Set Mapping Reflection

Could it be that the above questions have resisted solution because we haven’t tapped into the full strength of “proper?” Let us focus on the second question for a moment. Consider the following analogy. Recall that a forcing Q satisfies the countable chain condition (c.c.c.) if every uncountable collection of conditions in Q contains two compatible conditions. Similarly, a forcing Q satisfies Knaster’s Condition (Property K) if every uncountable collection of conditions contains an uncountable subcollection of pairwise compatible conditions. It is easily verified that the product of a c.c.c. forcing and one with Property K is c.c.c.. A consequence of this is that a Property K forcing cannot destroy a counterexample to Souslin’s Hypothesis. Hence while the forcing axiom for c.c.c. forcings (a.k.a. MAא1) does imply Souslin’s Hypothesis, the forcing axiom for Property K forcings is consistent with the failure of Souslin’s Hypothesis. What if the common and widely successful methods for building proper forcings inadvertently satisfied a stronger form of properness and that counterexamples to Question 1.2 were preserved by this stronger condition? It turns out that this is indeed the case and we will now formulate a combinatorial obstruction of this sort. A f-sequence is a sequence 〈fα : α < ω1〉 of continuous functions fα : α → ω such that if E ⊆ ω1 is closed and unbounded, there is a δ such that fδ takes all values in ω on E ∩ δ. Notice that if f : δ → ω is continuous and δ is a limit ordinal, then there is a cofinal C ⊆ δ of ordertype ω such that f(ξ) depends only on |C∩ξ|. That is f is obtained by coloring the intervals in δ between points of C. Jensen’s principle ♦ easily implies the existence of a f-sequence. Since only the club filter is quantified over in the definition of a f-sequence, f-sequences are preserved by c.c.c. forcing. This is because if E is a club in a c.c.c. forcing extension, E contains a club from the ground model (this appears as an exercise in [9]). In fact a much broader class of proper forcings preserve f-sequences; this will be discussed more later. In [11] it was shown that the existence of a f-sequence implies the existence of an Aronszajn line with no Countryman suborder.