A Theory of Particular Sets

TL;DR

This paper introduces TOPS, a set theory that restricts quantification to particular sets to avoid the indeterminacy issues present in traditional ZFC, using Natural Deduction.

Contribution

It proposes a novel set theory, TOPS, which limits quantification to particular sets, addressing semantic concerns in set theory.

Findings

TOPS successfully avoids indeterminacy in set quantification.

TOPS is formulated using Natural Deduction.

The approach offers a new perspective on set-theoretic semantics.

Abstract

ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sentences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets.

Equations

$$\begin{array}[]{lrcl}&V_{\alpha+1}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\mathcal{P}V_{\alpha}\\ \text{If $\alpha$ is a limit, }&V_{\alpha}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{\beta<\alpha}V_{\beta}\\ \text{In particular, }&V_{0}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\emptyset\end{array}$$

$${\mathsf{WR}}_{\in A}^{R/F}(a)$$

$$\begin{array}[]{lrcl}\text{Terms}&r,s,t,A,B&::=&x\hskip 5.0pt|\hskip 5.0pt\emptyset\hskip 5.0pt|\hskip 5.0ptA\cup B\hskip 5.0pt|\hskip 5.0pt\bigcup_{a\in A}B(a)\hskip 5.0pt|\hskip 5.0pt\{{{r}\mid\mid{\phi}}\}\hskip 5.0pt|\hskip 5.0pt\mathsf{UE}({A})\\ &&&\hskip 5.0pt|\hskip 5.0pt\mathsf{IR}({s}/{x.r})\hskip 5.0pt|\hskip 5.0pt\mathsf{WR}_{\in\!{A}}^{{x,y.\phi}/{z,Y.r}}({s})\hskip 5.0pt|\hskip 5.0pt\mathcal{P}A\\ \lx@intercol\text{Propositional formulas}\quad\quad\phi,\psi\hfil\lx@intercol&::=&\mathsf{False}\hskip 5.0pt|\hskip 5.0pt\mathsf{True}\hskip 5.0pt|\hskip 5.0pt\phi\vee\psi\hskip 5.0pt|\hskip 5.0pt\phi\wedge\psi\hskip 5.0pt|\hskip 5.0pt\phi\Rightarrow\psi\hskip 5.0pt|\hskip 5.0pt\neg\phi\\ &&&\hskip 5.0pt|\hskip 5.0pt\exists x\!\in\!A.\,\phi\hskip 5.0pt|\hskip 5.0pt\forall x\!\in\!A.\,\phi\hskip 5.0pt|\hskip 5.0pts=t\\ &&&\hskip 5.0pt|\hskip 5.0pt\mathsf{IsSet}({A})\hskip 5.0pt|\hskip 5.0pts\in A\hskip 5.0pt|\hskip 5.0pt\mathsf{W}_{\in\!{A}}^{{x,y.\phi}}({s})\end{array}$$

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$$\begin{array}[]{lcl}&\vdash^{\mathsf{p}}&\phi_{0}\\ &\vdash^{\mathsf{t}}&A\\ x&\vdash^{\mathsf{p}}&\phi_{1}\\ x&\vdash^{\mathsf{t}}&B\\ x,y&\vdash^{\mathsf{t}}&C\\ x,y,z&\vdash^{\mathsf{p}}&\psi\end{array}$$

$${\varphi }_0,x:A,{\varphi }_1,y:B,z:C⊢\psi$$

$${\varphi }_0\Rightarrow \forall x\in A.({\varphi }_1\Rightarrow \forall y\in B.\forall z\in C.\psi )$$

