This matches The World of Null-A by A. E. van Vogt. (The title was originally written as The World of Ā, but the nonstandard Ā glyph was replaced in later printings.)
The Wikipedia plot summary includes all the elements that you describe:
Gilbert Gosseyn..., a man living in an apparent utopia where those with superior understanding and mental control rule the rest of humanity, wants to be tested by the giant Machine that determines such superiority. However, he finds that his memories are false. In his search for his real identity, he discovers that he has extra bodies that are activated when he dies (so that, in a sense, he cannot be killed), that a galactic society of humans exists outside the Solar system, a large interstellar empire wishes to conquer both the Earth and Venus (inhabited by masters of non-Aristotelian logic), and he has extra brain matter that, when properly trained, can allow him to move matter with his mind.
It is generally considered one of van Vogt's best works, and it spawned two sequels, The Players of Null-A and Null-A Three, which are not generally as well regarded. The novel version was published in 1948; it was a fix-up, heavily revised from three novellas that were published earlier in the decade. (Each novella covered a single one of Gosseyn's bodies, I believe.)