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Post Made Community Wiki by Stefan Kohl♦
fixed typo
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Joel David Hamkins
Joel David Hamkins
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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$$\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...

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...

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...

Source Link
Joel David Hamkins
Joel David Hamkins
  • 236.3k
  • 44
  • 777
  • 1.4k

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...