, quantifiers, (boolean and propositional) connectives, and parentheses. Expressions are grouped into: terms, names of sets, either a variable or has the form $(\tau \in \sigma)$, $(\tau = \sigma)$, or $℩ x A$, where $\tau$ and $\sigma$ are terms and $A$ is a formula; formulas, propositions, has the form $(A \leftrightarrow B)$, $(A \rightarrow B)$; $(A \lor B)$, $(A \land B)$, $\lnot A$; $\forall x A$, or $\exists x A$; where $A$ and $B$ are formulas and $x$ is a term. Notation $\Leftrightarrow$ is often used to define additional expressions.
Table: Reduction Orders for Term Rewriting. (b, boolean; x, term; A, B, formula)
| Operator Type | Operators | |
|---|---|---|
| 1 | parentheses | $()$ (operator precedence) |
| 2 | predicate (A o B ↦ b) | $\in$ (incidence), $=$ (equality) |
| 3 | definite descriptor (o x A ↦ x) | $℩$ ("the object such that") |
| 4 | quantifier (o x A ↦ b) | $\forall$ (for all), $\exists$ (there exists) |
| 5 | boolean connective (b ↦ b) | $\lnot$ (not); $\land$ (and); $\lor$ (or) |
| 6 | propositional connective (A o B ↦ b) | $\rightarrow$ (implies), $\leftrightarrow$ (equivalent) |
| 7 | notation | $\Leftrightarrow$ ("is a notation for") |
Common notations. Empty set, $\emptyset \Leftrightarrow ℩ x \forall y \lnot y \in x$. Set of all terms satisfying a formula, $\{x : A(x)\} \Leftrightarrow ℩ z \forall x (x \in z \leftrightarrow A(x))$. Set of two, $\{x, y\} \Leftrightarrow \{z : z = x \lor z = y\}$. Set of one, $\{x\} \Leftrightarrow \{x, x\}$. Ordered pair, $(x, y) \Leftrightarrow \{\{x\}, \{x, y\}\}$. Union, $x \cup y \Leftrightarrow \{z : z \in x \lor z \in y\}$. Intersection, $x \cap y \Leftrightarrow \{z : z \in x \land z \in y\}$. Union of elements, $\cup x \Leftrightarrow \{z : \exists v(z \in v \land v \in x)\}$. Cartesian product, $x \times y \Leftrightarrow \{z : \exists u v (z = (u, v) \land u \in x \land v \in y\}$. Predicate on mappings, $\mathrm{Fnc}(w) \Leftrightarrow \exists v (w \subset v \times v)$ $\land \forall u v_1 v_2 ((u, v_1) \in w \land (u, v_2) \in w \rightarrow v_1 = v_2)$. Value of a mapping $w$ at an element $x$, $w' x \Leftrightarrow ℩ y (x, y) \in w$. Predicate on a standard infinite set, $\mathrm{Inf}(z) \Leftrightarrow \emptyset \in z$ $\land \forall u (u \in z \rightarrow u \cup \{u\} \in z)$.
The principles of naive set theory can be expressed as:
Russell's paradox shows that these two axioms implicit in naive set theory are self-contradictory: let $A$ be $\lnot x \in x$, then $\forall x (x \in y \leftrightarrow \lnot x \in x)$ implies $y \in y \leftrightarrow \lnot y \in y$, which is a contradiction. The proposition that "a term is not an incident of itself" is almost universally true, so it describes any term. But if the collection of all terms is also a term, then it is an incident of itself, which contradicts the assumption.
Derivation rules and logical axioms (of Z, ZF, and NF):
Non-logical axioms of Z and ZF:
In NBG, class variables are introduced as capital letters, e.g. $X, Y, Z$. Proper class is a class that is not a set. Examples of proper classes: the class of all sets; the class of all ordinals; the class of all cardinals; the class of all groups; the class of all vector spaces. Class existence axioms are introduced, which imply the class existence theorem: any proposition defines a class if it has no bound class variable or definite description. Axiom of choice now states that there is a choice function defined on the class of all nonempty sets: $\exists X (\mathrm{Fnc}(X) \land \forall x (\lnot x = \emptyset \rightarrow X' x \in x))$.
Set $X$ is a collection of objects. Element $x$ of a set $X$ is an object in the set: $x \in X$.
