We follow through exercises 18 to 24 from Chapter IV of Kunen (1980) to show a result of Fraenkel-Mostowski.

Exercise 18

Let \(\mathbf{F}: \mathbf{V}\to \mathbf{V}\) be 1-1 and onto. Define \(\mathbf{E} \subseteq \mathbf{V}\times\mathbf{V}\) by \(x \mathbf{E} y\) iff \(x \in \mathbf{F}(y)\). Show (in ZFC) that \(\mathbf{V},\mathbf{E}\) is a model of \(\text{ZFC}^-\).

Sketch. Think of \(\mathbf{F}\) as a permutation on \(\mathbf{V}\), then relativising to \(\mathbf{V},\mathbf{E}\) basically means replacing the membership relation with \(\mathbf{E}\).

We take all the \(\text{ZFC}^-\) axioms, replace all occurrences of \(x\in y\) with \(x\in\mathbf{F}(y)\) and show that they are theorems of \(\text{ZFC}^-\).

• \(\text{extensionality}^{\mathbf{V},\mathbf{E}}\) is \[ \forall x~\forall y \left( \forall z \left(z\in \mathbf{F}(x) \leftrightarrow z\in \mathbf{F}(y)\right) \rightarrow x = y \right),\] let \(x, y\in \mathbf{V}\), using extensionality, \(\mathbf{F}(x) = \mathbf{F}(y)\), and as \(\mathbf{F}\) is 1-1 we know \(x = y\),

• let \(\phi\) be a formula with free variables \(x,z,\overline{w}\), its instance of \(\text{Comprehension}^{\mathbf{V},\mathbf{E}}\) is \[ \forall z~\forall \overline{w}~\exists y~\forall x \left( x \in \mathbf{F}(y) \leftrightarrow x\in \mathbf{F}(z) \land \phi \right),\] let \(z,\overline{w}\in \mathbf{V}\), comprehension gives us existence of \(\mathbf{F}(y)\) and use fact that \(\mathbf{F}\) is a permutation to get existence of \(y\),

• pairing looks easy

• for \(\text{union}^{\mathbf{V},\mathbf{E}}\) which is \[ \forall \mathscr{F}~\exists A~\forall Y~\forall x\left(x\in \mathbf{F}(Y)\land Y\in\mathbf{F}(\mathscr{F}) \rightarrow x\in\mathbf{F}(A)\right) \] let \(\mathscr{F}\in\mathbf{V}\), apply union to \(\operatorname{Im}_{\mathbf{F}}(\mathbf{F}(\mathscr{F}))\) (appealing to replacement), then there exists \(B\) satisfying \[ \forall Z~\forall w\left(w\in Z\land Z\in \operatorname{Im}_{\mathbf{F}}(\mathbf{F}(\mathscr{F})) \rightarrow w\in B\right) \] Let \(A = \mathbf{F}^{-1}(B)\) and let \(Y,x\in\mathbf{V}\), assume \(x\in \mathbf{F}(Y)\) and \(Y\in\mathbf{F}(\mathscr{F})\), let \(Z = \mathbf{F}(Y)\) and \(w = x\), apply previous statement to conclude that \(x\in \mathbf{F}(A) = B\).

other axioms should be similar (read: I’m lazy).

maybe do replacement sometime

Exercise 19

Use the preceding exercise to show the consistency of \(\text{ZFC}^- + \exists x\left(x = \left\{x\right\}\right)\), assuming consistency of ZFC.

Follow hint, define \(\mathbf{F}\) such that \(\mathbf{F}(0) = 1\), \(\mathbf{F}(1) = 0\) and \(\mathbf{F}\) is identity everywhere else. Define \(\mathbf{E}\) similarly as previous exercise and we show that \(\exists x\left(x = \left\{x\right\}\right)\) relativised to \(\mathbf{V},\mathbf{E}\) holds.

Work in \(\mathbf{V},\mathbf{E}\), because \(0\in 0\) we can conclude both \(0 \subseteq \left\{0\right\}\) and \(\left\{0\right\} \subseteq 0\) and therefore \(0 = 1\).

Likewise show the relative consistency of \(\text{ZFC}^- + \exists x~\exists y\left(x = \left\{y\right\} \land y = \left\{x\right\} \land x \ne y\right)\).

Similarly, define \(\mathbf{F}\) such that \(\mathbf{F}\) swaps \(0\) and \(\left\{\left\{0\right\}\right\}\). Now working in \(\mathbf{V},\mathbf{E}\), \(0 \in \left\{0\right\}\) and \(\left\{0\right\} \in 0\), let \(x = \left\{0\right\}\) and \(y = 0\).

Exercise 20

Assume consistency of ZFC and show the consistency of \(\text{ZFC}^-\) plus the following modified Mostowski Collapsing Theorem: If \(R\) is an extensional relation on the set \(A\), then \(\left\langle A, R\right\rangle\) is isomorphic to \(\left\langle M, \in\right\rangle\) for some transitive set \(M\). Note \(R\) is not assumed to be well-founded.

TODO: Induct at \(V_\alpha\) and force everything out…

Exercise 21

Show that there is no finite \(S\subset \text{ZFC}^-\) such that one can prove in \(\text{ZFC}^-\) that “\(R\) well-orders \(A\)” is absolute for transitive models of \(S\).

Follow hint, we use previous exercise. Suppose not, then let \(S \subset \text{ZFC}^-\) be finite such that for all transitive models \(M\vDash S\), “\(R\) well-orders \(A\)” is absolute.

Do model theory.

Exercise 22

For any set \(A\), define \(R(\alpha,A)\) by:

• \(R(0, A) = A \cup \operatorname{trcl}(A)\),
• \(R(\alpha+1, A) = \mathcal{P}(R(\alpha, A))\), and
• \(R(\alpha, A) = \bigcup_{\xi<\alpha} R(\xi, A)\) when \(\alpha\) is a limit ordinal.

Let \(\mathbf{WF}(A) = \bigcup_{\alpha\in\mathbf{ON}} R(\alpha, A)\). Show that in \(\text{ZF}^-\) that \(\mathbf{WF}(A)\) is a transitive model of \(\text{ZF}^-\) and that AC implies \(\text{AC}^{\mathbf{WF}(A)}\).

• \(\mathbf{WF}(A)\) being transitive is obvious.
• All transitive models satisfy Extensionality.
• Comprehension follows from \(\mathbf{WF}(A)\) being closed under power set.
• Powerset follows from Lemma 2.9.
• Pairing and Union looks easy.
• Replacement - For any definable class function \(\mathbf{F}\) and any set \(X\in\mathbf{WF}(A)\), take minimum \(\alpha\) such that \(\mathbf{F}\) image of \(X\) is contained in \(R(\alpha, A)\).
• Infinity looks obvious.

Assume choice and \(\text{choice}^{\mathbf{WF}(A)}\) follows from Lemma 3.14 (universal statements relativise down).

Exercise 23

Use a nice enough \(\mathbf{F}\) to get a \(U\) satisfying \(\forall x\in U.~ x = \left\{x\right\}\) (argument in 18 should work in a model of just \(\text{ZF}^-\)), then plug 22.

Exercise 24

doable