Related papers: A Sacks Real out of Nowhere
We develop iterated forcing constructions dual to finite support iterations in the sense that they add random reals instead of Cohen reals in limit steps. In view of useful applications we focus in particular on two-dimensional "random"…
We prove that every finite partition of $\omega$ admit an infinite subset that does not compute a Schnorr random real. We use this result to answer two questions of Brendle, Brooke-Taylor, Ng and Nies and strength a result of Khan and…
In a self-contained way, we deal with revised countable support iterated forcing for the reals. We improve theorems on preservation of the property UP, weaker than semi proper, and we hopefully improve the presentation. We continue [Sh:b,…
We present a direct construction of stationary set preserving forcings that make $\omega$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $\kappa > \omega_1$. In addition, it is made possible to ensure that…
For every uncountable regular $\kappa$, we give two examples of proper posets which turn improper in some $\kappa$-closed forcing extension.
It is established that for any finite set of positive real numbers $A$, we have $$|A/A+A| \gg \frac{|A|^{\frac{3}{2}+\frac{1}{26}}}{\log^{1/2}|A|}.$$
Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $\theta$ be an irreducible character of $N$. We count the real irreducible characters of $G$ lying over $\theta$
Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to show that O_G can never be countably infinite. This paper retrieves correct parts of the…
We study the problem of whether a betting-strategy can be decomposed into an equivalent set of simpler betting-strategies, such as betting-strategies that bet on a restricted set of stages or bet on a restricted of favorable outcomes. We…
We use ``iterated square sequences'' to show: There is an L-definable partition n: L-singulars --> omega such that if M is an inner model without 0#: (a) For some n, M satisfies that {alpha | n(alpha)=n} is stationary. (b) For each n there…
The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
We present a general framework for forcing on $\omega_2$ with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial…
In this paper we calibrate the strength of the soundness of a Kripke-Platek set theory with the axioms of Infinity and \Pi_{1}-Collection with the assumption that`there exists an uncountable regular ordinal' in terms of the existence of…
As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a…
This is an overview about a method of constructing ccc forcings: Suppose first that a continuous, commutative system of complete embeddings between countable forcings indexed along $\omega_1$ is given. Then its direct limit satisfies ccc by…
Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight aleph_1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add…
It is shown that if $f$ or $1/f$ is a real entire function of infinite order of growth, with only real zeros, then $f''+\omega f$ has infinitely many non-real zeros for any $\omega > 0$.
Assume $\text{MA}(\kappa)$. We show that for every real chain of size $\kappa$ in the quotient Boolean algebra $P(\omega)/fin$ we can find an almost chain of representatives such that every $n\in\omega$ oscillates at most three times along…
A new class of languages of infinite words is introduced, called the max-regular languages, extending the class of $\omega$-regular languages. The class has two equivalent descriptions: in terms of automata (a type of deterministic counter…