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JoshuaZ comments on Open thread, Oct. 12 - Oct. 18, 2015 - Less Wrong
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I'm not sure that's strong evidence for the thesis in question. If ZFC had a low-lying inconsistency, ZFC+an inaccessible cardinal would still prove ZFC consistent, but it would be itself an inconsistent system that was effectively lying to you. Same remarks apply to any large cardinal axiom.