Abstract
Standard forcing axioms are usually stated in the form which asserts the existence of sufficiently generic filters in every partial order P which belongs to a given class κ of forcing notions. This approach, which is derived by “internalizing” generic extensions, has been very successful in providing strong forcing axioms and proving their consistency; in [FMS] a maximal axiom of this sort is proved consistent for the case when one wishes to consider only generic filters for families of at most N 1 dense sets. However, when applying these axioms we need to know when there is a partial order in the class κ which introduces the object we wish to find. Of course, there is no easy general answer to this question and even some of the most basic instances are still open.
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