A substantial portion of this text appears to be concerned with traditional Zermelo–Fraenkel set theory (and its extensions).
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.
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.
[1] MAA, Chauvenet Prize, https://www.maa.org/programs-and-communities/member-communit...
[2] Leinster, T., "Rethinking set theory", https://arxiv.org/abs/1212.6543
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.
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.
[1] MAA, Chauvenet Prize, https://www.maa.org/programs-and-communities/member-communit...
[2] Leinster, T., "Rethinking set theory", https://arxiv.org/abs/1212.6543