What does Nick see as his "cardinal virtue" in Chapter 3 of The Great Gatsby?
There is certainly a degree of irony in Nick's remark that honesty is his cardinal virtue. Nick is playing a part too, like Gatsby and, to a lesser extent, like Daisy. We cannot trust everything Nick says about himself just as Nick cannot trust everything said by Jordan (a cheater and a shirker), Daisy (a consummate performer) or Gatsby (the greatest "truth artist" of them all).
For instance, Nicks says of himself, "I’m inclined to reserve all judgments," yet he opens the novel with a series of judgments and clearly disdains Tom upon introducing him in the second chapter. Nick's self-descriptions of honesty and objectivity are dubious. What are we to make of this fact? What does it mean that Nick, like the others, is a dishonest person?
Dishonesty takes many forms in The Great Gatsby and the moral value of honesty is brought into question at several points.
Jay Gatsby is a thorough "pretender," yet he is the sole character capable of maintaining any innocence. He is even capable of acting on it. So despite the fact that he is a criminal and a fraud, Gatsby stands as perhaps the only "honest" character in the book. He does what he says he is going to do (tries to marry Daisy) and refuses to leave Daisy and run away while he can at the end of the novel, despite the fact that Daisy turns her back on him. This integrity in Gatsby, while it is demonstrative of his dedication to a dream/illusion, suggests a contrast to the "honesty" that Tom displays in the novel.
Tom does not hide his feelings. He has an affair with Myrtle and everyone knows about it. In a way, his brutality and philandering are proof of his honesty, yet Tom is certainly the most villainous character in the novel.
Truth-telling and transparency are not necessarily pure virtues then in Gatsby. Nick's "cardinal virtue" would not be worth much if he didn't also have a few things in common with Jay Gatsby - a sense of loyalty, a willingness to suspend disbelief in favor of a romantic illusion, etc.
The fact that Nick is not entirely honest and that he is often judgmental and even petty serves to make his claim of honesty ironic. This irony includes Nick in the multi-layered irony that surrounds Jay Gatsby. There are levels and layers of "truth" and meaning around each of these characters.
Not entirely pure in their motivations, they are nonetheless capable of real friendship and possess a sense of beauty that is not entirely materialistic.
Nick's intentions to become a bondman and climb the ladder of professional and self-improvement ends up coming to nothing - and actually comes to nothing rather quickly. The hollowness of his ambitions are readily apparent even as he shares them in the opening chapters.
Nick's journey, insofar as he is a figure on a quest, is to find something real that might supplant the frivolous and conventional ambitions he arrived with on the east coast. What he ends up finding is a kind of honesty in Jay Gatsby, though Nick has to sift through many layers of deceit to reach that final honesty. This is what Nick prepares us for at the outset in his initial description of Gatsby and it is where he leaves us in the end.
The quote you are looking for is in the last paragraph of the chapter in The Great Gatsby. The narrator, Nick, writes:
Everyone suspects himself of at least one of the cardinal virtues, and this is mine: I am one of the few honest people that I have ever known.
Nick of course is referring to his feeling that he needs to officially break off relations with a girl back home before he goes any farther in his relations with Jordan. He implies that it would be dishonest to continue with Jordan without first breaking his relationship off with the other woman.
Nick's statement here is important to the novel as a whole, because Nick keeps emphasizing how honest, nonjudgmental, and objective he is, when as a narrator he is unreliable, and is judgmental and subjective.