UDC 519.513/.514

MATHEMATICS

A. G. DRAGALIN, V. A. LYUBETSKII

CONSTRUCTION OF AN EFFECTIVELY UNATTAINABLE CARDINAL IN A NATURAL EXTENSION OF THE ZERMELO–FRAENKEL SYSTEM

(Presented by Academician P. S. Novikov on 26 XII 1968)

A natural extension of the Zermelo–Fraenkel system with or without the axiom of choice is considered, see \((^1)\) (denotation \(ZF\)), obtained by adding to the syntax of \(ZF\) a two-place relation \(\operatorname{Tr}(x,y)\) and a number of axioms such that \(\operatorname{Tr}(x,y)\) can be interpreted as the predicate: “the formula \(x\), under valuation \(y\), is true.” If the system \(ZF\) with the axiom on the existence of an unattainable cardinal is consistent, then the system \(ZF\operatorname{Tr}\) thus obtained is also consistent.

In this extension of \(ZF\) one constructs a cardinal \(a_0\), whose cofinality is \(\omega_0\), such that

\[ \forall a\!:\operatorname{Ord}(a), a<a_0 :\to : \overline{P}a<a_0 \]

and the ordinal \(a_0\) is not named by any term of \(ZF\) in which any sets of ranks less than \(a_0\) are taken as constants. Moreover, no sequence of ordinals cofinal in \(a_0\) can be named in this way either. In this sense \(a_0\) may be called an effectively unattainable cardinal. The cardinal \(a_0\) can be included as an element in some standard transitive model of \(ZF\), and it remains unnamed “inside” this model. The cardinal \(a_0\) refutes the conceivable hypothesis that for every cardinal whose cofinality is \(\omega_0\) there exists a term of \(ZF\) and such sets (of ranks smaller than this cardinal) as constants of this term that the term “names” this cardinal.

The proof follows naturally from the possibility of constructing in \(ZF\) an absolute (relative to the universe) natural model of \(ZF\).

Formalization of the syntax of \(ZF\) by means of \(ZF\). In what follows we shall regard the systems \(ZF\) and \(ZF\operatorname{Tr}\) as supplemented with Hilbert \(\iota\)-terms in the generally accepted way. \(\dot{x}, \dot{y}, \dot{M}, \dot{x}_1,\ldots,\dot{x}_n,\ldots\) are constant terms satisfying the formula expressing “to be a variable.” \(\varphi\) is a special variable ranging over objects satisfying the formula expressing “to be a formula.” \(t(x,n)\) is a formula expressing “\(x\) is a term with \(n\) variables.” \(\exists T\dot{x}_0\ldots\dot{x}_n\) and \(\forall T\dot{x}_0\ldots\dot{x}_n\) are abbreviations; for example, the first of them should be deciphered as follows: “there exists an \(x\) such that \(t(x,n)\) and the variables in \(x\) are the first \(n\) variables in the series \(\dot{x}_0\ldots\dot{x}_n\ldots\),” i.e. the existence is asserted of a term \(T\) with variables \(\dot{x}_0\ldots\dot{x}_n\). \(\exists\langle x_0\ldots x_n\rangle\) and \(\forall\langle x_0\ldots x_n\rangle\) are abbreviations; for example, the first of them is deciphered as follows: “there exists a tuple \(\langle x_0\ldots x_n\rangle\).” \(Ak_{ZF}(x)\) is the formula “saying” “\(x\) is an axiom of \(ZF\).” \(O\) is a special variable for a formula expressing: “to be a function defined on the set of all variables” (valuation).

\[ \binom{\dot{y}_1\ldots \dot{y}_n}{x_1\ldots x_n} \]

is a term (for fixed \(n\)) which, from the tuple \(\langle x_1\ldots x_n\rangle\), yields the set of pairs \(\{\langle \dot{y}_1 x_1\rangle\ldots\langle \dot{y}_n x_n\rangle\}\) (special valuation). \(xy\) is a two-place term; if \(x\) is a valuation and \(y\) is a special valuation of the form

\[ \binom{y_1\ldots y_n}{x_1\ldots x_n}, \]

then its value is such a valuation that its values on the

\(\dot y_1\ldots \dot y_n\) are \(x_1\ldots x_n\), and on the other variables are the same as for \(y\) in \(x\). We shall use this term when, instead of \(x\), an arbitrary valuation \(O\) is substituted, and instead of \(y\) a certain concrete special valuation

\[ \begin{pmatrix} \dot y_1\ldots \dot y_n\\ x_1\ldots x_n \end{pmatrix} \]

