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Sy David Friedman

Publications and source records attributed to Sy David Friedman.

9 recordsLinked to original sources
arXiv2025

Good projective witnesses

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and $\mathfrak{c}$, and there is a $Π^1_2$-definable MCG of this cardinality. Here $Π^1_2$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

math.LO↗
arXiv2022

Coding over Core Models

Early in their careers, both Peter Koepke and Philip Welch made major contributions to two important areas of set theory, core model theory and coding, respectively. In this article we aim to survey some of the work that has been done which combines these two themes, extending Jensen's original Coding Theorem from $L$ to core models witnessing large cardinal properties.

math.LO↗
arXiv2020

A null ideal for inaccessibles

In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was also addressed by Shelah ([11, Problem 0.5]) and concerns the definition of a forcing which is $κ^kappa$-bounding, $< κ$-closed and $κ^+$-cc, for $κ$ inaccessible.

math.LO↗
arXiv2016

On the set-generic multiverse

The forcing method is a powerful tool to prove the consistency of set-theoretic assertions relative to the consistency of the axioms of set theory. Laver's theorem and Bukovský's theorem assert that set-generic extensions of a given ground model constitute a quite reasonable and sufficiently general class of standard models of set-theory. In sections 2 and 3 of this note, we give a proof of Bukovský's theorem in a modern setting (for another proof of this theorem see Bukovský [4]). In section 4 we check that the multiverse of set-generic extensions can be treated as a collection of countable transitive models in a conservative extension of ZFC. The last section then deals with the problem of the existence of infinitely-many independent buttons, which arose in the modal-theoretic approach to the set-generic multiverse by J.Hamkins and B.Loewe [12].

math.LO↗
arXiv2016

Definable maximal cofinitary groups

Using countable support iteration of $S$-proper posets, for some appropriate stationary set $S$, we obtain a generic extension of the constructible universe, in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and there is a maximal cofinitary group with a $Π^1_2$-definable set of generators.

math.LO↗
arXiv2016

Collapsing the cardinals of $HOD$

Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.

math.LO↗
arXiv2015

Independence of higher Kurepa hypotheses

We study the Generalized Kurepa Hypothesis introduced by Chang. We show that relative to the existence of an inaccessible cardinal the Gap-$n$-Kurepa hypothesis does not follow from the Gap-$m$-Kurepa hypothesis for $m$ different from $n$. The use of an inaccessible is necessary for this result.

math.LO↗
arXiv2015

Killing the GCH everywhere with a single real

Shelah-Woodin investigate the possibility of violating instances of $GCH$ through the addition of a single real. In particular they show that it is possible to obtain a failure of $CH$ by adding a single real to a model of $GCH$, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate $GCH$ at all infinite cardinals by adding a single real to a model of $GCH.$ Our assumption is the existence of an $H(κ^{+3})$-strong cardinal, by work of Gitik and Mitchell it is known that more than an $H(κ^{++})$-strong cardinal is required.

math.LO↗