UDC 517.12 : 164.02 + 519.5

MATHEMATICS

V. N. GRISHIN

CONSISTENCY OF A CERTAIN FRAGMENT OF QUINE’S SYSTEM NF

(Presented by Academician P. S. Novikov on 11 IV 1969)

Let us denote by \(\mathrm{NF}_n\) the fragment of NF (New Foundations) containing those axioms of NF (see \((^1)\)) whose stratifiability (stratification) can be established using numbers not exceeding \(n\), i.e. the numbers \(1, 2, 3, \ldots, n\). For example, to check the stratification of the formula

\[ \forall z \exists y \forall x (x \varepsilon y \equiv \forall t (t \varepsilon x \to t \varepsilon z)) \]

the numbers \(1, 2, 3\) are required (the variables \(t, x, z, y\) are assigned the numbers \(1, 2, 2, 3\), respectively). The axiom of NF mentioned will therefore be an axiom of \(\mathrm{NF}_3\).

The consistency of \(\mathrm{NF}_3\) is proved. We note without proof that from the consistency of \(\mathrm{NF}_5\) the consistency of NF follows.

Let \(T_n\) denote type theory without the axiom of infinity, having \(n\) type variables, i.e. containing \(n\) alphabets of variables. In addition to the logical axioms, the axioms of \(T_n\) are the formulas

\[ \forall x^i (x^i \varepsilon y^{i+1}) \equiv (x^i \varepsilon z^{i+1}) \to y^{i+1} = z^{i+1}, \]

\[ \exists y^{i+1} \forall x^i (x^i \varepsilon y^{i+1} \equiv A(x^i)) \quad (i = 1, 2, \ldots, n - 1), \]

where \(x^i, y^{i+1}, z^{i+1}\) are variables of the \(i\)-th and \((i+1)\)-st alphabets (types), respectively, and the variable \(y^{i+1}\) does not occur freely in the formula \(A(x^i)\). We note that the special (nonlogical) axioms of \(\mathrm{NF}_n\) are the formulas obtained from the stated axioms of \(T_n\) by erasing the upper, i.e. type, index.

By a result of Specker \((^2)\), in order to prove the consistency of \(\mathrm{NF}_3\) it suffices to find a model \(M = (X_1, X_2, X_3, \in, =)\) (where \(X_2 \subseteq \{A : A \subseteq X_1\}\) and \(X_3 \subseteq \{A : A \subseteq X_2\}\)) for \(T_3\) admitting an \(\varepsilon\)-isomorphism, i.e. admitting the existence of such one-to-one functions \(f_1 : X_1 \to X_2\) and \(f_2 : X_2 \to X_3\) that

\[ a \in b \equiv f_1(a) \in f_2(b) \]

for any \(a \in X_1,\ b \in X_2\). The existence of an \(\varepsilon\)-isomorphism is equivalent to the existence of an isomorphism \(\varphi\) with respect to inclusion \((\subseteq)\) between the domains \(X_2\) and \(X_3\). If one defines \(f_1\) by the equivalence

\[ b = f_1(a) \equiv \{b\} = \varphi(\{a\}) \]

for any \(a \in X_1\) and \(b \in X_2\) (where \(\{a\}\) and \(\{b\}\) denote one-element sets containing \(a\) and \(b\), respectively) and sets \(f_2 = \varphi\), then the system \(f_1, f_2\) is an \(\varepsilon\)-isomorphism.

The following lemma gives some sufficient conditions for two families of subsets \(\mathfrak{A}\) and \(\mathfrak{B}\) of sets \(X\) and \(Y\) to be isomorphic with respect to inclusion.

Lemma. Let two families of subsets \(\mathfrak{A}\) and \(\mathfrak{B}\) of sets \(X\) and \(Y\) satisfy the following conditions:

\(1^\circ.\ |\mathfrak{A}| = |\mathfrak{B}| = \aleph_0,\) where \(|\mathfrak{A}|\) and \(|\mathfrak{B}|\) are the cardinalities of \(\mathfrak{A}\) and \(\mathfrak{B}\), respectively.

\(2^\circ\). \(\mathfrak A\) and \(\mathfrak B\) are closed under the operations of intersection and complement to \(X\) and \(Y\), respectively.

\(3^\circ\). \(\mathfrak A\) and \(\mathfrak B\) contain all one-element subsets of \(X\) and \(Y\), respectively.

\(4^\circ\). For every infinite \(A \in \mathfrak A\) (or \(B \in \mathfrak B\)) there exists \(A_1 \in \mathfrak A\) \((B_1 \in \mathfrak B)\) such that \(A_1 \subseteq A\) and \(|A_1| = |A - A_1|\) \((B_1 \subseteq B\) and \(|B_1| = |B - B_1|)\).

Then the families \(\mathfrak A\) and \(\mathfrak B\) are isomorphic with respect to inclusion.

