### Recent Writing

### Mathoverflow Activity

- Comment by Victoria Gitman on A model of ZF without a well-ordering of the reals in which any two sets of reals are comparableHow about the easy symmetric models? Are there obvious incomparable sets in those? Say, force with ${\rm Add}(\omega,\omega_2)$ and symmetric names are fixed by automorphisms fixing initial segments.Victoria Gitman
- Comment by Victoria Gitman on A model of ZF without a well-ordering of the reals in which any two sets of reals are comparableThanks, Asaf! This is very nice, somehow I didn't think about the perfect set property! But I realized that what I really need is that any two subsets of $H_{\omega_1}$ are comparable, and the Solovay model doesn't have this property. So I am still stuck.Victoria Gitman
- A model of ZF without a well-ordering of the reals in which any two sets of reals are comparableLet us say that two sets $A$ and $B$ are comparable if there is an injection from $A$ to $B$ or there is an injection from $B$ to $A$. Obviously, in a model of ${\rm ZFC}$ any two sets are comparable by comparing their cardinalities. But this is not necessarily the case in a model […]Victoria Gitman

- Comment by Victoria Gitman on A model of ZF without a well-ordering of the reals in which any two sets of reals are comparable
### Cantor’s Attic

- ProjectiveRegularity properties ← Older revision Revision as of 19:40, 20 February 2018 Line 66: Line 66: The following statements are equivalent: The following statements are equivalent: * For every real $a$, $\aleph_1^{L[a]}$ is countable. * For every real $a$, $\aleph_1^{L[a]}$ is countable. −* For every real $a$, $\aleph_1^{L[a]}$ is inaccessible in $L[a]$.+* For every real $a$, $\aleph_1^V$ […]Julian Barathieu
- Second-orderModels of $\text{MK}$ ← Older revision Revision as of 12:15, 16 February 2018 Line 45: Line 45: == Models of $\text{MK}$ == == Models of $\text{MK}$ == −In consistency strength, $\text{MK}$ is stronger than [[ZFC|$\text{ZFC}$]] and weaker than the existence of an [[inaccessible]] cardinal. It directly implies the consistency of $\text{ZFC}$. However, if a cardinal […]Julian Barathieu

- Projective

# Monthly Archives: October 2013

## Forcing to add proper classes to a model of ${\rm GBC}$: The technicalities

In the previous post Forcing to add proper classes to a model of ${\rm GBC}$: An introduction, I made several sweeping assertions that will now be held up to public scrutiny.

## Forcing to add proper classes to a model of ${\rm GBC}$: An introduction

If you are interested in a mathematical universe whose ontology includes both sets and classes, you might consider for its foundation the ${\rm GBC}$ (Gödel-Bernays) axioms.