Preserving P-points in definable forcing
I isolate a simple condition that is equivalent to preservation of P-points in definable proper forcing.
Proper forcings and absoluteness in
We show that in the presence of large cardinals proper forcings do not change the theory of with real and ordinal parameters and do not code any set of ordinals into the reals unless that set has already been so coded in the ground model.
Properties of forcing preserved by finite support iterations
We shall investigate some properties of forcing which are preserved by finite support iterations and which ensure that unbounded families in given partially ordered sets remain unbounded.
Properties of measure and category in generalized Cohen's and Silver's forcing
Provident sets and rudimentary set forcing
Using the theory of rudimentary recursion and provident sets expounded in [MB], we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s . Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We...
Ramseyan ultrafilters
We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...
Sacks forcing collapses to
We shall prove that Sacks algebra is nowhere -distributive, which implies that Sacks forcing collapses to .
Some recent results on Cohen algebras
Stranger things about forcing without AC
Typically, set theorists reason about forcing constructions in the context of Zermelo--Fraenkel set theory (ZFC). We show that without the axiom of choice (AC), several simple properties of forcing posets fail to hold, one of which answers Miller's question from the work: Arnold W. Miller, {Long Borel hierarchies}, MLQ Math. Log. Q. {54} (2008), no. 3, 307--322.
Strolling through paradise
Sums of Darboux and continuous functions
It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent - the model used is iterated Sacks forcing - that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of set theory there are universally bad Darboux functions, Darboux functions whose sum with any nowhere...
The distributivity numbers of finite products of P(ω)/fin
Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o., is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).
The smallest common extension of a sequence of models of ZFC
In this note, we show that the model obtained by finite support iteration of a sequence of generic extensions of models of ZFC of length is sometimes the smallest common extension of this sequence and very often it is not.
Unramified forcing preserving the law of double negation.
α-Properness and Axiom A
We show that under ZFC, for every indecomposable ordinal α < ω₁, there exists a poset which is β-proper for every β < α but not α-proper. It is also shown that a poset is forcing equivalent to a poset satisfying Axiom A if and only if it is α-proper for every α < ω₁.
Из принципа селектора для аналитических отношений эквивалентнисти не следует существование полного -упорядочения континуума
Интуиционистская теория алгебраических систем и гейтинговозначный анализ.
О комбинировании нестандартных методов. I. Монaдология.
Письмо в редакцию:Форсинг в допустимых множествах
Порядково непрерывные функционалы в булевозначных моделях теории множеств.