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• ccc forcing. Preserves all cardinals. Preserves stationary subsets of $\omega_1$. Closed under finite support iterations.
• countably closed forcing. Preserves $\omega_1$. Preserves stationary subsets of $\omega_1$. Closed under countable support iterations.
And what was wanted was a class of forcing notions in which these two kinds of forcing could be mixed together in iterations, while still preserving $\omega_1$. That is, what is wanted is a class of forcing that contains all ccc forcing, all countably closed forcing, which all preserve $\omega_1$ and which support some kind of iteration theorem. The full class of all $\omega-1$-preserving \omega_1$-preserving forcing does not fit the bill, since it is not closed under iterations. But meanwhile, the class of proper forcing does have the desired features... 1 My understanding is that part of the early motivation was the observation of the attractive features of two main classes of forcing: • ccc forcing. Preserves all cardinals. Preserves stationary subsets of$\omega_1$. Closed under finite support iterations. • countably closed forcing. Preserves$\omega_1$. Preserves stationary subsets of$\omega_1$. Closed under countable support iterations. And what was wanted was a class of forcing notions in which these two kinds of forcing could be mixed together in iterations, while still preserving$\omega_1$. That is, what is wanted is a class of forcing that contains all ccc forcing, all countably closed forcing, which all preserve$\omega_1$and which support some kind of iteration theorem. The full class of all$\omega-1\$-preserving forcing does not fit the bill, since it is not closed under iterations. But meanwhile, the class of proper forcing does have the desired features...