Class $\mathcal{C}$ (or set class) is a collection of sets. Power set of a set is the class $\mathcal{P}(X)$ of subsets of a set $X$: $\mathcal{P}(X) = \{A : A \subset X\}$. Exponential object $Y^X$ in the category of sets is the set of all mappings $f: X \mapsto Y$ between two given sets $X$ and $Y$. The power set $\mathcal{P}(X)$ of a set $X$ can be seen as the set of all mappings $f: X \mapsto 2$ from $X$ to a given set of two elements $2 = \{0, 1\}$, so $\mathcal{P}(X)$ is sometimes also denoted as $2^X$.
Space $(X, \cdots)$ is a set with certain mathematical structures (algebraic, topological, measure).
Subset of a set $X$ is a set $A$ whose elements are also elements of $X$: $A \subset X$ iff $x \in A \Rightarrow x \in X$. Superset of a set $X$ is a set $Y$ of which $X$ is a subset: $Y \supset X$ iff $X \subset Y$. A set $X$ is said to include or contain another set $A$ if $A$ is a subset of $X$.
Two sets $X$ and $Y$ are equal if they are subsets of each other: $X = Y$ iff $X \subset Y$ and $Y \subset X$.
Proper subset or strict subset of a set $X$ is a subset $A$ of $X$ which is not equal to $X$: $A \subsetneq X$ iff $A \subset X$ but not $X \subset A$. Proper superset or strict superset of a set $X$ is a set $Y$ of which $X$ is a proper subset: $Y \supsetneq X$ iff $X \subsetneq Y$.
Union $\cup$ of a class $\mathcal{C}$ of sets is the set $\cup \mathcal{C}$ of elements that belong to at least one of the sets in $\mathcal{C}$: $\cup \mathcal{C} = \{x : x \in A \in \mathcal{C}\}$. The union of two sets $A$ and $B$ can be denoted as $A \cup B$.
Intersection $\cap$ of a class $\mathcal{C}$ of sets is the set $\cap \mathcal{C}$ of elements that belong to all sets in $\mathcal{C}$: $\cap \mathcal{C} = \{x : x \in A, \forall A \in \mathcal{C}\}$. The intersection of two sets $A$ and $B$ can be denoted as $A \cap B$.
Difference $\setminus$ between two sets $A$ and $B$ is the set of elements in $A$ that is not in $B$: $A \setminus B = \{x : x \in A, x \notin B\}$. Relative complement of a subset $B$ of a set $A$ is the difference between $A$ and $B$: $\complement_A B = A \setminus B$. (Absolute) complement $\complement$ of a subset $A$ in a default underlying set $X$ is the relative complement of $A$ in $X$: $\complement A = \complement_X A$. Symmetric difference $\Delta$ of two sets $A$ and $B$ is the set of elements in either $A$ or $B$, but not both: $A \Delta B = (A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)$.
Direct product $\times$, Cartesian product, or product set of a class $\{X_i : i \in I\}$ of sets indexed by a well-ordered set $(I, \le)$ is the set of all tuples $(x_i)_{i \in I}$ of elements of the sets: $\prod_{i \in I} X_i = \{ (x_i)_{i \in I} : x_i \in X_i \}$. The direct product of two sets $X$ and $Y$ can be denoted as $X \times Y$. Cartesian power $X^n$ of a set $X$ is the direct product of $n$ copies of the set: $X^n = \prod_{i=1}^n X$; Cartesian square $X^2$ of a set $X$ is its second Cartesian power.
Disjoint union $\sqcup$ (or discriminated union) of an indexed class $\{A_i : i \in I\}$ of sets is the set of elements in each set retaining their index: $\sqcup_i A_i = \{ (i, x_i) : i \in I, x_i \in A_i \}$.
Correspondence $(R, X, Y)$ between two sets $X$ and $Y$ is a subset $R \subset X \times Y$ of their Cartesian product. Correspondence generalizes the concept of mapping and binary relation.
Involution $R^\sharp$, inverse $R^{-1}$, or transpose $R^T$ of a correspondence $(R, X, Y)$ is the correspondence $(R^\sharp, Y, X)$ consisting of all switched pairs $(y, x)$ in $R$: $R^\sharp = \{(y, x) : (x, y) \in R\}$.
