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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 ?)

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up vote 9 down vote accepted

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.

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See also Simon Thomas' answer here: mathoverflow.net/questions/30408/… –  Joel David Hamkins Apr 27 '12 at 11:36

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