Adding a lot of Cohen reals by adding a few

The purpose of the paper is to produce models V_1 \subset V_2 such that adding kappa-many Cohen reals to V_2 adds lambda Cohen reals to V_1. Some of the results: 1. Suppose that V satisfies GCH, kappa = \cup kappa_n= \cup o(kappa_n). Then there is a cardinal preserving generic extension V_1 of V satisfying GCH and having the same reals as V does , so that adding kappa many Cohen reals over V_1 produces kappa^+ Cohen reals over V. 2. Suppose that V is a model of GCH. Then there is a cofinality preserving extension V_1 satisfying GCH so that adding a Cohen real to V_1 produces aleph_1 Cohen reals over V. 3. There is a pair (W,W_1) of generic cofinality preserving etensions of L such that W is contained in W_1 and W_1 contains a perfect set of W-reals which is not in W. The last statement is a slight improvement of a result of B.Velickovic and H.Woodin on the Prikry problem

On hidden extenders

A model with a sequence of indiscernibles depending on a particular precovering set is constructed.The initial assumption is as follows: for every n<omega the set {alpha | o(alpha)=alpha^+n } is unbounded in kappa

Blowing up the power of a singular cardinal

Suppose that kappa is a singular cardinal of cofinality omega and GCH holds. Assume that for every n= alpha^{+n} is unbounded in kappa.Then there is a cardinal preserving extension satisfying 2^kappa=kappa^++ and GCH below kappa. By a result of W. Mitchell and the author the assumptions are optimal

Weak Width of Subgroups

We say that the weak width of an infinite subgroup $H$ of $G$ in $G$ is $n$ if there exists a collection of $n$ strongly essentially distinct conjugates $\{ H, g_1^{-1} H g_1,\cdots, g_{n-1}^{-1} H g_{n-1} \}$ of $H$ in $G$ such that the intersection $H \cap g_i^{-1} H g_i$ is infinite for all $1 \leq i \leq n-1$ and $n$ is maximal possible. We prove that a quasiconvex subgroup of a negatively curved group has finite weak width in the ambient group. We also give examples demonstrating that height, width, and weak width are different invariants of a subgroup

A Proof of Simon's Conjecture

We prove Simon's conjecture for 3-manifolds

Some results on nonstationry ideal 2

This is a continuation of "Some results on nonstationry ideal". The upper bound on precipitousness of NS_lambda^+ for a regular lambda given in this paper is proved to be exact.It is shown that saturatedness of NS_kappa^aleph_0 over inaccessible kappa requires at least o(kappa)=kappa^++.The upper bounds on the strength of NS_kappa precipitous for inaccessible kappa are reduced quite close to the lower bounds

Cardinal preserving ideals

We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength ``NS_lambda is aleph_1-preserving'', for lambda > aleph_2

Wide gaps with short extenders

Let kappa be the limit of (1) if each kappa_n carries an extender of the length of the first Mahlo above kappa_n, then for every ld above kappa there is a generic extension with power of kappa above ld. (2) if each kappa_n carries an extender of the length of the first fixed point of the aleph function above kappa_n of order n then for every ld between kappa and the first inaccessible above kappa there is a generic extension satisfying 2^kappa>ld.Comment: 15 pages, LaTe

The power set function

We survey old and recent results on the problem of finding a complete set of rules describing the behavior of the power function, i.e. the function which takes a cardinal $\kappa$ to the cardinality of its power $2^\kappa$

No bound for the first fixed point

Our aim is to show that it is impossible to find a bound for the power of the first fixed point of the aleph function.Comment: 63 pages, LaTe