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Let $(X_i, d_i, e_i)$ be a sequence of pointed metric spaces, let $\prod _\omega (X_i, d_i, e_i)$ be the ultraproduct of said spaces with respect to a nonprincipal ultrafilter $\omega$, and let $(\hat{X}, \hat{d})$ be the nonstandard hull of $\prod _\omega (X_i, d_i, e_i)$ (namely, the quotient under infinitesimal equivalence of the nearstandard subset of $\prod _\omega (X_i, d_i, e_i)$). Of course, $(\hat{X},\hat{d})$ is a bona fide standard metric space, so in particular we can consider the nonstandard extension/ultrapower $(^*\hat{X},^*\hat{d})$.

My question is: are $(^*\hat{X},^*\hat{d})$ and $\prod _\omega (X_i, d_i, e_i)$ isomorphic?

pseudocydonia
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  • Even in the case that the original ultraproduct was an ultrapower, I would guess that if the first ultrapower and the second ultrapower were saturated with respect to different cardinalities then the two would be non-isomorphic. But I'm guessing the implicit assumption is that they would not be non-isomorphic for that trivial reason, i.e. both the original ultraproduct and the second ultrapower are implicitly assumed to be saturated with respect to the same cardinality. In that case idk, although I'd guess the answer is more likely to be no when the original ultraproduct is not an ultrapower. – Chill2Macht Feb 13 '23 at 18:47

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