Foundations of Mathematics (2015) [pdf]
A substantial portion of this text appears to be concerned with traditional Zermelo–Fraenkel set theory (and its extensions).<p>I have gradually come to believe that ZF theory has received a disproportionate amount of attention on account of the fact that it serves as the "official" foundations of mathematics, but that it is not an especially beautiful, or useful, structure.<p>I believe that ZF theory is an interesting object, worthy of mathematical study, but _not_ the best candidate for the foundations of mathematics! I am very happy to see that this year's Chauvenet Prize [1] was won by Tom Leinster's "Rethinking set theory", in which he highlights that Lawvere set theory looks like a much better candidate. I cannot do better than recommend you look at Leinster's superb article.<p>[1] MAA, Chauvenet Prize, <a href="https://www.maa.org/programs-and-communities/member-communities/maa-awards/writing-awards/chauvenet-prizes" rel="nofollow">https://www.maa.org/programs-and-communities/member-communit...</a><p>[2] Leinster, T., "Rethinking set theory", <a href="https://arxiv.org/abs/1212.6543" rel="nofollow">https://arxiv.org/abs/1212.6543</a>