= A semantic Gödel fixed point need not be canonical
Assuming PA is <consistent first-order theory>[consistent], the <first-order theory>[theory] $S=\operatorname{PA}+\neg\operatorname{Con}_{\operatorname{PA}}$ is <consistent first-order theory>[consistent] by second incompleteness. Its false <axiom> $G=\neg\operatorname{Con}_{\operatorname{PA}}$ is provable, so the standard semantic biconditional $G\leftrightarrow\neg\operatorname{Pr}_S(\ulcorner G\urcorner)$ is true with both sides false. Thus a true biconditional alone does not make an arbitrary sentence a true canonical <Gödel sentence>. A canonical <Pi-1 formula>[Pi-1] construction adds the necessary syntactic or sound-base information. Truth, internal equivalence and canonical construction must be distinguished.
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