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Description: Transfer theorems

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The results of Jech/Sochor [1966a] and Jech/Sochor [1966b] and Pincus [1971], Pincus [1972a], and Pincus [1971] and Pincus [1977a] show that many independence results in $ZF^0$ will transfer automatically to $ZF$. These transfer theorems are given in this note.  We use the notation of Pincus.

Definition: Assume $x$ is a set, $A$ is a class and $\Phi (x_1,\ldots, x_n)$ is a formula in the language of $ZF$ with variables among $x_1, \ldots , x_n$.

1. $|x|$ is the least ordinal $\alpha$ such that there is a one to one function from $x$ onto $\alpha$. (Note that this definition of $|x|$ is for this note only and differs from the definition of $|x|$ used elsewhere in this document.  Also note that $|x|$ is defined only for well ordered sets $x$.)
2. $|x|^- = \sup\{|y|:$ there is a surjection from $x$ onto $y\}$.
3. $|x|_- = \sup\{|y|:$ there is an injection from $y$ to $x \}$.
4. $R_\beta(x)$ is defined for ordinals $\beta$ by induction: $R_0(x) = x$, $R_{\alpha + 1}(x) = R_\alpha (x) \cup \cal P (R_\alpha(x))$ and for limit ordinals $\lambda$, $R_\lambda (x) = \bigcup_{\gamma< \lambda} R_\gamma (x)$.
5. $\Phi^A (x_1,\ldots,x_n)$ is $\Phi$ with quantifiers restricted to $A$. Similarly, if $\sigma$ is a term in $ZF$ then $\sigma^A$ is the term defined by the same formula that defines $\sigma$ but with its quantifiers restricted to $A$.
6. Assume that $V$ is a model of $ZF$ and $V_0 \subseteq V$ is a substructure of $V$ which is a model of $ZF^0$ such that every element of a $V_0$ set is an element of $V_0$.  (But elements of $V_0$ atoms are not in $V_0$.)  Assume also that $V$ and $V_0$ have the same ordinals, the same cofinality function, the same surjective cardinal function $|x|^-$ (on $V_0$ sets), the same injective cardinal function $|x|_-$ (on $V_0$ sets) and the same cardinality function $|x|$ (on$V_0$ sets.)  Then the formula $\Phi(x_1,\ldots,x_n)$ is absolute if for all $x_1, \ldots , x_n$ in $V_0$, $\Phi^{V_0}(x_1, \ldots ,x_n)\leftrightarrow \Phi(x_1, \ldots ,x_n)$.  (This is sometimes called absolute for $V_0$, $V$.)
7. A term $\sigma (x_1, \ldots ,x_n)$ is ordinal valued if ZF$^0 \vdash \sigma(x_1, \ldots ,x_n) \in On$.
8. A formula $\Phi(x_1, \ldots ,x_n)$ is boundable if for some absolute ordinal valued term $r$, $ZF^0 \vdash \Phi(x_1, \ldots ,x_n)\leftrightarrow \Phi^{R_r(x_1 \cup \cdots \cup x_n)}(x_1, \ldots ,x_n)$. Similarly, the term $\sigma(x_1, \ldots ,x_n)$ is boundable if $$ZF^0\vdash \sigma(x_1, \ldots ,x_n) = \sigma^{R_r(x_1 \cup\cdots \cup x_n)}(x_1, \ldots ,x_n).$$
9. A sentence is boundable if it is the existential closure of a boundable formula.

The theorem of Jech/Sochor [1966b] is A boundable sentence is transferable. Many applications of the theorem use the following corollary: If $\Phi$ is a sentence such that there is an ordinal $\alpha$ such that $ZF^0 \vdash \Phi \leftrightarrow\Phi^{R_\alpha (A)}$ where $A$ is the set of atoms, then $\Phi$ is transferable.  (Note that $\alpha$ must be a constant absolute ordinal valued term.) For example, Form 5, $C(\aleph_0,\aleph_0,{\Bbb R})$ is the statement, "If $D$ is a denumerable collection of denumerable subsets of $\Bbb R$ then $\exists f: D \to \bigcup D$ such that $\forall d \in D$, $f(d)\in D$." This is equivalent to $(C(\aleph_0,\aleph_0,{\Bbb R}))^{R_{\omega_2}(A)}$. Similarly, the following forms are boundable and therefore transferable: 6, 13, 34, 35, 37, 38, 70, 74, 79, 92, 93, 94, 130, 142, 169, 170, 194, 197, 199, 203, 206, 211, 212, 222, 223, 251, 252, 272, 273, 280, 289, 305, 306, 307, 309, 313, 361, 362, 363, 364, 366, 367, 368, and 369. In addition all of these forms are true in all Fraenkel-Mostowski models since they make assertions about the standard part of the model only.

