&gt; The axioms don&#x27;t have to be true. You can still talk about their implications in absolute terms.<p>No, you can&#x27;t. Two mathematicians using different rules of inference (say, those of classical vs. intuitionistic logic) will arrive at different theorems, even if they begin from the same axioms.<p>&gt; &quot;Assuming a=0 implies a=0&quot; is absolutely true.<p>Assuming that “a = 0” and implication are both expressible in our formal system, it would indeed be very weird (though not a priori impossible) if “a = 0” didn&#x27;t imply “a = 0”. But I have no problem imagining a formal system where “a = 0” isn&#x27;t expressible (e.g., because equality is inexpressible), or implication isn&#x27;t expressible (e.g., geometric logic), or implication has strange properties to someone only familiar with the classical and&#x2F;or intuitionistic interpretation of “implication” (e.g., linear logic).