# Relative consistency proofs using proper class models

Are there any easy relative consistency proofs in set theory that can be done using proper class models, rather than set models?

The only easy one I can think of is proving the consistency of $\mathsf{ZF}$ relative to $\mathsf{ZF} - \text{Foundation}$ using the class $\text{WF} = \bigcup_{\alpha \in \text{Ord}} V_\alpha$ of well-founded (i.e. hereditary) sets.

The next ones I can think of are the consistency of $\mathsf{ZFC}$ relative to $\mathsf{ZF}$, using $\text{L}$ or $\text{HOD}$, and the consistency of $\mathsf{ZF} + \mathsf{CH}$ relative to $\mathsf{ZF}$, using $\text{L}$. But these are a bit harder. Are there any more that are as easy as the one using $\text{WF}$?

EDIT: By "easy" here I mean that it does not involve encoding formulas as natural numbers, or formalizing the satisfaction relation. I consider showing one theory to be consistent relative to another theory using proper class models and relativization to be easier than proving formalized consistency statements using set models and the formalized satisfaction relation. For this reason, I am teaching it first in my class. However, this advantage of using proper class models is nullified if I have to use the formalized satisfaction relation to construct $\text{L}$ or $\text{HOD}$.

• You cannot do this for sentences $\sigma$ false in $L$, as if you could create a proper class model $M$ in $\mathsf{ZF}$ of $\sigma$, we could still create it in $\mathsf{ZF+V=L}$, but as $Ord\subseteq M$, $M=L$. – Camilo Arosemena-Serrato Nov 25 '13 at 4:43
• @CamiloArosemena Good point. But perhaps one can still do something interesting where the desired sentence $\sigma$ is simply an axiom of $\mathsf{ZF}$ but we are only assuming some fragment of $\mathsf{ZF}$, as in the example with $\text{WF}$. – Trevor Wilson Nov 25 '13 at 4:49