Some questions regarding a theorem of Paul J. Cohen

Question

In his paper "Automorphisms of Set Theory", Paul Cohen proved the following theorem:

"There exist models of $ZF$ admitting automorphisms of order two. More exactly, If $M$ is any countable model with a nonstandard (non-well-founded) ordinal $\alpha_0$, then there is a model $M^{'}$ with the same ordinals as $M$, which admits a nontrivial automorphism of order two. If $\omega$ is nonstandard in $M$, then for some $x$ in $M^{'}$ of rank less than $\omega$+$\omega$, we will have $\sigma$$x$$\ne$$x$. If $\omega$ is standard in $M$, then, for every nonstandard $\alpha$$\le$$\alpha_0$, there will be a set $x$ in $M^{'}$ of rank $\alpha$ and $\sigma$$x$$\ne$$x$. Moreover, the cardinals in $M$ are the cardinals in $M^{'}$."

The answers to the following questions will greatly help me understand its import:

(1). Is the nontrivial automorphism $\sigma$ of order two defined on $M^{'}$ defined on the entire domain of $M^{'}$?

(2). Consider Cohen's two remarks; one immediately following the theorem, the other immediate preceeding the theorem:

a) "Clearly $M$ cannot be well-founded and admit any automorphism $\sigma$$\ne$$I$, as seen by induction on the rank."

b) "When we speak of models of $ZF$ we will always mean that the axioms of regularity and extnsionality are included."

Question 2.1: Since $M^{'}$ admits a nontrivial automorphism of order two, then $M^{'}$ would be non-well-founded. Would $M^{'}$ being non-well-founded be the view of $M^{'}$ 'outside' $M^{'}$ while the axioms of regularity and extensionality hold 'inside' $M^{'}$?

2.2: If $M$ is a countable model of $ZF$ with a nonstandard (non-well-founded) ordinal $\alpha_0$, does $M$ necessarily need be non-well-founded?

2.3: If $M^{'}$ is non-well-founded ('outside' of $M^{'}$), then it would seem to (possibly) be a model of $ZF^{-f}$. Could there then exist models of $ZF^{-f} $+$BAFA$ of the type Cohen talks about in his paper? Could his method of constructing models of $ZF$+$\lnot$$AC$ discussed in his paper allow one to construct a nontrivial elementary embedding $j$:$M^{'}$$\rightarrow$$M^{'}$ of the following type:

'There is a nonstandard (non-well-founded) ordinal $\kappa$, such that for every transitive set $M^{'}$ that includes $\kappa$, there is a nontrivial elementary embedding of $M^{'}$ into $M^{'}$ (in some forcing extension $M^{'}[G]$) with critical point (also a nonstandard ordinal) below $\kappa$.'?

Answer 1

For $(1)$, I'm not sure what you mean - yes, it's an automorphism of the whole structure $M'$.

For $(2.1)$: yes, $M'$ is "internally well-founded" even though it is ill-founded externally. Basically, what's going on is that well-foundedness is not a first-order-expressible property (thanks to compactness), and the axiom of foundation is a first-order approximation of well-foundedness.

For $(2.2)$: yes, this is immediate - a model of set theory is well-founded iff all its ordinals are well-founded.

For $(2.3)$, since $M'\models ZF$ we certainly have $M'\models ZF^{-f}$; as to incorporating BAFA, though, I have no idea.

Finally, your question about critical points. First of all, you seem to be using $M'$ both as a name for the model and as a variable ("for every transitive set $M'$. . ."). This, together with the introduction of BAFA earlier, makes me confused about what you are asking for. For instance, do you want:

• A model $W$ of $ZF$, with an externally ill-founded ordinal $\kappa$, such that for every $M\in W$ which $W$ thinks is transitive and contains $\kappa$, there is an elementary embedding $j: M\rightarrow M$ - definable in a forcing extension of $M$ - such that $crit(j)<\kappa$?

Or:

• A model $W$ of $ZF$, with an externally ill-founded ordinal $\kappa$, such that for every $M\in W$ which $W$ thinks is transitive and contains $\kappa$, there is an elementary embedding $j: M\rightarrow M$ - definable in a forcing extension of $W$ - such that $crit(j)<\kappa$?

Or what, exactly?