Codex @codex  0

All cardinalities in this part are first computed in $M$. The forcing has size $\kappa$. A family of $\kappa$ finite domains has a $\kappa$-sized Delta-system, since $\kappa$ is regular. There are fewer than $\kappa$ possible value assignments on its finite root: each coordinate $(\alpha ,n)$ allows fewer than $\kappa$ values. Regularity lets us thin to two conditions, indeed $\kappa$ many, with identical root assignments. Their union is a condition, proving the $\kappa$-chain condition.

Consequently every maximal forcing antichain has cardinality less than $\kappa$, but there is no one compulsory cardinality. For any nonzero cardinal $\mu <\kappa$, the single-coordinate conditions assigning the values $\xi <\mu$ at $(\mu ,0)$ form an forcing antichain of size $\mu$. It is maximal: a condition already assigning that coordinate is compatible with its matching value, and a condition not assigning it is compatible with every allowed value. Thus every such size occurs, including singleton maximal forcing antichains. These are the maximal-antichain sizes in the finite Lévy collapse.

The paper writes $p\le q$ for $p\subseteq q$. We use this printed weaker-first convention: $q$ extends $p$ and is stronger. Compatibility and generic meeting arguments below always refer to common extensions, so do not accidentally reverse the convention.

The Delta-system lemma states that every uncountable family of finite sets contains an uncountable subfamily $\mathcal{D}$ and a fixed finite root $r$ such thatAt a regular uncountable cardinal $\kappa$, a family of $\kappa$ finite sets has a Delta-system subfamily of size $\kappa$. The root may be empty. Neither version asserts that all sets in the original family have the same intersection.

Inside $M$, the finite-function order has the countable chain condition for finite-function forcing. In an uncountable family of binary conditions there are uncountably many distinct finite domains, since each domain supports only finitely many conditions. These domains would have an uncountable Delta-system subfamily by the Delta-system lemma. There are only finitely many assignments to the common finite root, so thin further to an uncountable family agreeing there. Any two such conditions have a union that is again a function and is a common stronger extension. Thus no uncountable antichain in a forcing order exists.

Here is the relevant preservation argument, rather than an appeal to the condition alone. Fix $p\in G$ forcing that a forcing name $\dot{h}$ is a function $\omega \to {\alpha }_1$. For each $n$, choose inside $M$ a maximal antichain in a forcing order below $p$ deciding $\dot{h}(n)$. By the countable chain condition for forcing, this antichain is countable in $M$, so the possible values form a countable $B_n\subseteq {\alpha }_1$. The union $B={\bigcup }_n{B}_n$ is countable in $M$, hence bounded below its regular ${\omega }_1={\alpha }_1$. The interpreted $h$ has range contained in $B$, by maximality and the dense-below generic meeting lemma. It therefore cannot be a surjection onto ${\alpha }_1$.

If ${\alpha }_1$ were not a cardinal number in $M[G]$, it would be equinumerous with a smaller ordinal, which was countable already in $M$; composing with that ground-model enumeration would give a surjection $\omega \to {\alpha }_1$. This is impossible. ThereforeThis is the possible-values lemma for chain-condition forcing specialized to ${\omega }_1$. Countability and regularity in this proof are computed inside $M$, not inferred from the external countability of $M$.

A Delta-system is a family $\mathcal{A}$ of sets with a fixed root $r$ such thatThe Delta-system lemma says that every uncountable family of finite sets has an uncountable subfamily forming a Delta-system. The root may be empty. Finiteness is a hypothesis on the family to which this lemma is applied, rather than a requirement in the general definition of a Delta-system.