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Refinement-minimal intersecting covers
Disclaimer: This is not an answer, and just too long for a comment.
Here is an example why the chain condition does not hold (which does not mean that there are no refinement-minimal covers, just that ...
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Harvey Friedman: The expanding mind
Friedman has made public on his website a 2016 draft titled "Expanding Mind Theory". On page 4 there is a definition of a theory $\mathrm{EM}$ in first-order logic which formalizes the ...
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