Pincus has strengthened the theorem of Jech/Sochor by the following:
Definition: A formula $\Phi(y_1, \ldots ,y_k)$ is surjectively boundable if it is a conjunction of formulas of the form $$(*) \; \begin{array}{r}\Phi_i = \forall x_1, \ldots ,x_n\big[\big(|\bigcup_{j=1}^n x_j|^- \le\sigma_i(y_1, \ldots ,y_k) \land \bigcup_{j=1}^n x_j\, \cap\, TCL(\bigcup_{m=1}^k y_m) = \emptyset \big)\\ \to \Psi(x_1, \ldots ,x_n,y_1, \ldots ,y_k)\big], \end{array}$$ where $\sigma(y_1, \ldots ,y_k)$ and $\Psi (x_1, \ldots ,x_n,y_1, \ldots,y_k)$ are boundable. A sentence is surjectively boundable if it is the existential closure of a surjectively boundable formula.
A formula is injectively boundable if it is a conjunction of formulas of the form (*) but with $|\bigcup_{j=1}^n x_j|^-$ replaced by $|\bigcup_{j=1}^n x_j|_-$ and a sentence is injectively boundable if it is the existential closure of an injectively boundable formula.

  An injectively boundable statement is transferable. Note that boundable statements are surjectively boundable and surjectively boundable statements are injectively boundable.

In many cases the following corollary suffices: A formula $\Phi$ is transferable if $\Phi$ is a conjunction of sentences of the form $$\forall x_1, \ldots, x_n(|\bigcup_{j=1}^n x_j|_- \le \sigma_i \to \Psi_i (x_1, \ldots ,x_n) )$$ where $\sigma_i$ is a boundable constant term and $\Psi$ is boundable. For example, Form 9, Dedekind finite $\cong$ finite, is injectively boundable since it is equivalent to $\forall x (|x|_- \le \omega \to x$is finite$)$.  Similarly, Form 18 and Form 128 are injectively boundable. Form 166 ($PC(\infty,2,\infty)$) is injectively boundable since it is equivalent to $(\forall x) (|x|_-\le\omega \to$ every infinite family of pairs whose union is $x$ has an infinite subfamily with a choice function$)$.  Similarly, Forms 132, 336, 342, 376, 377, 378 are injectively boundable.  Other injectively boundable forms are Forms 10, 31, 32, 80, 119, 350, 357, 358, 373, 374 (all using Howard/Solski [1993], lemma 3.5), Form 17 (Blass [1977a]), Form 64 (Pincus [1972a], 2B3), Forms 71, 82, 83 (Pincus [1972a], 2B4), Forms 98, 163, 216, (Pincus [1972a], 2B5), Form 217 (Pincus [1972a] 2B6), and Form 325 (Note 46).
In Pincus [1972a] the following four definitions and the transfer theorem that follow them also appear:

Definition: A formula $\Psi(x,w_1, \ldots ,w_n)$ is an onto property if it is provable that

1. If there is a function $f$ from $x$ onto $y$ and $\Psi(x,w_1,\ldots ,w_n)$ then $\Psi(y,w_1, \ldots w_n)$.
2. $x \in V_0 \to (\Psi(x,w_1, \ldots, w_n)\leftrightarrow\Psi^{V_0}(x,w_1, \ldots ,w_n) )$.

Definition: A term $\sigma(x,w_1, \ldots w_n)$ is invariant if

1. $\sigma(x,w_1, \ldots ,w_n) = \{z\in R_\alpha(x):\Phi^{R_\alpha(x)}(z,x,w_1, \ldots ,w_n)\}$ for some $\Phi$.
2. $\sigma(x,w_1, \ldots ,w_n) \ne \emptyset$ if $x$ is infinite
3. There is a definable operator $\hat { }$ such that if $f: y\to x$ is one to one then $\hat f : \sigma(y,w_1,\ldots,w_n) \to\sigma(x,w_1, \ldots ,w_n)$ is one to one.

Definition: A formula $Z(\overrightarrow{\beta})$ with ordinal parameters $\overrightarrow{\beta}$ is a term-choice formula if it is equivalent to a formula in the form $$\begin{array}{rl}(\forall y)&[(\Psi(y,\overrightarrow{\beta})\land (\forall x \in y)[x \hbox{ is well orderable}]) \\ &\to (\{\sigma(x,\overrightarrow{\beta}): x\in y\land\sigma(x,\overrightarrow{\beta}) \ne \emptyset\} \hbox{ has a choice function})] \end{array}$$ where $\Psi(y,\overrightarrow{\beta})$ is onto and $\sigma(y,\overrightarrow{\beta})$ is invariant.

