Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-11T16:31:30.249Z Has data issue: false hasContentIssue false
  • English
  • Français

A Comment on “ p < t ”

Published online by Cambridge University Press:  20 November 2018

Saharon Shelah
Saharon Shelah*
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA e-mail: shelah@math.huji.ac.il
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 $\text{M}{{\text{A}}_{{{\aleph }_{1}}}}$ (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 $\mathfrak{p}\,<\,\mathfrak{t}$.

Keywords

03E1703E0503E50
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 2 , 01 June 2009 , pp. 303 - 314
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Abraham, U. and Shelah, S., Lusin sequences under CH and underMartin's Axiom. Fund. Math. 169(2001), no. 2, 97–103.Google Scholar
[2] Abraham, U. and Shelah, S., Ladder gaps over stationary sets. J. Symbolic Logic 69(2004), no. 2, 518–532.Google Scholar
[3] Bartoszyński, T. and Judah, H., Set Theory. On the structure of the real line. A K Peters, Wellesley, MA, 1995.Google Scholar
[4] Bell, M. G., On the combinatorial principle P(c) . Fund. Math. 114(1981), no. 2, 149–157.Google Scholar
[5] Fremlin, D. H., Consequences of Martin's Axiom. Cambridge Tracts in Mathematics 84, Cambridge University Press, Cambridge, MA, 1984.Google Scholar
[6] Piotrowski, Z. and Szymański, A., Some remarks on category in topological spaces. Proc. Amer. Math. Soc. 101(1987), no. 1, 156–160.Google Scholar
[7] Rothberger, F., Sur un ensemble toujours de première catégorie qui est dépourvu de la propriété λ . Fund. Math. 32(1939), 294–300.Google Scholar
[8] Rothberger, F., On some problems of Hausdorff and Sierpiński. Fund. Math. 35(1948), 29–46.Google Scholar
[9] Shelah, S.. Large continuum, oracles.arXiv:LO.0707.1818.Google Scholar
[10] Todorčević, S. and Veličković, B., Martin's axiom and partitions. Compositio Mathematica 63(1987), 391–408.Google Scholar