\((x,y)\) is also a two-place term of \(ZF\ Tr\) such that if \(x\) is a term with variables \(\dot x_1\ldots \dot x_n\) (denote it by \(T\dot x_1\ldots \dot x_n\)), and \(y\) is a special valuation for these variables

\[ \begin{pmatrix} \dot x_1\ldots \dot x_n\\ x_1\ldots x_n \end{pmatrix}, \]

then the value of the term is such a set \(z\) that

\[ Tr\left(\dot x=T\dot x_1\ldots \dot x_n,\, \begin{pmatrix} \dot x x_1\ldots \dot x_n\\ x x_1\ldots x_n \end{pmatrix}\right). \lim_{z} T\dot x_0\ldots \dot x_n, \begin{pmatrix} \dot x_1\ldots \dot x_n\\ x_1\ldots x_n \end{pmatrix} \]

is a three-place term whose value is the least upper bound of the ranks of the elements

\[ \left\{T\dot x_0\ldots \dot x_n,\, \begin{pmatrix} \dot x_0\ldots \dot x_n\\ x_0\ldots x_n \end{pmatrix}\ \middle|\ x_0\in z\right\}. \]

\(x'y\) is a two-place term such that if \(x\) is a valuation and \(y\) is a formula with free variables \(\dot y_1\ldots \dot y_n\), then the value of the term is the set of the values of the valuation \(x\) on \(\dot y_1\ldots \dot y_n\).

\(x_y\) is a two-place term; if \(x\) is a formula and \(y\) is a variable, then the value of this term is the formula obtained from \(x\) by relativizing all variables bound in \(x\) to \(y\).

\(\operatorname{cf}(\alpha)\) is a term which gives, for the ordinal \(\alpha\), the least ordinal cofinal with \(\alpha\). \(\operatorname{cf}(\gamma,\alpha)\) is a formula saying “\(\gamma\) is cofinal with \(\alpha\).” \(\operatorname{Ist}_M(x,y)\) is a \(ZF\) formula defining truth on the set \(M\) for the formulas \(x\) under valuations \(y\) from elements of \(M\).

In the formulas \(Tr(x,y)\) and \(\operatorname{Ist}_M(x,y)\), if \(x\) is a closed formula, then the argument \(y\) is omitted: \(Tr(x)\) and \(\operatorname{Ist}_M(x)\). The system \(ZF\ Tr\) is obtained from the system \(ZF\) by extending the syntax of the latter with the new relation \(Tr(x,y)\) (the definition of formula is changed correspondingly). The axioms of \(ZF\ Tr\) are the axioms of \(ZF\), with the axiom schema of substitution strengthened so that it applies not only to all formulas of \(ZF\), but also to formulas of \(ZF\ Tr\), and the following 7 axioms (syntactic axioms of \(ZF\ Tr\)):

\[ Tr\left(\dot x\in \dot y,\, \begin{pmatrix} \dot x\ \dot y\\ x\ y \end{pmatrix}\right)\equiv x\in y, \]

\[ Tr(\dot{\neg}\varphi,O)\equiv \neg Tr(\varphi,O),\qquad Tr(\varphi\&\psi,O)\equiv Tr(\varphi,O)\&Tr(\psi,O), \]

\[ Tr(\dot{\exists}\dot x\varphi\dot x,O)\equiv \exists x\,Tr\left(\varphi\dot x,\, O\begin{pmatrix} \dot x\\ x \end{pmatrix}\right) \]

and so on.

A cardinal \(\alpha\) is called inaccessible if it has the following property:

\[ \operatorname{cf}(\alpha)=\alpha\ \&\ \forall\alpha'.\,\alpha'<\alpha\to \overline{\overline{P\alpha'}}<\alpha. \]

Let us consider two ways of naming an ordinal by \(ZF\) terms:

\[ H_{\lim}(\alpha)\Leftrightarrow \exists n\exists T\dot x_0\ldots \dot x_n\exists\langle x_1\ldots x_n\rangle \exists\gamma.\, \operatorname{cf}(\gamma,\alpha)\ \&\ \lim_{\{\beta\mid \beta<\gamma\}} T\dot x_0\ldots \dot x_n, \]

\[ \begin{pmatrix} \dot x_0\ldots \dot x_n\\ \beta\ldots x_n \end{pmatrix} =\alpha, \]

\[ H(\alpha)\Leftrightarrow \exists n\exists T\dot x_1\ldots \dot x_n\exists\langle x_1\ldots x_n\rangle\, T\dot x_1\ldots \dot x_n,\, \begin{pmatrix} \dot x_1\ldots \dot x_n\\ x_1\ldots x_n \end{pmatrix} =\alpha. \]

An ordinal \(\alpha\) is called effectively inaccessible if

\[ \neg H(\alpha)\ \&\ \neg H_{\lim}(\alpha) \]

and

\[ \forall\alpha'.\,\alpha'<\alpha\to \overline{\overline{P\alpha'}}<\alpha. \]

Denote the conjunction of the last three formulas by \(In(\alpha)\). By a model here we always mean a standard and transitive model (see \((^2)\)). If a model, as a set, has the form

\[ \bigcup_{\beta<\alpha} R_\beta,\quad \text{where}\quad R_\beta=\bigcup_{\gamma<\beta} P(R_\gamma), \]

it is called natural. A model \(M\) is called absolute if

\[ \forall\varphi\forall O.\,O'\varphi\subseteq M\to Tr(\varphi,O)\equiv Tr\left(\varphi_M,\,O\begin{pmatrix} \dot M\\ M \end{pmatrix}\right). \]

As will follow from the proof of Lemma 2, absolute natural models form a class and, consequently, can be numbered (in increasing order with respect to \(\subseteq\)) by all ordinals: \(M_0,\ldots,M_\beta,\ldots\). The least ordinal not belonging to \(M_\beta\) will be denoted by \(\alpha_\beta\) and called the boundary of the model \(M_\beta\). Let \(M_\alpha, M_\beta, M_\gamma\), etc. be special variables for objects satisfying the formula expressing “to be an absolute natural model of ZF,” and let \(\alpha_\beta, \alpha_\gamma\), etc. be special variables for objects satisfying the formula expressing “to be the boundary of an absolute natural model of ZF.” \(M_0, M_1, M_2,\ldots\), etc. will be used as designations of constant terms that “define” the models \(M_0, M_1, M_2,\ldots\), etc. (the possibility of constructing such terms of \(\mathrm{ZF}'\mathrm{Tr}\) follows from Lemma 2), and \(\alpha_0,\alpha_1,\ldots,\alpha_\omega\) as designations of constant terms naming the boundaries of the models \(M_0,M_1,\ldots,M_\omega\).

Proposition 1. If the system ZF with the additional axiom on the existence of an inaccessible cardinal is consistent, then the system \(\mathrm{ZF}'\mathrm{Tr}\) is also consistent.

Proposition 2. It is consistent relative to \(\mathrm{ZF}'\mathrm{Tr}\) that if \(\alpha\) is the boundary of a natural model of ZF, then \(\operatorname{cf}(\alpha)<\alpha\).

Proposition 3. If \(\beta\) is of the second kind, then
\[ M_\beta=\bigcup_{\gamma<\beta} M_\gamma . \]
If \(\beta\) is of the first kind, then \(M_\beta\) is obtained by applying the process described in Lemma 2 to the set \(M_{\beta-1}\). \(\operatorname{cf}(\alpha_\beta)=\omega_0\) if \(\beta<\omega_1\). \(\operatorname{cf}(\alpha_\beta)\geq \operatorname{cf}(\beta)\).

Theorem. One can name by terms of \(\mathrm{ZF}'\mathrm{Tr}\) such ordinals \(\alpha_0\) and \(\alpha_1\) that they will be the boundaries of absolute natural models of ZF, \(M_0\) and \(M_1\) respectively, and \(\alpha_0\in M_1\), \(\operatorname{cf}(\alpha_0)=\omega_0\), \(\operatorname{cf}(\alpha_1)=\omega_0\), and \(\operatorname{In}(\alpha_0)\), \((\operatorname{In}(\alpha_0))_{M_1}\).

The proof follows from the lemmas.

Lemma 1.
\[ \underset{\mathrm{ZF}\,\mathrm{Tr}}{\vdash}\ \forall\varphi\, \operatorname{Ak}_{\mathrm{ZF}}(\varphi)\to \operatorname{Tr}(\varphi). \]

Lemma 2.
\[ \underset{\mathrm{ZF}'\mathrm{Tr}}{\vdash}\ \exists M\alpha:\ M_\alpha=\bigcup_{\beta<\alpha}R_\beta\ \&\ \operatorname{cf}(\alpha)=\omega_0\ \&\ \forall\varphi\,\forall O\, O'\varphi\subseteq M\to \]
\[ \to \operatorname{Tr}\left(\varphi_{\dot M}\equiv \varphi,\ O\binom{\dot M}{M}\right). \]

We number all terms \(t_1\ldots t_n\ldots\). For some ordinal \(\gamma_0\) we form the set of values of the term \(t_1\) on the set
\[ \bigcup_{\beta<\gamma_0} R_\beta \]
and denote the least upper bound of this set by \(\gamma_1\); consider the union of the sets of values of the terms \(t_1\) and \(t_2\) on the set
\[ \bigcup_{\beta<\gamma_1} R_\beta \]
and denote the least upper bound of this set by \(\gamma_2\), and in this way we construct \(\gamma_3\ldots\gamma_n\ldots\). The ordinal \(\gamma\) is \(\sup_n\{\gamma_n\}\). The set
\[ \bigcup_{\beta<\gamma} R_\beta \]
is the set required in the lemma.

