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# A Comment on $\mathfrak{p} < \mathfrak{t}$''

Published:2009-06-01
Printed: Jun 2009
• Saharon Shelah
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## Abstract

Dealing with the cardinal invariants ${\mathfrak p}$ and ${\mathfrak t}$ of the continuum, we prove that ${\mathfrak m}={\mathfrak p} = \aleph_2\ \Rightarrow\ {\mathfrak t} =\aleph_2$. In other words, if ${\bf MA}_{\aleph_1}$ (or a weak version of this) holds, then (of course $\aleph_2\le {\mathfrak p}\le {\mathfrak t}$ and) ${\mathfrak p}=\aleph_2\ \Rightarrow\ {\mathfrak p}={\mathfrak t}$. The proof is based on a criterion for ${\mathfrak p}<{\mathfrak t}$.
 MSC Classifications: 03E17 - Cardinal characteristics of the continuum 03E05 - Other combinatorial set theory 03E50 - Continuum hypothesis and Martin's axiom [See also 03E57]