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Since 'conscious' indicates mental awareness of something existing and not just aberrant mental fluctuations, one can be conscious without being self conscious and thus have consciousness.
He seems to be studying something for which he has no initial definition. I have some trouble with consciousness being an axiom and not the result of the axiom of existence as though it is separate from existence. Same for making identity into an axiom. Usually an axiom has to be independent from any other axioms. Consciousness and identity are derivative from existence. They imply existence.