Forcing with matrices of countable elementary submodels

Abstract

We analyze the forcing notion P of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form H θ . We show that forcing with this poset adds a Kurepa tree T . Moreover, if Pc is a suborder of P containing only continuous matrices, then the Kurepa tree T is almost Souslin, i.e. the level set of any antichain in T is not stationary in ω1.

A minimal Kurepa tree with respect to club embeddings

“…Remark 3.4. The Kurepa tree constructed in [3] also satisfies the hypothesis of the last proposition. So it can be made minimal in the same way as above.…”

“…So it can be made minimal in the same way as above. It is shown in [3] that this tree has no Aronszajn subtree, so by Fact 2.6, this tree has no Aronszajn subtree after embeddings are added either.…”

Few new reals

“…We need to ensure in that case that for any I ∈ I w ξ such that ǫ = b w ξ (δ) < min(I) < δ there is some w ∈ P ξ extended by w| ξ such that w P ξ min(I) / ∈ Ċξ (δ) and with the property that w ∈ Q for some Q ∈ N ∆ * ξ+1 of minimal height such that δ Q > δ. 14 To see that this will indeed happen, and hence that no problem arises after all in this case, we start by observing that, by definition of Pξ +1condition, there is some w weaker than u * |ξ, forcing min(I) / ∈ Ċ ξ(δ),…”

“…For each ξ we can pick N ξ to be a sufficiently correct countable model 4 A slightly enhanced form of the notion of T -symmetric system is defined in Section 2. 5 See also [14]. 6 Incidentally, P 0 is in fact strongly proper, and so each new real it adds is in fact contained in an extension of V by some Cohen real.…”

“…P 0 has received some attention in the literature. For example, Todorčević proved that P 0 adds a Kurepa tree (s. [14]). Also, [14] presents a mild variant of P 0 which not only preserves CH but actually forces ♦.…”