You CAN Change Your Axioms
Just wanted to point out that the unacceptability of coming to a conclusion that can't be reached from your initial premises (while its negation can) does not mean that you have to abandon that conclusion. If in your heart that conclusion feels like it must be true, all you have to do to remain consistent, is appropriately alter you initial premise or axiom. Of course by doing that you may wind up having to alter OTHER conclusions that stemmed from the original, but now altered premise. But if the conclusion that forced this alteration is especially clearcut or important to your values, you have no choice.
This whole procedure is second nature to world class scientists but takes some getting used to by your average Joe.
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