Let $U$ and $V$ be two non-principal ultrafilters over **N**, and $R\_1$ and $R\_2$ the non-standard extensions of **R** given by $R\_1=R^N/U$ and $R\_2=R^N/V$. Are they always isomorphic (I think not, but could not prove it), and, if not, what axioms must be added to ZFC to ensure they are (or is it a consequence of AC that they are not ?)



Isomorphisms between non-standard reals.
If the continuum hypothesis holds, then both of these ultrapowers are saturated models of cardinality $\omega\_1$, and one can see that they are isomorphic by a back-and-forth argument. 


When the CH fails, then they need not be isomorphic. 


