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Mikhail Katz
Mikhail Katz
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As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try the following approach. 

In a universe $V$ satisfying CH$CH$, we can take two inequivalent filters and obtain fields that are automatically isomorphic by the result of Erdos &Co mentioned in the MSE post linked in the comments above. Now the idea is to take a forcing extension $V^F$ satisfying $\neg CH$. 

One can't transfer naively the construction of the hyperreal field to $V^F$ because the ultrafilter is not definable, but perhaps one can work with the definable hyperreal field of Kanovei and Shelah (which exploits a huge index set using all ultrafilters simultaneously, thereby defeating non-definability). Perhaps one can specify two variants of the Kanovei-Shelah construction whether the ideals are not equivalent but the quotient fields will be by the Erdos-type argument, and then take a forcing extension to exhibit a similar phenomenon under $\neg CH$.

As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try the following approach. In a universe $V$ satisfying CH, we can take two inequivalent filters and obtain fields that are automatically isomorphic by the result of Erdos &Co mentioned in the MSE post linked in the comments above. Now the idea is to take a forcing extension $V^F$ satisfying $\neg CH$. One can't transfer naively the construction of the hyperreal field to $V^F$ because the ultrafilter is not definable, but perhaps one can work with the definable hyperreal field of Kanovei and Shelah (which exploits a huge index set using all ultrafilters simultaneously, thereby defeating non-definability). Perhaps one can specify two variants of the Kanovei-Shelah construction whether the ideals are not equivalent but the quotient fields will be by the Erdos-type argument, and then take a forcing extension to exhibit a similar phenomenon under $\neg CH$.

As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try the following approach. 

In a universe $V$ satisfying $CH$, we can take two inequivalent filters and obtain fields that are automatically isomorphic by the result of Erdos &Co mentioned in the MSE post linked in the comments above. Now the idea is to take a forcing extension $V^F$ satisfying $\neg CH$. 

One can't transfer naively the construction of the hyperreal field to $V^F$ because the ultrafilter is not definable, but perhaps one can work with the definable hyperreal field of Kanovei and Shelah (which exploits a huge index set using all ultrafilters simultaneously, thereby defeating non-definability). Perhaps one can specify two variants of the Kanovei-Shelah construction whether the ideals are not equivalent but the quotient fields will be by the Erdos-type argument, and then take a forcing extension to exhibit a similar phenomenon under $\neg CH$.

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Mikhail Katz
Mikhail Katz
  • 16.6k
  • 2
  • 54
  • 127

As far as I understand the question is still open under $\neg CH$, namely whether isomorphism of hyperreal fields implies equivalence of filters (up to permutation of index set). Perhaps one can try the following approach. In a universe $V$ satisfying CH, we can take two inequivalent filters and obtain fields that are automatically isomorphic by the result of Erdos &Co mentioned in the MSE post linked in the comments above. Now the idea is to take a forcing extension $V^F$ satisfying $\neg CH$. One can't transfer naively the construction of the hyperreal field to $V^F$ because the ultrafilter is not definable, but perhaps one can work with the definable hyperreal field of Kanovei and Shelah (which exploits a huge index set using all ultrafilters simultaneously, thereby defeating non-definability). Perhaps one can specify two variants of the Kanovei-Shelah construction whether the ideals are not equivalent but the quotient fields will be by the Erdos-type argument, and then take a forcing extension to exhibit a similar phenomenon under $\neg CH$.