# Class Forcing in Class Theory

@article{Antos2018ClassFI, title={Class Forcing in Class Theory}, author={Carolin Antos}, journal={arXiv: Logic}, year={2018}, pages={1-16} }

In this article we show that Morse-Kelley class theory (MK) provides us with an adequate framework for class forcing. We give a rigorous definition of class forcing in a model \((M,\mathcal {C})\) of MK, the main result being that the Definability Lemma (and the Truth Lemma) can be proven without restricting the notion of forcing. Furthermore we show under which conditions the axioms are preserved. We conclude by proving that Laver’s Theorem does not hold for class forcings.

## 10 Citations

CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

- Computer Science, MathematicsThe Journal of Symbolic Logic
- 2016

It is shown that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing, and the forcing theorem is equivalent to the existence of a Boolean completion.

Boolean-valued class forcing

- Fundamenta Mathematicae
- 2021

We show that the Boolean algebras approach to class forcing can be carried out in the theory Kelley-Morse plus the Choice Scheme (KM + CC) using hyperclass Boolean completions of class partial…

Hyperclass forcing in Morse-Kelley class theory

- Mathematics
- 2017

In this article we introduce and study hyperclass-forcing (where the conditions of the forcing notion are themselves classes) in the context of an extension of Morse-Kelley class theory, called MK∗∗.…

HYPERCLASS FORCING IN MORSE-KELLEY CLASS THEORY

- Computer Science, MathematicsThe Journal of Symbolic Logic
- 2017

The main result combines hyperclass forcing with coding methods of [3] and [4] to show that every β-model of MK** can be extended to a minimal such model ofMK** with the same ordinals.

Characterizations of pretameness and the Ord-cc

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 2018

The results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications ofclass forcing.

Forcing and the Universe of Sets: Must We Lose Insight?

- Computer ScienceJ. Philos. Log.
- 2020

It is argued that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse that seems to necessitate the addition of subsets to V.

UNIVERSISM AND EXTENSIONS OF V

- Computer Science, MathematicsThe Review of Symbolic Logic
- 2020

A method of interpreting extension-talk (V-logic) is presented, and it is shown how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories.

Kelley–Morse set theory does not prove the class Fodor principle

- MathematicsFundamenta Mathematicae
- 2021

We show that Kelley-Morse set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a…

MODERN CLASS FORCING

- 2019

We survey recent developments in the theory of class forcing formalized in the second-order set-theoretic setting.

The Structure of Models of Second-order Set Theories

- Mathematics
- 2018

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the…

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