Definition: A sentence is a term-choice sentence if it is equivalent to a statement in the form $\forall \overrightarrow{\beta} [\Omega(\overrightarrow{\beta}) \land Z(\overrightarrow{\beta})]$ where $Z(\overrightarrow{\beta})$ is term choice and $\Omega(\overrightarrow{\beta})$ is absolute.

For example, if we take $\Psi(y)$ to be the formula which asserts that $y$ is well orderable and $\sigma(x) =\{ z\in R_0 : (|x|=2 \land z\in x)\lor(|x|\ne 2 \land z=0)\}$ then $$(\forall y)\left[\Psi(y)\land (\forall x\in y)(\hbox{\(x\) is well orderable}) \to \left(\{ \sigma(x) : x\in y \land \sigma(x)\ne\emptyset\}\hbox{ has a choice function}\right)\right]$$ is a term choice statement and is equivalent to $C(WO,2)$ (Form 111).

Class 2 statements are transferable where a class 2 statement is a finite conjunction of

1. surjectively boundable statements,
2. "every set can be linearly ordered",
3. term-choice statements.

The following forms are class 2 statements: forms 30, 45, 46, 47, 48, 60, and 165 (all from Pincus [1972a], pp 736-737), 61, 62, 85, 88, 111, 122, 178, 185, 213, 250, 288, and 295.

Another transfer theorem from Pincus [1972a]: If $\Phi$ is a class 2 statement which is true in the Mostowski linear order model ($\cal N 3$) then $\Phi$ is $ZF$ consistent with the Boolean prime ideal theorem (Form 14).

Finally a transfer theorem from Pincus [1977a]: If $\Phi$ is a conjunction of any of the following seven kinds of statements and $\Phi$ has a Fraenkel-Mostowski model, then $\Phi$ has a $ZF$ model:

1. Injectively boundable statements.
2. For any ordinal number $\alpha$, $C(< \aleph_\alpha,\infty)$. (This includes, for any ordinal $\alpha$, $C(\aleph_\alpha,\infty)$ which is equivalent to $C(< \aleph_{\alpha +1},\infty)$.)
3. For every ordinal $\eta,\ DC^{< \eta}$. See the note below for definitions.
4. The ordering principle, OP (Form 30).
5. Term choice statements
6. The Boolean prime ideal theorem (Form 14).
7. The Hahn-Banach theorem (Form 52).

Note: $DC^{< \eta}$ is the statement $\forall \lambda < \eta$,$DC^\lambda$ and $DC^\lambda$ is "If for every $\alpha < \lambda$ every$R$-admissible $\alpha$ sequence extends to an $R$-admissible $\alpha + 1$ sequence then there is an $R$-admissible $\lambda$ sequence." (An $\alpha$ sequence is a function with domain $\alpha$  and an $\alpha$ sequence $\sigma$ is $R$-admissible if for all $\beta  <\alpha$, $\sigma \vert \beta\mathrel R \sigma(\beta)$. Note also that $DC^{\aleph_\alpha}$ is equivalent to Form 87 ($DC(\aleph_\alpha)$). We show that $DC(\aleph_\alpha)$ implies$DC^{\aleph_\alpha}$. The proof of the other implication is easier. Assume $R$ is a binary relation such that for every $\beta <\aleph_\alpha$ every $R$-admissible $\beta$ sequence of elements of $X$ extends to an $R$-admissible $\beta +1$ sequence of elements of $X$. Define the relation $R'$ by $\sigma \mathrel{R'} t$ iff ($\sigma$ is an $R$-admissible $\beta$ sequence for some $\beta < \aleph_\alpha$ and the $\beta + 1$ sequence $\sigma'$ which extends $\sigma$ for which$\sigma'(\beta) = t$ is $R$-admissible) or ($\sigma$ is not$R$-admissible). $R'$ satisfies the hypotheses of $DC(\aleph_\alpha)$ and therefore there is an $R'$-admissible $\aleph_\alpha$ sequence $f$. $f$ is $R$-admissible for if not, let $\beta$ be the least ordinal $<\aleph_\alpha$ for which $\lnot(f\vert\beta\mathrel R f(\beta))$. Since $f\vert\beta\mathrel{R'} f(\beta)$ it must be the case that $f\vert\beta$ is not $R$-admissible. Hence for some $\gamma < \beta$, $\lnot\left( (f\vert\beta)\vert \gamma\mathrel R f(\gamma)\right)$.  Since $(f\vert\beta)\vert\gamma =  f\vert\gamma$, this contradicts the definition of $\beta$. Forms that can be transferred using this theorem and which have not been mentioned in this note include forms 8, 14, 39, 40, 43, 44, 52, 86, and 87

In Pincus [1974c], Postscript 3, it is shown that Form 123 is transferable and according to Pincus [1976], Form 120 and Form 121 are transferable.

Howard-Rubin number: 103

Type: Transfer theorems

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