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Foundations of Set Theory by Fraenkel, Bar-Hillel and Levy is a classic that provides what it sounds like you're after. It surveys ZF and its milieu, type-theoretic approaches (including Quine's New F …

Chen and Lih call such a $G(n,k,t)$ (with the same notation) a uniform subset graph; see "Hamiltonian uniform subset graphs" (1987). As you point out, the Johnson graph $J(n,k)$ is then $G(n,k,k-1)$. …

Regarding (2), some evidence that Baldwin refers to some sort of $L_{\omega_1 \omega}$ version of Morley's theorem, rather than just an alternate proof making use of $L_{\omega_1 \omega}$ machinery, c …

You would probably enjoy checking out homotopy type theory and Vladimir Voevodsky's corresponding program of univalent foundations for mathematics. Steve Awodey's survey article (linked to from that s …

I am trying to pin down: who first proved that $\mathsf{WKL}_0$ has an $\omega$-model in which every set is of low degree? As shown in Simpson's Subsystems of Second Order Arithmetic (Theorem IX.2.17) …

Yes, it appears e.g. as Theorem 4.19 in Chapter I of Kechris' Classical Descriptive Set Theory. (The relevant page is visible in Google Books if it's not in your library.)

Richard Kaye's book Models of Peano Arithmetic is good and accessible. And I know that what Frank said in his comment, about its availability as a pdf online, is indeed true; though like Frank, I shan …

NF does prove Cantor's theorem in the sense you indicate, $|\mathscr{P}_1(X)|<|\mathscr{P}(X)|$ for any set $X$. The usual ZF proof goes through, because definitions in that proof which need to be st …

The following remarks may not speak to what you're really after, but given your explicit reference to Hilbert's Program still being years away when wondering what Zermelo might have in mind, they may …

I'm late to the party, but hope that the following might usefully complement Ali's nice answer and references, first by elaborating slightly on the historical side, and second by briefly indicating an …

Benacerraf and Putnam's Philosophy of Mathematics: Selected Readings is a pretty standard (as these things go) collection of seminal papers in the philosophy of mathematics generally, and in the philo …