Mapping (in short, map) or function $f: X \mapsto Y$ from set $X$ to set $Y$ is a subset of $X \times Y$ such that $\forall x \in X$, $\exists! y \in Y$: $(x,y) \in f$; in other definitions, $f(x) = y$ is an equivalent notation for $(x, y) \in f$. Transformation $u: X \mapsto X$ is a mapping from a set to itself. (Algebraic) operation $\omega: X^n \mapsto X$ is a mapping from a Cartesian product $X^n$ of a set to the set $X$ itself. Operator $A: X \mapsto Y$ is a mapping from a space $X$ to another space $Y$, especially when $X$ and $Y$ are both vector spaces; the mapping is commonly denoted as $Ax = y$. Functional is an operator from a vector space $X$ of functions to a scalar field $\mathbb{F}$, usually $\mathbb{R}$ or $\mathbb{C}$. Morphism $\alpha \in H_{\mathcal{K}}(A, B)$ of a category $\mathcal{K}$ is a mapping from an object $A$ in the category into another object $B$.
For a mapping $f: X \mapsto Y$: its domain is $X$; codomain, $Y$, graph, $f$; value at an element $x \in X$ of the domain, $f(x)$ such that $(x, f(x)) \in f$; image of a subset $A \subset X$ of the domain, $f(A) = \{y : \exists x \in A, (x, y) \in f\}$; image $f(X)$ of the domain is also alled the image or range of the function; preimage (or inverse image) of a subset $B \subset Y$ of the codomain, $f^{-1}(B) = \{x : \exists y \in B, (x, y) \in f\}$; extension to a superset $W \supset X$ of the domain, $g: W \mapsto Z$ such that $f \subset g$; restriction to a subset $A \subset X$ of the domain, $f|A = \{(x, y) : x \in A, y = f(x)\}$. Composition $\circ$ of two mappings $f: X \mapsto Y$ and $g: Y \mapsto Z$ is the mapping $g \circ f = \{(x, z) : \exists y \in Y, (x, y) \in f, (y, z) \in g\}$.
Sequence $\{x_n\}$ of elements of a set $X$ is a mapping $\{x_n\}: \mathbb{N} \mapsto X$. Term $x_n$ of a sequence $\{x_n\}$ is an element of the mapping: $x_n = (n, x) \in f$, where $n$ is its index and $x$ is its value. Series $\sum_{n \in \mathbb{N}} x_n$ is a pair $(\{x_n\}, \{s_n\})$ consisting of a sequence $\{x_n\}$ of elements of a topological vector space $(X, +, d(\cdot, \cdot))$ and the sequence $\{s_n\}$ where $s_n = \sum_{i=1}^n x_i$. For a series $(\{x_n\}, \{s_n\})$, $x_n$ is its $n$-th term and $s_n$ is its partial sum of order $n$. The study of series is equivalent to the study of sequences.
Binary relation $R$ is a correspondence on a set $X$: $R \subset X \times X$. ($n$-place or $n$-ary) relation $R$ on a set $X$, $n \in \mathbb{N}$, is a subset $R \subset X^n$ of the $n$-th Cartesian power of the set.
Partial order on a set $X$ is a binary relation $\le$ (less than or equal to) on the set that satisfies: (1) reflexivity: $x \le x$; (2) anti-symmetry: $x \le y, y \le x \Rightarrow x = y$; (3) transitivity: $x \le y, y \le z \Rightarrow x \le z$. Partially ordered set or poset $(X, \le)$ is a set $X$ with a partial order $\le$. Any subset of a partially ordered set is also a partially ordered set. Chain of a partially ordered set $(X, \le)$ is a totally ordered subset $(A, \le)$. Antichain of a partially ordered set $(X, \le)$ is an incomparable subset $(A, \le)$: $\{(x, y) : x \le y, x, y \in A\} = \emptyset$. Length/Width of a partially ordered set $(X, \le)$ is the largest size of its chains/antichains. Minimal/Maximal element of a partially ordered set $(X, \le)$ is an element $x$, if exists, no greater/less than any other element: $\nexists y \in X: (y, x) \in R$; for the latter, $\nexists y \in X: (x, y) \in R$. Least/Greatest element, bottom/top $\bot$ / $\top$, or zero/unit $0$ / $1$ of a partially ordered set $(X, \le)$ is the element, if exists, less/greater than all other elements: $\forall x \in X, (\bot, x) \in R$; for the latter, $\forall x \in X, (x, \top) \in R$. Lexicographic order of (a subset of) the direct product $\prod_{m \in M} A_m$ of a well-ordered class $\{(A_m, \le)\}_{m \in (M, \le)}$ of partially ordered sets is the partial order $\le$ such that $(a_m)_{m \in M} < (b_m)_{m \in M}$ iff $\exists n \in M$: $\forall m \in M, m < n, a_m = b_m$, $a_n < b_n$; lexicographic product is a direct product with lexicographic order.
