A COMMENT ON “p < t”
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A COMMENT ON " p < t " SAHARON SHELAH Abstract. Dealing with the cardinal invariants p and t of the continuum we prove that m ≥ p = ℵ 2 ⇒ t = ℵ 2. In other words, if MA ℵ 1 (or a weak version of this) holds, then (of course ℵ 2 ≤ p ≤ t and) p = ℵ 2 ⇒ p = t. The proof is based on a criterion for p < t. 0. Introduction We are interested in two cardinal invariants of the continuum, p and t. The cardinal p measures when a family of infinite subsets of ω with finite intersection property has a pseudo–intersection. A relative is t, which deals with towers, i.e., families well ordered by almost inclusion. These are closely related classical cardinal invariants. Rothberger [7], [8] proved (stated in our terminology) that p ≤ t and p = ℵ 1 ⇒ p = t, and he asked if p = t. Our main result is Corollary 2.5 stating that m ≥ p = ℵ 2 ⇒ p = t, where m is the minimal cardinal λ such that Martin Axiom for λ dense sets fails (i.e. ¬MA λ). Considering that m ≥ ℵ 1 is a theorem (of ZFC), the parallelism with Rothberger's theorem is clear. The reader may conclude that probably m = p ⇒ p = t; this is not unreasonable but we believe that eventually one should be able to show CON(m = λ + p = λ + t = λ +). In the first section we present a characterization of p < t which is crucial for the proof of 2.5, and which also sheds some light on the strategy to approach the question of p < t presented in [9]. We thank Andreas Blass for detailed comments on an earlier version and David Fremlin for historical information. Notation Our notation is rather standard and compatible with that of classical textbooks (like Bartoszy´nski and Judach [3]). In forcing we keep the older convention that a stronger condition is the larger one. (1) Ordinal numbers will be denoted be the lower case initial letters of the Greek alphabet (α, β, γ, δ. . .) and also by i, j (with possible sub-and superscripts). (2) Cardinal numbers will be called κ, κ i , λ; λ will be the value of p.
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