Thinking about the theory of directed graphs T I came up to the thought that T may have no proper axioms. Thus, T is the first order predicate calculus with the signature consisting of a single predicate $P(x,y)$. This implies that T is complete. I don't understand how this could be true since T may have models satisfying opposite properties. Say, directed graphs may contain or not the source.
2026-04-13 23:49:09.1776124149
Is the theory of directed graphs complete?
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The first-order theory with no non-logical axioms is never complete, no matter what the language you're looking at. For instance, it does not decide the truth value of the sentence $\exists x\exists y~x \neq y$.