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Good morning

I try to understand the proof in Jech Lemma 23.1. Does $S_\alpha$ is a set or a subset of $\alpha$ ? I really don't "see" the core of the proof. Can somebody help me ?

Thanks a lot.

Regards.

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    You could add some more to the question.2012-03-26

1 Answers 1

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Each $S_\alpha$ is a countable set consisting of subsets of $\alpha$. The difference between $\lozenge$ and the combinatorial principle in the lemma is that for each $\alpha<\omega_1$, $\lozenge$ guesses a single subset of $\alpha$ while the principle in the lemma guesses countably many subsets.

In the $\lozenge$ case, for each $X\subseteq\omega_1$, the sequence stationarily often guesses $X\cap\alpha$ correctly. With the other principle, stationarily often one of the countably many guesses of $X\cap\alpha$ is correct. So on the surface, $\lozenge$ is stronger.
However, it turns out that the two are actually equivalent.

It is clear that the new principle follows from $\lozenge$, which gives one guess for each $\alpha$ while the new principle allows countably many.

Given a sequence $(S_\alpha)_{\alpha\lt\omega_1}$ witnessing the principle in the lemma, we replace $\omega_1$ by $\omega\times\omega_1$ using some coding function $f:\omega_1\to\omega\times\omega_1$. Now each $S_\alpha$ gives countably many guesses $a_\alpha^n$, $n\in\omega$, for subsets of $\omega\times\omega_1$. Now we construct countably many sequences that are candidates for $\lozenge$ sequences.
Namely, $S_\alpha^n$ is the $n$-th section of the set $a_\alpha^n$. Note that each $S_\alpha^n$ is now a (single) subset of $\alpha$.

The proof now proceeds by showing that we get a contradiction if none of the sequences $(S_\alpha^n)_{\alpha\lt\omega_1}$ is a $\lozenge$-sequence. This depends on the fact that if the union of countably many sets is stationary, then one of the sets if stationary, which follows from the fact that the intersection of countably many clubs in $\omega_1$ is again club.

The real use of this lemma is that it provides a version of $\lozenge$ that can be strengthened by replacing "stationary" by "club". The usual $\lozenge$ becomes contradictory when we demand that the guessing is correct on club many places.