We show that in the model obtained by iteratively pseudo-intersecting a Ramsey ultrafilter via a length- ω 2 countable support iteration of restricted Mathias forcing over a ground model satisfying CH , there is a unique Q -point up to isomorphism. In particular, it is consistent that there is only one Q -point while there are 2 c -many near-coherence classes of ultrafilters.

Abstract For every natural number n we introduce a new weak choice principle n upper R upper C Subscript f i n $\mathrm {nRC_{fin}}$ n R C f i n : Given any infinite set x , there is an infinite subset y subset of or equal to x $y\subseteq x$ y ⊆ x and a selection function f that chooses an n-element subset from every finite z subset of or equal to y $z\subseteq y$ z ⊆ y containing at least n elements. By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of n upper R upper C Subscript f i n $\mathrm {nRC_{fin}}$ n R C f i n to the weak choice principles upper R upper C Subscript m $\mathrm {RC_m}$ R C m (that has already been studied in [3] and [6]): Given any infinite set x , there is an infinite subset y subset of or equal to x $y\subseteq x$ y ⊆ x with a choice function f on the family of all m-element subsets of y . Moreover, we prove a stronger analogue of the results in [6] when we study the relation between

For a set $M$, $\operatorname{fin}(M)$ denotes the set of all finite subsets of $M$, $M^2$ denotes the Cartesian product $M\times M$, $[M]^2$ denotes the set of all $2$-element subsets of $M$, and $\operatorname{seq}^{1-1}(M)$ denotes the set of all finite sequences without repetition which can be formed with elements of $M$. Furthermore, for a set $S$, let $|S|$ denote the cardinality of $S$. Under the assumption that the four cardinalities $|[M]^2|$, $|M^2|$, $|\operatorname{fin}(M)|$, $|\operatorname{seq}^{1-1}(M)|$ are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF.

Abstract In this article we give a forcing characterization for the Ramsey property of -Sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.

Abstract In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals and , either ≤ or ≤ . However, in ZF this is no longer so. For a given infinite set A consider seq1-1(A), the set of all sequences of A without repetition. We compare |seq1-1(A)|, the cardinality of this set, to | |, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF ⊢ ∀A(| seq1-1(A)| ≠ | |), and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then | fin(B)| < | (B*)| even though the existence for some infinite set B* of a function ƒ from fin(B*) onto (B*) is consistent with ZF.