# If $M$ is a transitive set and $R$ well orders $A$ in $M$, then $R$ well orders $A$ universally

I am asked to prove the following: Working in ZF - P.

If $$M$$ is (Edit) a transitive model of ZF - P, and $$R, A \in M$$, then

$$(R \ well \ orders \ A) ^M \rightarrow (R \ well \ orders \ A)$$

I am struggling with the minimal element. Since $$R$$ well orders A relative to $M$ we have that: $\forall X \in M s.t. X \subset A (X \neq 0 \rightarrow \exists y \in X (\lnot \exists z \in X (zRy))]$

So I need to show that for any $$X$$ (not necessarily in $$M$$) s.t. $$X \subset A$$ and $$X \neq 0$$, X has a minimal element.

My idea was to suppose there is such an X with no minimal element and arrive at a contradiction by showing this would imply A has no minimal element but the details of this are tripping me up.

Considering any element $$a \in A$$ we try to show $$\exists b \in A$$ s.t. $$b R a$$.

If $$a \in A$$ then $$a \in X$$ or $$a \in A-X$$. If $$a \in X$$ then we are done since X has no minimal element so we can always find something smaller.

Now if $$a \in A-X$$ then...I want to say something in $$X$$ is smaller then it? Or possibly compare $$a$$ and elements in $$X$$ since those are both in $$M$$ and hence well ordered there, but I'm getting stuck.

Any assistance would be awesome.

• Do you know any absoluteness theorems? – Asaf Karagila Apr 18 at 13:00
• Are you given any information about $M$ beyond just that it's a transitive set? For example, is it a model of ZF (or some reasonable fragment of ZF)? Without some such information, I wouldn't expect the claim to be true. – Andreas Blass Apr 18 at 13:42
• I have that M is a transitive model of ZF - P. I'll update the original post. I also have some basic absoluteness theorems, (for example the one about $\Delta_0$ formulas). – Math Lady Apr 18 at 14:48
• Since I guess (not really sure) $ZF^-$ is enough to prove the theorem that states that any wellordered set is iso to a unique ordinal, let $F:\langle A, R\rangle \rightarrow \langle \alpha , \in \rangle$ be an order iso for some ordinal $\alpha$ in $M$. The same $F$ is an order iso in $V$. – Shervin Sorouri Apr 18 at 15:52
• By $ZF^-$ I meant ZF - Powerset. – Shervin Sorouri Apr 18 at 15:53