205 research outputs found
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube
We compare the forcing related properties of a complete Boolean algebra B
with the properties of the convergences (the algebraic convergence)
and on B generalizing the convergence on the Cantor and
Aleksandrov cube respectively. In particular we show that is a
topological convergence iff forcing by B does not produce new reals and that
is weakly topological if B satisfies condition
(implied by the -cc). On the other hand, if is a
weakly topological convergence, then B is a -cc algebra or in
some generic extension the distributivity number of the ground model is greater
than or equal to the tower number of the extension. So, the statement "The
convergence on the collapsing algebra B=\ro
((\omega_2)^{<\omega}) is weakly topological" is independent of ZFC
Universality properties of forcing
The purpose of this paper is to investigate forcing as a tool to construct
universal models. In particular, we look at theories of initial segments of the
universe and show that any model of a sufficiently rich fragment of those
theories can be embedded into a model constructed by forcing. Our results rely
on the model-theoretic properties of good ultrafilters, for which we provide a
new existence proof on non-necessarily complete Boolean algebras
A Note on Direct Products, Subreducts and Subvarieties of PBZ*--lattices
PBZ*--lattices are bounded lattice--ordered structures arising in the study
of quantum logics, which include orthomodular lattices, as well as
antiortholattices. Antiortholattices turn out not only to be directly
irreducible, but also to have directly irreducible lattice reducts. Their
presence in varieties of PBZ*--lattices determines the lengths of the subposets
of dense elements of the members of those varieties. The variety they generate
includes two disjoint infinite ascending chains of subvarieties, and the
lattice of subvarieties of the variety of pseudo--Kleene algebras can be
embedded as a poset in the lattice of subvarieties of its subvariety formed of
its members that satisfy the Strong De Morgan condition. We obtain
axiomatizations for all members of a complete sublattice of the lattice of
subvarieties of this latter variety axiomatized by the Strong De Morgan
identity with respect to the variety generated by antiortholattices.Comment: 18 page
Absoluteness via Resurrection
The resurrection axioms are forcing axioms introduced recently by Hamkins and
Johnstone, developing on ideas of Chalons and Velickovi\'c. We introduce a
stronger form of resurrection axioms (the \emph{iterated} resurrection axioms
for a class of forcings and a given
ordinal ), and show that implies generic
absoluteness for the first-order theory of with respect to
forcings in preserving the axiom, where is a
cardinal which depends on ( if is any
among the classes of countably closed, proper, semiproper, stationary set
preserving forcings).
We also prove that the consistency strength of these axioms is below that of
a Mahlo cardinal for most forcing classes, and below that of a stationary limit
of supercompact cardinals for the class of stationary set preserving posets.
Moreover we outline that simultaneous generic absoluteness for
with respect to and for with respect to
with is in principle
possible, and we present several natural models of the Morse Kelley set theory
where this phenomenon occurs (even for all simultaneously). Finally,
we compare the iterated resurrection axioms (and the generic absoluteness
results we can draw from them) with a variety of other forcing axioms, and also
with the generic absoluteness results by Woodin and the second author.Comment: 34 page
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