One example of non-standard model of set theory different of ultrapowers to obtain independence results

Question

In chapter 3 of Jech book about set theory (1978),he mention that is possible use models (M,E) nonstandard (E not well-founded) to get independence results in set theory (ZF or ZFC). Anybody knows some example different than ultrapowers?

Answer 1

Forcing and inner model techniques are far more powerful at the moment for proving independence results - we don't know many ways to build non-well-founded models which are useful. E.g. the ultrapower construction isn't useful at all here, since the ultrapower has the same theory as the original structure!

One nice result, however, is the following:

If $M\models ZFC$ is countable, then $M$ has an (illfounded, possibly) end extension $N$ satisfying $ZFC+V=L$. (See e.g. Theorem 3.4 of this paper of Hamkins.)

$N$ is an end extension of $M$ if $M\subseteq N$ and for all $m\in M$, we have $\{n\in N: N\models n\in m\}=\{n\in M: M\models n\in m\}$, that is, elements of $M$ don't "gain new elements" in passing to $N$. For example, forcing extensions are end extensions.

Now, this lets us prove the following:

Let $\varphi$ be a $\Sigma_1$ sentence consistent with ZFC. Then $\varphi$ is consistent with ZFC+V=L.

Since if $\varphi$ is $\Sigma_1$ and true in $M$, and $N$ is an end extension of $M$, then $\varphi$ is true in $N$ as well.

But this is a very limited trick, and in general I think current applications of illfounded models for independence results are few and far between.