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Isolating cardinal invariants. (English) Zbl 1025.03046

Since the Continuum Hypothesis was shown by Cohen to be unprovable in ZFC, a new subfield of set theory emerged, dealing with combinatorial cardinal characteristics of the continuum [see, e.g., A. Blass, Combinatorial cardinal characteristics of the continuum,
http://www.math.lsa.umich.edu/ablass/set.html
for a survey].
These combinatorial cardinals, sometimes (and below) called “cardinal invariants (of the continuum)” lie between \(\aleph_1\) and \({\mathfrak c}\) (inclusive), and are usually defined as the minimal cardinality of a set of reals which is large in some sense. Some inequalities are provable between these cardinals but it is more often the case that an inequality (not to mention equality) is not provable.
This paper is concerned with a delicate study of this phenomenon. In fact, this is a study of the forcing machinery used to enlarge some cardinal invariant while leaving some other cardinal invariant small.
To get a flavor of this paper, we quote some results: The author defines the notion of a “tame invariant”, which is general enough to cover most classical cardinal invariants (including \(b, d, a,\) add(meager), \(t, u\), and \(s\), but not including \(h\) and \(g\) – see Blass’ paper cited above for the definitions). Under some large cardinals hypothesis (namely, the existence of a proper class of measureable Woodin cardinals), the author proves that if \(x\) is a tame invariant and \(x< c\) holds in some forcing extension, then this happens in the iterated Sacks extension. Generally, a cardinal invariant \(y\) can be “isolated” if there exists a forcing \(P\) such that for each tame invariant \(x\) with \(x< y\) in some forcing extension, \(x< y\) holds in the \(P\) extension. Thus the Sacks forcing witnesses that \({\mathfrak c}\) can be isolated (under the mentioned large cardinals assumption). The author shows, under the same assumption, that the cardinal invariants \(b, d, h,\) cov(meager), cov(null), and non(strong measure zero) can also be isolated, and that this is witnessed by the iterated Sacks, Laver, Miller, Mathias, Cohen, Solovay, and \(PT_g\) forcing, respectively. Moreover, the author introduces a new forcing notion witnessing that add(null) can be isolated.
Some extensions and related results are also discussed. One of the more interesting ones is related to the long-lasting failure to solve the minimal tower problem (asking whether \(p< t\) is consistent). With regard to \(t\), an isolating forcing is found only for tame invariants \(x\), with \(\aleph_1=x< t\) holding in some forcing extension, which is not the case for \(x=p\).
It is left open whether the large cardinals hypothesis is necessary to obtain the above mentioned results.

MSC:

03E17 Cardinal characteristics of the continuum
03E55 Large cardinals
03E40 Other aspects of forcing and Boolean-valued models
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References:

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