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Properness without elementaricity
, 2003
"... We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(χ), ∈). This leads to forcing notions which are “reasonably ” definable. We present two specific properties materializing ..."
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Cited by 16 (9 self)
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We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(χ), ∈). This leads to forcing notions which are “reasonably ” definable. We present two specific properties materializing this intuition: nep (nonelementary properness) and snep (Souslin nonelementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is “preservation by iteration”, but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.
A PARALLEL TO THE NULL IDEAL FOR INACCESSIBLE λ
, 2012
"... Abstract. It is well known to generalize the meagre ideal replacing ℵ0 by a (regular) cardinal λ> ℵ0 and requiring the ideal to be λ +complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ0 by λ, so ..."
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Abstract. It is well known to generalize the meagre ideal replacing ℵ0 by a (regular) cardinal λ> ℵ0 and requiring the ideal to be λ +complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ0 by λ, so requiring it to be (< λ)complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of λBorel sets modulo the ideal. Surprisingly we get a positive = existence answer for λ a “mild ” large cardinal: the weakly compact one. We apply this to get consistency results on cardinal invariants for such λ’s. We shall deal with other cardinals more properties related forcing notions in a continuation. (1004) revision:20120524 modified:20120528
SWEET & SOUR AND OTHER FLAVOURS OF CCC FORCING NOTIONS
, 2001
"... We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them. ..."
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We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.