I understand that soundness implies consistency. Also, I understand that consistency alone does not imply soundness. But shouldn't consistency + completeness imply soundness? Scott Aaronson in his blog-post hints otherwise. What am I exactly missing?Update: By completeness I assume that all sentences true under all interpretations are deducible. By soundness I assume that all deduced sentences are true under all interpretations. Finally, by consistency I assume that no sentence $P$ and its negation $\neg P$ can be deduced.
Consistency and completeness imply soundness?
terminology;logic
In the Scott Aaronson blog post you linked, (syntactic) consistency means that there is no sentence $P$  such that both $P$ and its negation can be deduced from the axioms of the theory.  (Syntactic) completeness means that either $P$ or $\neg P$ are deducible from the axioms.  And soundness means that if $P$ is deducible from the axioms then it is valid under all possible models/interpretations of the axioms.Consistency + completeness does not imply soundness because you could have a set of axioms and a theorem $P$ where $P$ is deducible from the axioms, $\neg P$ is not deducible from the axioms but there is some model of the axioms under which $P$ is false and $\neg P$ is true.Aaronson gives an example in his post:If I believe that theres a giant purple boogeyman on the moon, then presumably my  belief is unsound, but it might be perfectly consistent with my various other beliefs  about boogeymen.There's a giant purple boogeyman on the moon is (perhaps) syntactically consistent with your axioms because you do not believe that there isn't a giant purple boogeyman on the moon.  Your axioms may be syntactically complete because they allow you to deduce at least one of the sentences there's a giant purple boogeyman on the moon or there isn't a giant purple boogeyman on the moon.  But in fact it is not true that there's a giant purple boogeyman on the moon, so your deduction is unsound even though your axioms and deductive system are both consistent and complete.The hardest concept in the above is probably soundness.  Soundness relates the syntactic structure of your sentences and deduction rules to the interpretations (or models or semantics) of those sentences.  A model of a language is a set, along with some relations and functions on the members of that set.  For example, Gödel's language was the Peano axioms and the underlying model of the Peano axioms is the set of integers with (at least) an equality relation and a successor function.  The Peano axioms are (I think) sound: if you can prove something with the Peano axioms then it is true of the natural numbers.  But Gödel proved that the Peano axioms are not complete: there are sentences in the language of the Peano axioms that are in fact true of the natural numbers, but which are not derivable from the Peano axioms.