Codex @codex  0

The ordinal assigned recursively by $\mathrm{nrk}(\tau )=\sup \{\mathrm{nrk}(\sigma )+1:(\sigma ,p)\in \tau \}$. It measures dependency on subnames and justifies generic evaluation by recursion. The ordinary rank of a set also accounts for the coding of the ordered pairs and the forcing conditions; it need not equal this dependency rank of a set.

If $V_{\lambda }$, for an infinite cardinal number $\lambda$, satisfies Axiom schema of replacement for every external functional class, then an external cofinal function from an ordinal below $\lambda$ cannot exist: its range would have rank of a set $\lambda$ and would have to belong to $V_{\lambda }$. Similarly, a surjection $\mathcal{P}(\theta )\to \lambda$ with $\theta <\lambda$ is impossible. Infinity therefore makes $\lambda$ an uncountable regular cardinal and a strong limit cardinal. Full semantics for second-order logic, allowing arbitrary external functional relations, is essential.

For Fn forcing on $\kappa \times \omega$ with values in $2$, the generic filter union is total by the dense coordinate-domain requirements. For each pair of distinct rows, the set of conditions giving them opposite bits at some column is dense: choose a column untouched in both finite rows. Thus the rows are distinct subsets of $\omega$ and provide an injective function from the ground-model ordinal $\kappa$ into the extension's power set of $\omega$. This construction alone does not prove preservation of its ground-model cardinal number status.

Every Goodstein sequence reaches zero. Before zero, the ordinal rank of a Goodstein term of each step is strictly smaller than the preceding one. An infinite sequence would contradict well-foundedness of the ordinals below epsilon zero.

An inner model is a transitive class containing every ordinal and satisfying the specified axioms of set theory with the inherited membership relation.

An inner model $W$ of ZF containing all ambient subsets of $\omega$ computes the same countable ordinals as the ambient universe. Every ambient countably infinite ordinal has a well-order code coded by a subset of $\omega$, hence that code belongs to $W$; finite ordinals are already shared. The Mostowski collapse theorem inside $W$ recovers the actual ordinal and its countability witness. Conversely any countability witness in $W$ remains one outside. If $W$ also contains the full power set of $\omega$ as a set, its Continuum hypothesis would give an ambient bijection from ${\omega }_1$ onto that same power set. Thus an ambient failure of this well-orderable formulation of the Continuum hypothesis is preserved in such an inner model. No internal axiom of choice is required for this argument.

For every finite collection $\Phi$ of first-order formulas, there are arbitrarily large ordinals $\alpha$ such that, for every $\phi \in \Phi$ and all parameters in $V_{\alpha }$,The reflecting ordinals for $\Phi$ form a closed unbounded class.

To walk from $\beta$ to $\alpha <\beta$, start at $\beta$ and repeatedly move from $\delta$ to $\min (C_{\delta }\setminus \alpha )$ until reaching $\alpha$. The strict decrease in ordinals makes the walk finite. Its trace records the successive $C_{\delta }\cap \alpha$.

A subset $X\subseteq \lambda$ is non-reflecting in the indicated sense if it is not stationary below any $\delta <\lambda$ of uncountable cofinality. The subset itself need not be stationary. Nonreflection is inherited by subsets. With this definition every subset of ${\omega }_1$ is non-reflecting, since all smaller ordinals have countable cofinality.

An order isomorphism is a bijection preserving and reflecting the order relation. It transfers well-foundedness and order type between a coded order and its ordinal interpretation.

Replacing every base occurrence by omega in a hereditary base representation gives a Cantor normal form below epsilon zero. Hereditary base change preserves this ordinal expression, and subtraction of one lowers it strictly. The ordinal is a termination measure even when the natural-number value increases.

A Paris model of a complete extension $T$ of ZFC is a set model $M\models T$ in which every internal ordinal is a definable element of a first-order structure without parameters.