Completeness condition in Godel first incompleteness theorem superflous
Wikipedia says:
Theory is complete if it is a maximal consistent set of sentences.
Than it says:
Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and complete.
But completeness definition says that theory is consistent so above
definition says that
Any effectively generated theory capable of expressing elementary
arithmetic cannot be both consistent and (maximal and consistent).
So it should be enough to say:
Any effectively generated theory capable of expressing elementary
arithmetic cannot be complete.
Am I right or am I missing something?
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