I'm not familiar with locale theory. Thanks for the reference. I'm guessing there will be some non-intuitive results. My non-expert impression is that whatev

I'm not familiar with locale theory. Thanks for the reference. I'm guessing there will be some non-intuitive results. My non-expert impression is that whatever one chooses in terms of logic and set theory there will be bizarre results when dealing with sets of cardinality of the continuum and not dealing with the continuum leaves out too much. Here is a reference to a mathiverflow comment. https://mathoverflow.net/a/22935 https://mathoverflow.net/a/22935