I'm studying the Levy collapse: For S ⊆ On and λ regular, Col(λ, S) is defined in the following way:
Col(λ, S) = {p | p is a function ∧ |p| < λ ∧ dom(p) ⊆ S × λ ∧ ∀ <α, ξ > ∈ dom(p)(p(α, ξ ) = 0 ∨ p(α, ξ ) ∈ α)} ordered by p ≤ q iff p ⊇ q.
I have a question concerning Kanamori, The Higher Infinite, Lemma 10.17 (c): If κ is regular, κ>λ, and either κ is inaccessible or λ = ω, then Col(λ, κ) has the κ-c.c.
I understand that we should apply the ∆-system lemma but I have troubles to write down a proper proof. Could anyone help me to formulate the proof?
Thank you very much for your help in advance!
The case $\lambda=\omega$ follows from a typical $\Delta$-system lemma argument: Suppose that $\{p_\xi \mid \xi<\kappa\}$ is an antichain over $\operatorname{Col}(\omega, S)$. Then consider the set $$\{\operatorname{dom} p_\xi \mid \xi<\kappa\}$$ and apply the $\Delta$-system lemma to this set to get index set $I\subseteq\kappa$ of size $\kappa$ such that $\{\operatorname{dom} p_\xi \mid \xi\in I\}$ forms a $\Delta$-system of root $r$.
If $p_\xi$ and $p_\eta$ are incompatible for $\xi,\eta\in I$, then $p_\xi\upharpoonright r$ and $p_\eta\upharpoonright r$ must be different. But you can see that there are at most less than $\kappa$ many possibilities for $p_\xi\upharpoonright r$ since $r$ is finite.
The case $\kappa$ inaccessible and $\lambda<\kappa$ regular is also similar, but you would need a different form of the $\Delta$-system lemma (that for a family of infinite sets.)