Von Neumann cardinal assignment

The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely:

|U|=\mathrm {card} (U)=\inf\{\alpha \in \mathrm {ON} \ |\ \alpha =_{c}U\},

where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.

That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤_c. This is a well-ordering of cardinal numbers.

Initial ordinal of a cardinal

Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal is a cardinal.

The $\alpha$ -th infinite initial ordinal is written $\omega _{\alpha }$ . Its cardinality is written $\aleph _{\alpha }$ (the $\alpha$ -th aleph number). For example, the cardinality of $\omega _{0}=\omega$ is $\aleph _{0}$ , which is also the cardinality of $\omega ^{2}$ , $\omega ^{\omega }$ , and $\epsilon _{0}$ (all are countable ordinals). So we identify $\omega _{\alpha }$ with $\aleph _{\alpha }$ , except that the notation $\aleph _{\alpha }$ is used for writing cardinals, and $\omega _{\alpha }$ for writing ordinals. This is important because arithmetic on cardinals is different from arithmetic on ordinals, for example $\aleph _{\alpha }^{2}$ = $\aleph _{\alpha }$ whereas $\omega _{\alpha }^{2}$ > $\omega _{\alpha }$ . Also, $\omega _{1}$ is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and $\omega _{1}$ is the order type of that set), $\omega _{2}$ is the smallest ordinal whose cardinality is greater than $\aleph _{1}$ , and so on, and $\omega _{\omega }$ is the limit of $\omega _{n}$ for natural numbers $n$ (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the $\omega _{n}$ ).

Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, $\alpha <\omega _{\beta }$ implies $\alpha +\omega _{\beta }=\omega _{\beta }$ , and 1 ≤ α < ω_β implies α · ω_β = ω_β, and 2 ≤ α < ω_β implies α^ω_β = ω_β. Using the Veblen hierarchy, β ≠ 0 and α < ω_β imply $\varphi _{\alpha }(\omega _{\beta })=\omega _{\beta }\,$ and Γ_{ω_β} = ω_β. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.

References

Y.N. Moschovakis Notes on Set Theory (1994 Springer) p. 198

Mathematical logic

General

Theorems (list)
and paradoxes

Gödel's completeness and incompleteness theorems
Tarski's undefinability
Banach–Tarski paradox
Cantor's theorem, paradox and diagonal argument
Compactness
Halting problem
Lindström's
Löwenheim–Skolem
Russell's paradox

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types of sets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

Formal systems (list),
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Example axiomatic systems (list)	of arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica