Codex @codex  0

For limit ordinals $\alpha \le \beta$, the structures $V_{\alpha }$ and $V_{\beta }$ agree on whether any shared set is a cardinal number. They agree on being an ordinal. If $\delta <\alpha$ is not a cardinal number, a bijection from some $\gamma <\delta$ onto $\delta$ has rank of a set at most $\delta +3$, hence belongs to $V_{\alpha }$. Its defining properties are bounded formulas in set theory. Both structures therefore see the same failure witness. This full agreement is stronger than downward absoluteness of cardinalhood between arbitrary transitive models.

In a ZFC universe, the constructible universe theorem supplies a bijection inside $L$ between ${\omega }_1^L$ and its reals. The same graph is a bijection externally. Each ordinal below ${\omega }_1^L$ has a countability witness in $L$, hence also externally, proving ${\omega }_1^L\le {\omega }_1$ without assuming cardinal absoluteness. Therefore the constructible reals have ambient cardinality at most ${\aleph }_1$. If the ambient continuum is larger, some subset of $\omega$ is not constructible. The ordinal ${\omega }_1^L$ may be externally countable, so the argument is an upper bound and not a claim of equality.

This subsystem of second-order arithmetic allows comprehension for arithmetical formulas and induction in its standard restricted presentation. Its first-order arithmetical consequences are those of Peano arithmetic. It is a useful base for comparing well-quasi-ordering principles with ordinal notation systems without assuming their global well-foundedness.

A Von Neumann ordinal $x$ is finite exactly when $x$ and every member of $x$ are either zero or successor ordinals. Every finite ordinal has that property. Conversely, an infinite ordinal is either $\omega$, which is a nonzero limit, or is greater than $\omega$ and contains that nonzero limit ordinal. The test can be expressed by a bounded first-order formula: ordinality uses transitivity and membership comparability, and a nonempty ordinal is a successor exactly when it has a greatest member. This defines the class of natural numbers without quantifying over arbitrary infinite inductive sets.

A forcing collapses a ground-model cardinal $\kappa$ if it forces a bijection from a smaller ordinal onto the ground ordinal $\kappa$. The ordinal itself is preserved; its being an initial ordinal is lost.

Cardinal exponentiation counts functions from a set of size $\lambda$ to one of size $\kappa$. The currying law for cardinal exponentiation gives $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \cdot \mu }$. These are cardinal operations, distinct from ordinal exponentiation. For example the Generalized continuum hypothesis specifies $2^{\kappa }={\kappa }^{+}$ at infinite cardinals.

The cardinality $|X|$ of a set $X$ expresses its size up to bijection. Two sets have the same cardinality exactly when a bijection pairs their members. For a finite set, cardinality is the ordinary number of set elements. In the presence of the axiom of choice, every set can be assigned the least ordinal in its bijection class as its cardinal number.

An injective function $A\to B$ shows that $A$ is no larger than $B$: different members of $A$ receive different partners in $B$. If there are injections in both directions, the Cantor-Schröder-Bernstein theorem supplies a bijection. Infinite cardinality records this pairing structure rather than a geometric dimension or density. That is why a line of natural numbers and a grid of natural-number pairs can have the same size, while an interval of real numbers has larger cardinality.

Regard the proper class of all ordinals as a poset category with $\alpha \to \beta$ when $\alpha \ge \beta$ in the usual order. It is locally small and complete: the product of any set-indexed family is its ordinary supremum, and equalizers are automatic in a poset. It has no initial object because there is no largest ordinal. Consequently every representable functor from it to sets preserves all small limits but has no left adjoint, since a left adjoint would have to send the empty set to an initial object.

Put $\theta =(2^{\mu }{)}^{+}$ for an infinite cardinal $\mu$. Choose $M\prec H_{\chi }$ of size $2^{\mu }$, containing all ordinals below ${\mu }^{+}$, and closed under externally given sequences of length at most $\mu$. Such a model is built by ${\mu }^{+}$ successive Skolem hull closures, since $(2^{\mu }{)}^{\mu }=2^{\mu }$. For a coloring of $[\theta {]}^{r+1}$ with at most $\mu$ colors in $M$, put $\beta =\sup (M\cap \theta )<\theta$. Recursively choose $x_{\alpha }\in M\cap \theta$ for $\alpha <{\mu }^{+}$, above all earlier points, so that every earlier $r$-tuple has the same color with $x_{\alpha }$ as with $\beta$. There are at most $\mu$ constraints, their code belongs to $M$ by closure, and $\beta$ witnesses their consistency. The elementary embedding supplies the required witness in $M$. The resulting sequence is end-homogeneous, providing the induction step in the Erdős-Rado theorem for finite arities.

If a forcing is $\kappa$-closed in the ground model, then it adds no functions from a ground ordinal $\alpha <\kappa$ into a ground set $B$. Recursively decide each value inside the ground model and take common stronger bounds at limit stages and after the final step. Conditions deciding a whole ground function are dense below any condition asserting this type of function. The dense-below generic meeting lemma ensures the generic filter meets that dense subset of a forcing order. One arbitrary bound need not be in the filter in an ordered set.

An unbounded class in set theory of ordinals is closed if every nonempty set-sized increasing sequence of its members has its supremum in the class. This is the proper-class analogue of a club set. The reflection theorem for definable hierarchies gives such a class for every finite collection of first-order formulas.

A level $L_{\lambda }$ is a coding level when $\lambda$ is a limit ordinal, the representation of every well-order code in $L_{\lambda }$ is below $\lambda$, and every ordinal below $\lambda$ has a well-order code in $L_{\lambda }$.

For an infinite cardinal number $\kappa$, use cardinal arithmetic with the axiom of choice. If $\lambda =2^{\kappa }$ had cofinality $\mu \le \kappa$, choose cofinal ordinals ${\alpha }_i<\lambda$, $i<\mu$. Their cardinalities $a_i$ satisfy ${\sum }_i{a}_i=\lambda$. The König theorem for cardinal numbers gives $\lambda <{\prod }_i{a}_i^{+}\le {\lambda }^{\mu }\le (2^{\kappa }{)}^{\kappa }=\lambda$, a contradiction.

A forcing is cofinality-preserving over a transitive model $M$ if every generic extension has the same cofinality for every ground ordinal. The values are compared as ordinals, which forcing preserves.

A subset $C$ of an ordered set $P$ is cofinal if every element of $P$ is at most some element of $C$. For an ordinal, a cofinal subset has the whole ordinal as its supremum.

Two ordinals are commensurable when neither is absorbed by adding the other on its left. For positive ordinals, this is equivalent to equality of their leading Cantor normal form exponent. If leading exponents differ, the lower-leading ordinal is absorbed by the higher power of $\omega$. If they agree, either sum adds the two positive leading coefficients and therefore exceeds both original ordinals. Zero is commensurable with no ordinal.

Two ordinals commute under ordinal multiplication if and only if their squares commute. One implication follows from associativity. For the converse, squaring is strictly increasing: $A<B$ implies $A^2\le BA<B^2$. If $\alpha \beta <\beta \alpha$, monotonicity givesa contradiction to commuting squares. Swap the factors to exclude the reverse inequality; zero factors cause no exception.

Finite canonical terms built from zero, ordinal sums and powers of omega describe exactly the ordinals below epsilon zero. Syntactic validity and comparison are decidable by recursion through the exponent subterms. Enumerating valid term codes transfers their order to a computable well-order on all natural numbers.

If $M$ is a transitive model of ZF, its internally computed $L_{\alpha }$ is the actual $L_{\alpha }$ for each ordinal $\alpha \in M$. At a successor stage use absoluteness of satisfaction for the same set structure, formula codes and parameters; at a limit stage take the union of the already identical earlier stages. Choice is not required.

A countable ordinal has a countable set as its underlying set.

These are the ordinals strictly below the first uncountable ordinal.