Total order of a set $X$ is a partial order on the set that satisfies the comparability condition (aka trichotomy law): $\forall a, b \in X$, $a \le b$ or $b \le a$; or equivalently, $R \cup R^\sharp = A \times A$. Totally ordered set or chain $(X, \le)$ is a set $X$ with a total order $\le$. Any subset of a totally ordered set is also a totally ordered set. Maximum/Minimum of a totally ordered set $(X, \le)$ is its greatest/least element, if exists. A totally ordered set cannot have a maximal/minimal element that is not its maximum/minimum.
Well-ordered set is a totally ordered set $(X, \le)$ such that every nonempty subset of $X$ has a minimum. Any subset of a well-ordered set is also a well-ordered set. Finite totally ordered sets are well-ordered. The set of integers $\mathbb{Z}$ with the total order $\le$ is not a well-ordered set. The lexicographic product of a finite collection of well-ordered sets is well-ordered. By axiom of choice, every set can be well-ordered.
The problem of equivalence classes of infinite sets, first considered by Cantor during 1871--1883.
Order isomorphism (序同构) between two partially ordered sets $(X, R_X)$ and $(Y, R_Y)$ is an order-preserving bijection $f: X \mapsto Y$: $R_Y = \{ (f(x_1), f(x_2)) : (x_1, x_2) \in R_X \}$. Order type $\overline{X}$ of a partially ordered set $(X, R_X)$, in Cantor's notation, is the property associated with classes of order isomorphic sets such that: $\overline{X} = \overline{Y}$ iff the partially ordered sets $(X, R_X)$ and $(Y, R_Y)$ are order isomorphic. Ordinal number $\alpha$ (序数; or ordinal for short) is the order type of a well-ordered set $(X, \le)$: $\alpha = \overline{X}$ [@Cantor1883]. In von Neumann's definition, an ordinal $\alpha$ is a set such that: (1) $x \in \alpha \Rightarrow x \subsetneq \alpha$; (2) $x, y \in \alpha \Rightarrow (x = y)$, $(x \in y)$, or $(y \in x)$; and (3) $\forall A \subsetneq \alpha, \exists x \in \alpha : x \cap A = \emptyset$. From this definition, we know that: the empty set is an ordinal; any element of an ordinal is an ordinal.
Algebraic structures on ordinals. Binary relations: equal to, $\alpha = \beta$ iff there are well-ordered sets $X$ and $Y$ such that $\overline{X} = \alpha$, $\overline{Y} = \beta$, and $\overline{X} = \overline{Y}$; less than, $\alpha < \beta$ iff $\exists \gamma \in \beta: \alpha = \gamma$. The class $\mathrm{Ord}$ of ordinals is well-ordered via $\le$, i.e. either $<$ or $=$. Binary operations (lexicographic order applies): addition, $\overline{X} + \overline{Y} = \overline{X \sqcup Y}$, (associative, commutative with identity $0 = \overline{\emptyset}$); multiplication, $\overline{X} \times \overline{Y} = \overline{X \times Y}$, (associative, commutative with identity $1 = \overline{\{\emptyset\}}$, left distributive with addition); exponentiation (via transfinite induction), $\alpha^0 = 1$, $\alpha^\beta = \alpha^{\beta - 1} \times \alpha$, and $\alpha^\lambda = \lim_{\beta < \lambda} \alpha^\beta$, where $\beta$ is not a limit ordinal and $\lambda$ is a limit ordinal other than $0$.
The class of ordinals. The order relation defined on ordinals implies that each ordinal is the set of all ordinals less than it: $\alpha = \{\beta : \beta < \alpha\}$. In von Neumann's representation, ordinal 0 is the empty set $\emptyset = \{\}$; ordinal 1 is the set of one ordinal (the empty set) $\{\emptyset\}$; ordinal n, $n \in \mathbb{N}$, is the set of $n$ ordinals $\{0, \cdots, n-1\}$. Written in pure set notation (braces and commas only), the finite ordinal $n$ would have $2^n$ pairs of braces. Transfinite ordinal (超限序数) is an ordinal larger than any finite ordinal. Ordinal $\omega$ is the set of all finite ordinals, $\omega = \{0, \cdots\}$. For any set of ordinals $A$, there is an ordinal $\gamma = \cup_{\alpha \in A} \{\beta : \beta \le \alpha\}$ greater than all ordinals in the set, so the class $\mathrm{Ord}$ of all ordinals is a proper class.
