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Philosophy

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    @controlgroup Please note that it took a long time to prove that the axiom of parallels is independent from the other axioms of Euclidean geometry. The proof was by constructing non-Euclidean geometries which violate the axiom of parallels. Commented Jan 4 at 0:47
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    @controlgroup a half-baked version of that was my motivation for including the words 'of interest' in an earlier version of my answer. I was wondering, for example, whether you could prove anything with, say, just two of Euclid's axioms, and if so, and if they were the only things you were interested in, the other axioms would be superfluous . Commented Jan 4 at 7:49
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    This isn't a sufficient definition I think - if X0, X1... are statements, consider the axioms X0, X0^X1, X0^X1^X2... Commented Jan 4 at 15:28
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    Any proper subset unbounded above is sufficient to derive the full set, any finite set of statements can be safely removed so it meets your criterion, but you can't get smaller than the countably infinite subset, even by choosing different axioms (as each axiom can only refer to a finite number of the X_i). Commented Jan 4 at 15:31
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    Both axiom schemas (specification and replacement) in ZFC have similar properties, allowing you to safely remove any finite subset. Commented Jan 4 at 15:34