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COMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 2 / June 2018
- Published online by Cambridge University Press:
- 01 August 2018, pp. 790-803
- Print publication:
- June 2018
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- Article
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Assuming three strongly compact cardinals, it is consistent that
$${\aleph _1} < add\left( {\cal N} \right) < cov\left( {\cal N} \right) < \mathfrakb < \mathfrakd < non\left( {\cal N} \right) < cof\left( {\cal N} \right) < {2^{{\aleph _0}}}.$$Under the same assumption, it is consistent that
$${\aleph _1} < add\left( {\cal N} \right) < cov\left( {\cal N} \right) < non\left( {\cal M} \right) < cov\left( {\cal M} \right) < non\left( {\cal N} \right) < cof\left( {\cal N} \right) < {2^{{\aleph _0}}}.$$
MAD SPECTRA
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- Journal:
- The Journal of Symbolic Logic / Volume 80 / Issue 3 / September 2015
- Published online by Cambridge University Press:
- 22 July 2015, pp. 901-916
- Print publication:
- September 2015
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- Article
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The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets
${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption 
$\vartheta ^{ < \vartheta } = \vartheta $, where 
$\vartheta = {\rm{min}}\left( {\rm {\cal A}} \right) $.
