Let $A\subseteq\operatorname{Fn}(I,J)$ be uncountable. Apply the <Delta-system lemma> to the finite sets $\operatorname{dom}p$ for $p\in A$. After passing to an uncountable subset $A'$, there is a fixed finite root $R$ such that
$$
\operatorname{dom}p\cap\operatorname{dom}q=R
$$
for distinct $p,q\in A'$. Because $J$ is countable and $R$ is finite, there are only countably many functions $R\to J$. A further uncountable subset $A''\subseteq A'$ therefore has the same restriction to $R$.

Any two conditions in $A''$ agree on the intersection of their domains, so their union is a common stronger condition. Thus every uncountable family contains two compatible conditions, and no uncountable antichain exists. Therefore
$$
\boxed{\operatorname{Fn}(I,J)\text{ has the countable chain condition}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 2 / b / Solution

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Every uncountable family of finite sets has an uncountable subfamily whose distinct members have the same pairwise intersection, called the root. It is also known as the sunflower lemma.


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