Proof. Construct the following enumerations without repetitions

\[ A_1, A_2, A_3, \ldots, A_k, \ldots,\qquad B_1, B_2, B_3, \ldots, B_k, \ldots \]

of the families \(\mathfrak A\) and \(\mathfrak B\), respectively, so that the equality

\[ \left| A_1^{\sigma_1} \cap \ldots \cap A_k^{\sigma_k} \right| = \left| B_1^{\sigma_1} \cap \ldots \cap B_k^{\sigma_k} \right| \tag{1} \]

holds for every \(k\) and every set \(\sigma_1, \ldots, \sigma_k\) of zeros and ones, where

\[ A^\sigma = \begin{cases} A, & \text{if } \sigma = 0,\\ -A, & \text{if } \sigma = 1. \end{cases} \]

For this purpose fix some enumerations of the families \(\mathfrak A\) and \(\mathfrak B\). In view of condition \(2^\circ\), \(X \in \mathfrak A\) and \(Y \in \mathfrak B\). Put \(A_1 = X\) and \(B_1 = Y\). Obviously, equality (1) holds for \(k = 1\). Suppose that \(A_1, \ldots, A_n, B_1, \ldots, B_n\) have already been defined, contain no repetitions, and satisfy equality (1).

Let \(n\) be odd. Put \(A_{n+1}\) equal to the first element, in the fixed enumeration of the family \(\mathfrak A\), that is different from \(A_1, \ldots, A_n\). Denote by \(\mathfrak D_n\) the family consisting of sets of the form \(A_1^{\sigma_1} \cap \ldots \cap A_n^{\sigma_n}\), where \(\sigma_1, \ldots, \sigma_n\) runs through all possible sets of zeros and ones. Denote by \(\mathfrak D_n'\) the family of sets of the form \(B_1^{\sigma_1} \cap \ldots \cap B_n^{\sigma_n}\). An arbitrary element of \(\mathfrak D_n\) is denoted by \(\Delta\), and the corresponding element of \(\mathfrak D_n'\), i.e. the element with the same set \(\sigma_1, \ldots, \sigma_n\), is denoted by \(\Delta'\). For an arbitrary element \(\delta\) of the family \(\mathfrak D_n\) or \(\mathfrak D_n'\) and an arbitrary subset \(C\) of the set \(X\) or \(Y\), define

\[ \tau_\delta^C = |C \cap \delta|,\qquad \chi_\delta^C = |-C \cap \delta|, \]

where \(-C\) is the complement of the set \(C\). Consider the numbers \(\tau_\Delta^{A_{n+1}}\) and \(\chi_\Delta^{A_{n+1}}\) \((\Delta \in \mathfrak D_n)\). The numbers \(\tau_\Delta^{A_{n+1}}\) and \(\chi_\Delta^{A_{n+1}}\) characterize the “behavior” of the set \(A_{n+1}\) with respect to the “pieces” \(\Delta\) from \(\mathfrak D_n\). Define \(B_{n+1}\) as that element of \(\mathfrak B\) whose “behavior” with respect to \(\mathfrak D_n'\) coincides with the “behavior” of \(A_{n+1}\) with respect to \(\mathfrak D_n\). More precisely, \(B_{n+1}\) is equal to the first element, in the fixed enumeration of the family \(\mathfrak B\), such that

\[ \tau_{\Delta'}^{B_{n+1}} = \tau_\Delta^{A_{n+1}},\qquad \chi_{\Delta'}^{B_{n+1}} = \chi_\Delta^{A_{n+1}} \]

for every \(\Delta \in \mathfrak D_n\).

Let us prove the existence of such a \(B_{n+1}\). We have:

\[ \tau_\Delta^{A_{n+1}} + \chi_\Delta^{A_{n+1}} = |\Delta| \]

for every \(\Delta \in \mathfrak D_n\), and \(|\Delta| = |\Delta'|\). By \(1^\circ\) and \(3^\circ\), the numbers \(\tau_\Delta^{A_{n+1}}\) and \(\chi_\Delta^{A_{n+1}}\) do not exceed \(\aleph_0\). If both of the numbers mentioned are equal to \(\aleph_0\), then the existence of \(\Delta' \cap B_{n+1}\) follows from condition \(4^\circ\). If one of these numbers is less than \(\aleph_0\), then the existence of \(\Delta' \cap B_{n+1}\) follows from \(2^\circ\) and \(3^\circ\). Obviously, equality (1) holds for \(k = n + 1\). Let us prove that \(B_{n+1} \ne B_i\) for every \(i = 1, \ldots, n\). If \(B_{n+1} = B_i\) for some \(i \le n\), then

\[ \tau_\Delta^{A_{n+1}} = \tau_{\Delta'}^{B_{n+1}} = \tau_{\Delta'}^{B_i} = \tau_\Delta^{A_i}, \tag{2} \]

\[ \chi_\Delta^{A_{n+1}} = \chi_{\Delta'}^{B_{n+1}} = \chi_{\Delta'}^{B_i} = \chi_\Delta^{B_i}. \tag{3} \]

