Perfect set theorems for in the universe without choice
Perfect sets and collapsing continuum
Under Martin’s axiom, collapsing of the continuum by Sacks forcing is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if proving that under this hypothesis there are no small uncountable maximal antichains in . We also construct a partition of into perfect sets which is a maximal antichain in and show that -sets are exactly (subsets of) selectors of maximal antichains of perfect sets.
Perfect sets of independent functions
Periodic extensive measurement
Permitted trigonometric thin sets and infinite combinatorics
We investigate properties of permitted trigonometric thin sets and construct uncountable permitted sets under some set-theoretical assumptions.
Permutation models and topological groups
Pexider functional equations -- their fuzzy analogs.
Pinning countable ordinals
Pinning for pairs of countable ordinals
Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal
By an - tree we mean a tree of power and height . Under CH and we call an -tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between and . In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus that there only exist Kurepa trees with -many branches, which answers another...
Pointwise convergence and the Wadge hierarchy
We show that if is a separable metrizable space which is not -compact then , the space of bounded real-valued continuous functions on with the topology of pointwise convergence, is Borel--complete. Assuming projective determinacy we show that if is projective not -compact and is least such that is then , the space of real-valued continuous functions on with the topology of pointwise convergence, is Borel--complete. We also prove a simultaneous improvement of theorems of Christensen...
Polyadic spaces of arbitrary compactness numbers
Positive functionals and the axiom of choice
Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers
If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. The cardinal invariant A(G) is defined to be the least uncountable cardinal in the abelian subgroup spectrum of G. The value of A(G) is examined for various groups G which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the...
Possibly every real function is continuous on a non-meagre set.
Possibly there is no uniformly completely Ramsey null set of size
We show that under the axiom there is no uniformly completely Ramsey null set of size . In particular, this holds in the iterated perfect set model. This answers a question of U. Darji.
Postscript to "Built-up systems of fundamental sequences and hierarchies of number-theoretic functions".
Potential isomorphism and semi-proper trees
We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. ...
Powers and generalized cardinal numbers for HCH-objects --- basic notions.
Predicate calculus and naive set theory in pure combinatory logic.