# Sequential and distributive forcings without choice

@article{Karagila2021SequentialAD, title={Sequential and distributive forcings without choice}, author={Asaf Karagila and Jonathan Schilhan}, journal={Canadian Mathematical Bulletin}, year={2021} }

. In the Zermelo–Fraenkel set theory with the Axiom of Choice a forcing notion is “ κ -distributive” if and only if it is “ κ -sequential”. We show that without the Axiom of Choice this equivalence fails, even if we include a weak form of the Axiom of Choice, the Principle of Dependent Choice for κ . Still, the equivalence may still hold along with very strong failures of the Axiom of Choice, assuming the consistency of large cardinal axioms. We also prove that while a κ -distributive forcing…

## One Citation

### Forcing over choiceless models and generic absoluteness

- Mathematics
- 2022

. We develop a toolbox for forcing over arbitrary models of set theory without the axiom of choice. In particular, we introduce a variant of the countable chain condition and prove an iteration…

## References

SHOWING 1-10 OF 12 REFERENCES

### On generic extensions without the axiom of choice

- Economics, MathematicsJournal of Symbolic Logic
- 1983

Abstract Let ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[G] of ZF (a…

### THE AXIOM OF CHOICE

- Mathematics
- 2003

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show…

### DEPENDENT CHOICE, PROPERNESS, AND GENERIC ABSOLUTENESS

- MathematicsThe Review of Symbolic Logic
- 2020

Abstract We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of…

### Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

- Economics, MathematicsSpringer Proceedings in Mathematics & Statistics
- 2021

In this paper, without the axiom of choice, we show that if a certain downward Lowenheim-Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is…

### How to have more things by forgetting how to count them†

- MathematicsProceedings of the Royal Society A
- 2020

This paper answers the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its DedekInd-finiteness or not add new sets of ordinals by presenting a varied list of conditions each equivalent to the preservation of Dedeksind- finiteness.

### All uncountable cardinals can be singular

- Mathematics
- 1980

Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, we prove the consistency with ZF of the statement that every infinite set is a countable union of sets of…

### Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse

- MathematicsArchive for Mathematical Logic
- 2022

<jats:p>We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if…

### The Bristol model: An abyss called a Cohen real

- ArtJ. Math. Log.
- 2018

The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking us one step closer to understanding models of [Formula: see text], and the HOD Conjecture and its relatives.

### Bristol models satisfy dependent choice

### The third millennium edition, revised and expanded

- Springer Monographs in Mathematics
- 2003