Indexing by ordindals. Transfinite sequence of order type $\alpha$, aka $\alpha$-sequence, of elements of a set $X$ is a mapping $\{x_\alpha\}: \alpha \mapsto X$. As a special case, an (infinite) sequence is an $\omega$-sequence. Any well-ordered set $(X, \le)$ can be uniquely indexed by the bijective $\alpha$-sequence, $\alpha = \overline{X}$, such that $\alpha(x) = \overline{\{y : y < x\}}$. Countably infinite set (or countable set) is a set that can be indexed by an $\omega$-sequence. Transfinite induction (超限归纳法) [@Hausdorff1906] is a method to prove a well-ordered set $\{P_\alpha\}$ of propositions, by showing: (1) $P_0$ is true; (2) if $\{P_\beta : \beta < \alpha\}$ is true, then $P_\alpha$ is true. As a special case, mathematical induction is transfinite induction on an $\omega$-sequence of propositions. By transfinite induction, every well-ordered set is order isomorphic to exactly one ordinal. Because an ordinal is a well-ordered set, every ordinal represents the order type of the class of well-ordered sets order isomorphic to it. Thus, the ordinal of an ordinal is itself: $\alpha = \overline{\alpha}$.
Classification of ordinals. Successor of an ordinal $\alpha$ is the ordinal $\alpha + 1 = \alpha \cup \{\alpha\}$. Predesessor of an ordinal $\alpha$, if exists, is the ordinal $\beta$ such that $\beta + 1 = \alpha$. Limit ordinal $\lambda$ is an ordinal that has no predesessor. For example, $0$ and $\omega$ are limit ordinals. Any ordinal $\alpha$ has a unique representation $\alpha = \lambda + n$, where $\lambda$ is a limit ordinal and $n \in \mathbb{N}$ is a finite ordinal. Limit of a transfinite sequence $\{x_\alpha\}$ is the least of the ordinals greater than all terms in the sequence: $\lim_{\beta < \alpha} x_\beta = \min \{\gamma : \gamma > x_\beta, \forall \beta < \alpha\}$. Cofinal ordinal to a limit ordinal $\lambda$ is an ordinal $\mu$ that is the limit of an ascending $\lambda$-sequence: $\mu = \lim_{\alpha < \lambda} x_\alpha$; apparently, $\mu \ge \lambda$. Cofinality $\mathrm{cf}(\alpha)$ of an ordinal $\alpha$ is the least limit ordinal that $\alpha$ is cofinal to: $\mathrm{cf}(\alpha) = \min \{\lambda : \exists \{x_\lambda\}, \lim_{\beta < \lambda} x_\beta = \alpha\}$; this means $\mathrm{cf}: \mathrm{Ord} \mapsto \mathrm{LimOrd}$ is a mapping from the class of ordinals to the class of limit ordinals. For example, $\mathrm{cf}(0) = 0$, $\mathrm{cf}(\omega) = \omega$, $\mathrm{cf}(\omega_n) = \omega_n$, $\mathrm{cf}(\omega_\omega) = \omega$. Regular ordinal is an limit ordinal that is its own cofinality: $\lambda = \mathrm{cf}(\lambda)$. Singular ordinal is an ordinal that is not regular.
Equipotence (等势), equipollence, or equinumerosity is an equivalence relation on the class of sets such that two sets satisfy this relation if there is a bijection between them. Cardinal number (基数; or cardinal for short), cardinality, or power (势) $|X|$ of a set $X$ is the property associated with classes of equipotent sets such that: $|X| = |Y|$ iff the sets $X$ and $Y$ are equipotent. The class of cardinals is a subclass of ordinals.
Algebraic structures on cardinals. Binary relation: less than or equal to, $|X| \le |Y|$ iff $\exists Z \subset Y: |X| = |Z|$. By the Cantor–Bernstein theorem, the class of cardinals is totally ordered. Binary operations: addition (associative, commutative), $|X| + |Y| = |X \sqcup Y|$; multiplication (associative, commutative; distributive w.r.t. addition), $|X| \times |Y| = |X \times Y|$; exponentiation, $|Y|^{|X|} = |Y^X|$.
