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**Homework Statement:**See attached image.

**Homework Equations:**ZFC set theory.

Consider the text in the attached image. What is meant with "We require of an axiom system that it be possible to decide whether or not any given formula is an axiom."? Is consistency synonymous with soundness? Is there an axiom system that can be completely and consistently axiomatised, or is this what Gödel's incompleteness theorem states is not possible?