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I am reading chapter 2 of Elements of set theory by Herbert Enderton and I have a confusion.

Can we contruct a set from subset axiom of ZF set theory, such that the set of all sets which does not belongs to itself. I think its true ( $\{ x \mid x \text { does not belongs to itself} \}$ ) and if it can be constructed than such a set should exist as we are constructing the set from subset axiom.

But I have also heard that ZF set theory avoids Russell's paradox through its axioms.

Please explain, where am I going wrong. Thanks. Forgive my latex as I am typing this via phone.

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    $\begingroup$ See also my answer: math.stackexchange.com/questions/162/… $\endgroup$ – Asaf Karagila May 19 '18 at 8:38
  • $\begingroup$ And by the way, Enderton is still used as a text!? Not that it's a bad book, but it was the text I learned from in the 1980s and it was a generation old even then. It was OK for the basics but (as I recall) pretty useless when it came to comparing ZF to other systems, or putting set theory in the context of first-order logic, or giving a flavor of research topics in set theory (not that it could really serve that last purpose any more even if it once had). $\endgroup$ – C Monsour May 19 '18 at 15:14
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    $\begingroup$ Also, by the way, if you want to learn more about the (early) history of set theory, I highly recommend Ferreiros et al's Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics. The most surprising part for me was learning that Cantor's original motivation for his work was to describe the sets of points on the real line where various Fourier series diverged. I usually think of set theory and analysis as about as far apart as one can get, but not so! $\endgroup$ – C Monsour May 19 '18 at 15:19
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    $\begingroup$ If you just want to learn set theory as a foundation for other math, Enderton is probably fine. If you are interested in set theory for its own sake, Enderton is lacking by being so, so old. If you want to learn more than what is in Enderton, I'd suggest Ciesielski's Set Theory for the Working Mathematician, Jech's The Axiom of Choice, and perhaps most of all (since it's low effort) browsing the Wikipedia pages on alternatives to ZFC, such as Morse-Kelley en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory . $\endgroup$ – C Monsour May 22 '18 at 11:47
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    $\begingroup$ I should add that none of these books is great for set theory as a foundation for category theory. If you take an interest in category theory and care about its foundations, you'll eventually want to read Michael Shulman's "Set Theory for Catgeory Theory" arxiv.org/pdf/0810.1279v2.pdf $\endgroup$ – C Monsour May 22 '18 at 11:51
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Where you are going wrong is that the subset axiom requires that you start with a set. If there were such a thing as the set of all sets, you could take the subset that are not members of themselves, and get the problem of Russell's paradox. But first you would have to show that there is a set of all sets, which you can't do in ZF. The subset axiom doesn't allow you to construct the set of all objects that have property P. It allows you to construct the set of all elements in set X that have property P.

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    $\begingroup$ Wow, yep I missed that. Thanks. Good point. "It allows you to construct the set of all elements in set X that have the property P." $\endgroup$ – Jasser May 19 '18 at 6:35

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