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I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree.

Question 1:

What kinds of forcings add cofinal branches to $\omega_1$-trees? What kinds of forcings cannot?

The answer may depend on the type of tree. I don't expect a full characterization of such forcings, but would like to know what experience anyone has with forcing branches.

Question 2:

Is the following statement correct? If $\mathbb{P}$ is a notion of forcing which has the countable chain condition and $\mathbb{P} \times \mathbb{P}$ also has the countable chain condition, then $\mathbb{P}$ cannot force a cofinal branch in an $\omega_1$-tree.

If the statement is correct, what is the proof? A non-example is that a Souslin poset is ccc, but the product of two Souslin posets is $\it{not}$ ccc (since the set of pairs of immediate sucessors of each node in the tree gives an uncountable antichain) and a Souslin poset $\it{can}$ add a cofinal branch to a Souslin tree.

Question 3:

Sometimes (as in the case of a special Aronszajn tree) a notion of forcing which satisfies the countable chain condition cannot add a branch to an $\omega_1$-tree because any forcing which adds a cofinal branch to the tree will collapse $\omega_1$ and ccc posets preserve $\omega_1$.

Does anyone have any experience with an $\omega_1$-tree such that no ccc forcing can add a branch, but the reason is not because adding a cofinal branch will collapse $\omega_1$?

These questions are related to and arose because of Joel Hamkins' open question on mathoverflow: Can there be an almost-special not-fully-special Aronszajn tree?

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# Which notions of forcing add a cofinal branch to an $\omega_1$-tree?

I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree.

Question 1:

What kinds of forcings add cofinal branches to $\omega_1$-trees? What kinds of forcings cannot?

The answer may depend on the type of tree. I don't expect a full characterization of such forcings, but would like to know what experience anyone has with forcing branches.

Question 2:

Is the following statement correct? If $\mathbb{P}$ is a notion of forcing which has the countable chain condition and $\mathbb{P} \times \mathbb{P}$ also has the countable chain condition, then $\mathbb{P}$ cannot force a cofinal branch in an $\omega_1$-tree.

If the statement is correct, what is the proof? A non-example is that a Souslin poset is ccc, but the product of two Souslin posets is $\it{not}$ ccc (since the set of pairs of immediate sucessors of each node in the tree gives an uncountable antichain) and a Souslin poset $\it{can}$ add a cofinal branch to a Souslin tree.

Question 3:

Sometimes (as in the case of a special Aronszajn tree) a notion of forcing which satisfies the countable chain condition cannot add a branch to an $\omega_1$-tree because any forcing which adds a cofinal branch to the tree will collapse $\omega_1$ and ccc posets preserve $\omega_1$.

Does anyone have any experience with an $\omega_1$-tree such that no ccc forcing can add a branch, but the reason is not because adding a cofinal branch will collapse $\omega_1$?

These questions are related to and arose because of Joel Hamkins' open question on mathoverflow: Can there be an almost-special not-fully-special Aronszajn tree?