For any projective space that contains more than one point, is it possible to prove that it contains a hyperplane without using the Axiom of Choice?

It's easy enough to prove that there exists a projective space that would be isomorphic to a hyperplane if the hyperplane existed. The lines and planes (seen as points and lines respectively) passing through a specific point in the parent space do the job.

  • $\begingroup$ Where does the axiom of choice get in? $\endgroup$ – Asaf Karagila Apr 16 '16 at 10:22
  • $\begingroup$ Well, if the dimension is finite, there is no need f Axiom of Choice. $\endgroup$ – Crostul Apr 16 '16 at 10:29

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