TITLE: Finite list of formulas not absolute for any set
QUESTION [6 upvotes]: I'm having some trouble with exercise II.5.8 in Kunen (2013 edition). The problem is:
Work in $ZFC^-$ (i.e. without Foundation), plus the assumption that $U:=\{x : x=\{x\}\}$ is a proper class with no infinite subsets. Write down a finite list of formulas $\phi_1, \ldots, \phi_n$ such that there is no set $C$ with $\phi_1, \ldots, \phi_n$ absolute for $C$.
My guess is that we want to choose $\phi_1, \ldots, \phi_n$ so that the assumption of them being absolute would show that $ZFC^-$ proves it own consistency. However, I'm not sure how to do this. Thanks

REPLY [3 votes]: Let $\phi$ be the assertion that $\omega$ exists and that for every $n\in\omega$ there is a set $A$ containing $n$ distinct elements $x$ for which $x=\{x\}$. 
This statement is true, under your assumptions, since we can prove by induction on $n$ that $U$ has subsets of size $n$ for every $n$. But this statement is not true when relativized to any set $C$, for then $C\cap U$ would be an infinite subset of $U$, contrary to assumption. 
Let me also point out that if you have the collection axiom in $\text{ZFC}^-$, then your theory is inconsistent. The reason is that for every $n\in\omega$, you will have a set $A\subset U$ of size $n$, but you will not be able to collect these, since that would give rise to an infinite subset of $U$. So I assume that you are using only the replacement axiom, and not collection (and these are not equivalent without foundation), and this will break many arguments.