Lemma 3.
\[ \underset{\mathrm{ZF}'\mathrm{Tr}}{\vdash}\ \forall\varphi\, \operatorname{Ak}_{\mathrm{ZF}}(\varphi)\to \operatorname{Ist}_{M_1}(\varphi). \]

Lemma 4.
\[ \underset{\mathrm{ZF}'\mathrm{Tr}}{\vdash}\ \neg H(\alpha_0),\qquad \underset{\mathrm{ZF}'\mathrm{Tr}}{\vdash}\ \operatorname{In}(\alpha_0),\qquad \underset{\mathrm{ZF}'\mathrm{Tr}}{\vdash}\ (\operatorname{In}(\alpha_0))_{M_1}. \]

Corollary 1. If ZF is consistent, then the following formula can be adjoined to ZF without contradiction:
\[ \exists M.\ M=\bigcup_{\beta<\alpha}R_\beta\ \&\ \forall\varphi\,\operatorname{Ak}_{\mathrm{ZF}}(\varphi)\to \operatorname{Ist}_M\varphi\ \&\ \exists a.\ a\in M\ \&\ \operatorname{In}_M(a), \]
where \(\operatorname{In}_M(a)\) is the formula obtained from the formula \(\operatorname{In}(a)\) by replacing in it every occurrence of the formula \(\operatorname{Tr}(x,y)\) by the formula \(\operatorname{Ist}_M(x,y)\).

Corollary 2. If ZF is consistent, then it is consistent to extend it in the following way: extend the syntax of ZF by a new constant \(\alpha_0\) and add the axioms: 1) \(\operatorname{card}(\alpha_0)\), \(\operatorname{cf}(\alpha_0)=\omega_0\); 2) \(\forall\alpha'.\,\alpha'<\alpha\to P\alpha'<\alpha_0\); 3\(_n\)) a list of axioms of the following form:
\[ \neg\exists x_1\ldots x_n'\,(\text{for all }x_i<\alpha_0\ \&\ T(x_1\ldots x_n)=\alpha_0). \]
where \(T\) is an arbitrary term of ZF.

Corollary 3. For every \(a_\beta\) there exist \(a_\beta\) (in cardinality) distinct natural models of ZF preceding (in the sense of the relation \(\equiv\)) \(M_\beta\).

All \(M_\beta\) are elementarily equivalent to one another and to the universe ZF Tr (i.e., the set of closed formulas true in \(M_\beta\) coincides with the set \(\{\varphi \mid \operatorname{Tr}(\varphi)\}\)). As Corollary 3 shows, before \(M_\beta\) (in the sense of \(\equiv\)) there are many other natural models of ZF. Are there among them (and how many?) such natural models that are elementarily equivalent to \(M_\beta\)?

Corollary 4. For every \(a_\beta\) there exist \(a_\beta\) (in cardinality) distinct natural models of ZF ordered by the relation \(\equiv\) (all of them \(\equiv M_\beta\)), and each of them is an elementary extension of any preceding one and is elementarily equivalent to \(M_\beta\) with respect to formulas of ZF, and their ordinal numbers are cofinal in \(\omega_0\).

Remark. Many of the assertions made, although formulated here for absolute natural models, are in fact also true for natural models, and even simply for sets of the form \(\bigcup_{\beta<\alpha} R_\beta\). The same consideration could also have been carried out in a “weaker” system than ZF Tr, for example in an extension of ZF obtained by adding the axiom of the existence of some model of ZF of a special kind or of the existence of an inaccessible cardinal, etc.

The author does not know whether there exists a cardinal \(a\) such that \(\neg H_{\lim}(a)\) and \(H(a)\), \(\operatorname{cf}(a)=\omega_0\), and \(\forall a' .\, a' < a \to \overline{P a'} < a\).

Moscow State University
named after M. V. Lomonosov

Received
12 XII 1968

REFERENCES

1. A. Fraenkel, I. Bar-Hillel, Foundations of Set Theory, Moscow, 1966.
2. J. C. Shepherdson, J. Symb. Logic, 18, 145 (1953).