From these equalities it follows that \(A_{n+1}=A_i\). Indeed, let \(a\in A_i\). Then there is a \(\Delta\in\mathfrak D_n\) such that
\[ a\in \Delta=A_1^{\sigma_1}\cap\cdots\cap A_i^{\sigma_i}\cap\cdots\cap A_n^{\sigma_n}. \]
Since \(a\in A_i\), we have \(\sigma_i=0\), i.e. \(\Delta\subseteq A_i\), or \(\chi_\Delta^{A_i}=0\). But \(\chi_\Delta^{A_i}=\chi_\Delta^{A_{n+1}}\) (in view of (3)). Therefore \(\Delta\subseteq A_{n+1}\), i.e. \(a\in A_{n+1}\). Let \(a\in -A_i\). Then there is a \(\Delta\in\mathfrak D_n\) such that
\[ a\in \Delta=A_1^{\sigma_1}\cap\cdots\cap A_i^{\sigma_i}\cap\cdots\cap A_n^{\sigma_n}. \]
Next we derive: \(\sigma_i=1\), \(\Delta\subseteq -A_i\), \(\tau_\Delta^{A_i}=0\), and \(\Delta\subseteq -A_{n+1}\), and, finally, \(a\in -A_{n+1}\). However, by virtue of its choice, \(A_{n+1}\) is distinct from \(A_1,\ldots,A_n\).

Let \(n\) be even. Put: \(B_{n+1}\) equal to the first, in the fixed enumeration of the family \(\mathfrak B\), element distinct from \(B_1,\ldots,B_n\). \(A_{n+1}\) is defined in the same way as \(B_{n+1}\) was defined in the case of odd \(n\).

The sequences constructed are, obviously, enumerations, and moreover without repetitions, of the families \(\mathfrak A\) and \(\mathfrak B\), satisfying equality (1). From condition (1) it follows that
\[ A_i\subseteq A_j \equiv B_i\subseteq B_j \]
for any \(i\) and \(j\). The lemma is proved.

It remains to find such a model \(M=(X_1,X_2,X_3,\in,=)\) for \(T_3\) that the families of sets \(X_2\) and \(X_3\) satisfy the condition of the lemma proved. Then the domains \(X_2\) and \(X_3\) will be isomorphic with respect to inclusion, and the model \(M\) admits an \(\varepsilon\)-isomorphism.

Consider the theory denoted by \(T_5^\infty\), obtained from the theory of types \(T_5\) by extending both the language \(T_5\) by adding an infinite list of constants \(c_1,c_2,c_3,\ldots,c_m,\ldots\), belonging to type 1, and the axioms of \(T_5\) by introducing the following axioms:

(I) \(c_i\ne c_j\) for \(i\ne j\).

(II) \(\forall x^{i+1}\operatorname{Fin}(x^{i+1})\) \((i=1,2,3)\), where \(\operatorname{Fin}(x^{i+1})\) is the formula of \(T_5\) expressing the predicate “\(x^{i+1}\) is finite.”

Obviously, the theory \(T_5^\infty\) is consistent. From the consistency of the theory \(T_5^\infty\) there follows the existence of a countable model for it. The part of this model formed by the domains \(X_1,X_2,X_3\) is a model for \(T_3\). Obviously, conditions \(1^0,2^0,3^0\) of the lemma are fulfilled for the domains \(X_2\) and \(X_3\). It remains to verify condition \(4^0\). Denote by \(\operatorname{Zr}(x^{i+1})\) \((i=1,2)\) the formula of the theory \(T_5\) expressing the following judgment: “either the set \(x^{i+1}\), or \(x^{i+1}\) with one element removed, decomposes into two equipotent disjoint sets.” In the theory \(T_5\) the formula is provable
\[ \forall x^{i+1}\bigl(\operatorname{Fin}(x^{i+1})\to \operatorname{Zr}(x^{i+1})\bigr)\qquad (i=1,2). \]
In the theory \(T_5^\infty\), in view of axiom (II), the formula is provable
\[ \forall x^{i+1}\operatorname{Zr}(x^{i+1})\qquad (i=1,2). \tag{4} \]
In the model for \(T_5^\infty\) formula (4) is satisfied, and therefore for every \(A\in X_{i+1}\) \((i=1,2)\) either \(A\), or \(A\) with one element removed, decomposes into two equipotent disjoint sets. Hence the validity of condition \(4^0\) of the lemma follows, if \(X_2\) is taken as \(\mathfrak A\), and \(X_3\) as \(\mathfrak B\).

All-Union Institute
of Scientific and Technical Information
Moscow

Received
13 III 1969

REFERENCES

¹ W. V. O. Quine, Am. Math. Monthly, 44, 70 (1937). ² E. Specker, Typical Ambiguity, Proc. Intern. Congr., 1960, p. 116.