Thanks for the great review! Your tip to swap 24 and 25 was helpful, as was your warning about inconsistent notation. However, one benefit of "inconsistent notation" is that it really forces you to develop a clear understanding.

Anyway, I'll add some additional thoughts.

Overall, I got a lot out of this. Naive Set Theory clarified a lot of foundational concepts I had previously taken for granted. It also made me crack up at times; for example:

• The slight feeling of discomfort that the reader may experience in connection with the definition of natural numbers is quite common and in most cases temporary.
• We want to be told that the successor of 7 is 8, but to be told that 7 is a subset of 8 or that 7 is an element of 8 is disturbing.

I personally found the trickiest part to be the proof of Zorn's lemma. So for posterity, here's a sketch of the proof that might be helpful for following the full proof given in the text:

Zorn's lemma. Let X be a partially-ordered set such that every chain in X has an upper bound (in X); then X has a maximal element.

Proof sketch.

• Let S be collection of weak initial segments of elements of X, ordered by set inclusion; show that if S has a maximal set, then X has a maximal element
• Let C be the collection of all chains in X, ordered by set inclusion; show that if C has a maximal set, then S has a maximal set (the text uses a script X in place of C)
• Use the axiom of choice to construct an "extension" function g on C that extends a non-maximal set by one element, and leaves a maximal set unchanged
• Define a special kind of subset of C called a tower
  • The definition of a tower is incredibly clever, and it rigorously describes the intuitive idea of "keep adding elements until you get a maximal set"
• Let t be the "smallest possible tower" (i.e. the intersection of all towers), and let A be the union of every set in t; show that g leaves A unchanged
• Conclude that C has a maximal set (A); thus S has a maximal set; thus X has a maximal element

Finally, here's the full list of ingredients in the axiom soup (note that the Peano "axioms" are actually proved, not taken as axioms):

• Axiom of extension (page 2): Two sets are equal if and only if they have the same elements.
• Axiom of specification (page 6): To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds.
• Axiom of pairing (page 9): For any two sets there exists a set that they both belong to.
• Axiom of unions (page 12): For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection.
• Axiom of powers (page 20): For each set there exists a collection of sets that contains among its elements all the subsets of the given set.
• Axiom of infinity (page 44): There exists a set containing 0 and containing the successor of each of its elements.
• Axiom of choice (page 59): The Cartesian product of a non-empty family of non-empty sets is non-empty.
• Axiom of substitution (page 75): If S(a, b) is a sentence such that for each element a in the set A the set {b : S(a, b)} can be formed, then there exists a function F with domain A such that F(a) = {b : S(a, b)} for each a in A.