Right from the outset this is a mind-bending, roller coaster ride of twists and turns. Don't expect detailed world building and character development, that is not what Van Vogt is all about. He is instead concerned with exploring his crazy ideas and plot twists.
In the opening chapter we discover that the protagonist, Gilbert Gosseyn, is not who he thinks he is as his memories are proven false. Gosseyn (and the reader) are then thrown into a state of confusion which lasts throughout the book. A little later Gosseyn is killed, only to awaken in the next chapter in a brand new copy of his body, on Venus. But all this makes sense, as far as the author is concerned, because he is trying to make clear his point that our identity is our memories, whether true or false, whether everything else is changes or not.
"Null-A" stands for Non-Aristotelian and this book describes a world in which this new pholosophy and logic have superceded the Aristotelian philosophy and logic that dominates our thinking today. Those who have best integrated this way of thinking into their lives go on to do all the important jobs in society and the cream of the crop go on to live in a utopian society on Venus populated with like minded people. All this is coming under threat however as forces of the old Aristotelian order are attempting a coup, seeking to overthrow this new order. Somehow Gosseyn is in the middle of all this and is the key to stopping it but he doesn't know how and appears to be a mere pawn in someone elses game.
As the author states in the last sentence of chapter 14: "It was all quite incomprehensible.". Yes, that sums it up quite succinctly. Van Vogt just flies off the handle a little too much here, the plot developments just too crazy and unbelievable but it's still quite an enjoyable read.