# Madness and regularity properties

@article{Horowitz2017MadnessAR, title={Madness and regularity properties}, author={Haim Horowitz and Saharon Shelah}, journal={arXiv: Logic}, year={2017} }

Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $\omega^{\omega}$-bounding sufficiently absolute forcing notions $\mathbb Q$.

## 3 Citations

### The Ramsey property implies no mad families

- 2019

Mathematics

Proceedings of the National Academy of Sciences

If all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families, and the implication is proved in Zermelo–Fraenkel set theory with only weak choice principles.

### On the non-existence of $$\kappa $$-mad families

- 2023

Mathematics

Archive for Mathematical Logic

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

### Transcendence bases, well-orderings of the reals and the axiom of choice

- 2019

Mathematics

We prove that $ZF+DC+"$there exists a transcendence basis for the reals$"+"$there is no well-ordering of the reals$"$ is consistent relative to $ZFC$. This answers a question of Larson and Zapletal.

## 8 References

### CANONICAL MODELS FOR FRAGMENTS OF THE AXIOM OF CHOICE

- 2017

Economics

The Journal of Symbolic Logic

The technology reduces many questions about ZF implications between consequences of the Axiom of Choice to natural ZFC forcing problems.

### On measure and category

- 1985

Economics, Mathematics

We show that under ZF + DC, even if every set of reals is measurable, not necessarily every set of reals has the Baire property. This was somewhat surprising, as for the Σ21 set the implication holds.

### A model of set-theory in which every set of reals is Lebesgue measurable*

- 1970

Economics, Mathematics

We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF,…

### A barren extension

- 1985

Mathematics

It is shown that provided ω→(ω)ω, a well-known Boolean extension adds no new sets of ordinals. Under an additional assumption, the same extension preserves all strong partition cardinals. This fact…

### Can you take Toernquist's inaccessible away?

- 2016

Psychology

We prove that ZF + DC + ”There are no mad families” is equiconsistent with ZFC.

### E-mail address: haim.horowitz@mail.huji.ac.il (Saharon Shelah

- 1904

Einstein Institute of Mathematics Edmond J. Safra campus,