The class of cardinals. The cardinality of a finite set is a natural number. Aleph numbers are the cardinality of infinite well-ordered sets. Aleph null $\aleph_0$ is the cardinality of the set of natural numbers: $\aleph_0 = |\mathbb{N}|$. $\aleph_1$ is the cardinality of the set of ordinals with cardinality no greater than $\aleph_0$: $\aleph_1 = |\{\alpha: |\alpha| \le \aleph_0\}|$. By transfinite induction, the class of aleph numbers can be indexed by ordinals: $\{\aleph_\alpha\}_\alpha$, where $\aleph_\alpha = |\{\beta: |\beta| < \aleph_\alpha\}|$. The collection of all cardinals $\mathrm{Card} = \mathbb{N} \cup \{\aleph_\alpha\}_{\alpha}$ is a proper class. Cardinality of the continuum $\mathfrak{c}$ is the cardinality of the set of real numbers: $\mathfrak{c} = |\mathbb{R}|$. It can be shown that $\mathfrak{c} = \alpha^{\omega}$ for all $2 \le \alpha \le \mathfrak{c}$. The following sets all have cardinality $\mathfrak{c}$: $T_\infty$, $(0,1)$, $[0,1]$, $\mathbb{R} \setminus \mathbb{A}$, $\mathbb{R} \setminus \mathbb{Q}$, $\mathbb{R}$, $\mathbb{R}^n$, $\mathbb{R^N}$, $C^0(\mathbb{R})$, $\mathcal{T}_{\mathbb{R}^n}$, $\mathcal{B}_{\mathbb{R}}$. Continuum hypothesis (CH) [@Cantor1878]: There is no cardinal between $\omega$ and $\mathfrak{c}$, the cardinalities of the countable and the continuum; i.e., $\aleph_1 = \mathfrak{c}$. Generalized continuum hypothesis (GCH): There is no cardinal between an aleph number and its power; i.e. $\aleph_{\alpha + 1} = 2^{\aleph_\alpha}$.
Relating ordinals and cardinals. Initial ordinal $\omega(\aleph_\alpha)$ of cardinality $\aleph_\alpha$ is the least ordinal with cardinality $\aleph_\alpha$: $\omega(\aleph_\alpha) = \min_{|\beta| = \aleph_\alpha} \beta$. By transfinite induction, the class of initial ordinals can be indexed by ordinals: $\{\omega_\alpha\}_\alpha$, where $\omega_\alpha = \omega(\aleph_\alpha)$. For example, $\omega_0 = \omega$. Initial ordinal $\omega_\alpha$ whose index is $0$ or a non-limit ordinal is regular; in general, initial ordinal $\omega_\lambda$ whose index is a limit ordinal is singular. Weakly inaccessible ordinal is a limit ordinal-indexed initial ordinal $\omega_\lambda$ that is regular. ZF set theory cannot prove that there is any weakly inaccessible ordinal other than $\omega$. Assuming the axiom of choice, the cardinal of a set $X$ is the least ordinal $\alpha$ such that $X$ and $\alpha$ are equipotent, $|X| = \min_{|\alpha| = |X|} \alpha$, then we have: (1) cardinals are the ordinals that are their own cardinals, $\mathrm{Card} = \{\alpha : \alpha = |\alpha|\}$; (2) $\omega(\aleph_\alpha) = \aleph_\alpha$, and thus $\omega_\alpha = \aleph_\alpha$, which means aleph numbers and initial ordinals are identical.
Relations among some proper classes: $\mathrm{PO} \supsetneq \mathrm{TO} \supsetneq \mathrm{WO} \supsetneq \mathrm{Ord} \supsetneq \mathrm{Card}$; $\mathrm{TfOrd} \supsetneq \mathrm{TfLimOrd} \supsetneq \mathrm{TfCard} \supsetneq \mathrm{TfRegOrd}$. Mappings between some proper classes: $|X|: [X] \mapsto \mathrm{Card}$; $\overline{X}: \mathrm{PO} \mapsto \mathrm{PO}$. Surjective mappings between some proper classes: $\overline{X}: \mathrm{WO} \mapsto \mathrm{Ord}$; $|X|: \mathrm{Ord} \mapsto \mathrm{Card}$.