$$\mathsf{Decl}(\mathsf{nil})$$

$$\mathsf{Hyp}(\mathsf{nil})$$

$$\mathsf{Decl}(\Gamma ,x:A)$$

$$\mathsf{Hyp}(\Gamma ,x:A)$$

$$\mathsf{Decl}(\Gamma ,\varphi )$$

$$\mathsf{Hyp}(\Gamma ,\varphi )$$

$$F(\vec{t})$$

$$\begin{array}[]{rcl}\phi\Leftrightarrow\psi&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&(\phi\Rightarrow\psi)\wedge(\psi\Rightarrow\phi)\\ \exists!x\!\in\!A.\,P(x)&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\exists x\!\in\!A.\,(P(x)\wedge\forall y\!\in\!A.\,(P(y)\Rightarrow y=x))\\ A\subseteq B&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\forall x\!\in\!A.\,x\in B\\ \{{x\in A\mid P(x)}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{x\in A}\{{{x}\mid\mid{P(x)}}\}\\ \{{r}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\{{{r}\mid\mid{\mathsf{True}}}\}\\ \{{r_{0},\ldots,r_{n-1}}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\{{r_{0}}\}\cup\cdots\cup\{{r_{n-1}}\}\\ \{{F(x)\mid x\!\in\!A}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{x\in A}\{{F(x)}\}\\ \{{F(x,y)\mid x\!\in\!A,y\!\in\!B(x)}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{x\in A}\bigcup_{y\in B(x)}\{{F(x,y)}\}\hskip 10.00002pt\text{etc.}\\ \{{F(x)\mid x\!\in\!A\mid P(x)}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{x\in A}\{{{F(x)}\mid\mid{P(x)}}\}\\ \{{F(x,y)\mid x\!\in\!A,y\!\in\!B(x)\mid P(x,y)}\}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{x\in A}\bigcup_{y\in B(x)}\{{{F(x,y)}\mid\mid{P(x,y)}}\}\hskip 10.00002pt\text{etc.}\\ A\cap B&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\{{x\!\in\!A\mid x\in B}\}\\ \rotatebox[origin={c}]{180.0}{$\iota$}x\!\in\!A.\,P(x)&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\mathsf{UE}({\{{x\!\in\!A\mid P(x)}\}})\\ \bigcup\mathcal{A}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\bigcup_{X\in\mathcal{A}}X\\ {\phi}\;\rightarrowtriangle_{\mathsf{NC}}\;{r}&\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}&\phi\;\vee\;(\mathsf{IsSet}({r})\wedge\forall x\in r.\,\phi)\end{array}$$

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$$\begin{array}[]{llll}\lx@intercol\text{{Sethood}}\hfil\lx@intercol\\ \text{({Set Membership})}&\Gamma&\vdash&s\in A\ \Rightarrow\ \mathsf{IsSet}({A})\\ \text{({Set Extensionality})}&\Gamma&\vdash&(\mathsf{IsSet}({A})\wedge\mathsf{IsSet}({B})\wedge A\subseteq B\wedge B\subseteq A)\ \Rightarrow\ A=B\\ \vspace{-2ex}\\ \lx@intercol\text{{Empty set}}\hfil\lx@intercol\\ \text{({$\emptyset$\ Set})}&\Gamma&\vdash&\mathsf{IsSet}({\emptyset})\\ \text{({$\emptyset$\ Element})}&\Gamma&\vdash&t\in\emptyset\ \Leftrightarrow\ \mathsf{False}\\ \vspace{-2ex}\\ \lx@intercol\text{{Binary union}}\hfil\lx@intercol\\ \text{({$\cup$\ Set})}&\Gamma&\vdash&\mathsf{IsSet}({A\cup B})\\ \text{({$\cup$\ Element})}&\Gamma&\vdash&t\in A\cup B\ \Leftrightarrow\ (t\in A\vee t\in B)\\ \vspace{-2ex}\\ \lx@intercol\text{{Indexed union}}\hfil\lx@intercol\\ \text{({$\bigcup_{\in}$\ Set})}&\Gamma&\vdash&\mathsf{IsSet}({\bigcup_{x\in A}B(x)})\\ \text{({$\bigcup_{\in}$\ Element})}&\Gamma&\vdash&t\in\bigcup_{x\in A}B(x)\ \Leftrightarrow\ \exists x\!\in\!A.\,t\in B(x)\\ \vspace{-2ex}\\ \lx@intercol\text{{Conditional singleton}}\hfil\lx@intercol\\ \text{({CS\ Set})}&\Gamma&\vdash&\mathsf{IsSet}({\{{{r}\mid\mid{\phi}}\}})\\ \text{({CS\ Element})}&\Gamma&\vdash&t\in\{{{r}\mid\mid{\phi}}\}\ \Leftrightarrow\ (t=r\wedge\phi)\\ \vspace{-2ex}\\ \lx@intercol\text{{Unique element}}\hfil\lx@intercol\\ \text{({$\mathsf{UE}$\ $\mathsf{NC}$})}&\Gamma&\vdash&{(\exists!x\!\in\!A.\,\mathsf{True})}\;\rightarrowtriangle_{\mathsf{NC}}\;{\mathsf{UE}({A})}\\ \text{({$\mathsf{UE}$\ Specification})}&\Gamma&\vdash&(\exists!x\!\in\!A.\,\mathsf{True})\ \Rightarrow\ \mathsf{UE}({A})\in A\\ \vspace{-2ex}\\ \lx@intercol\text{{Iterative reach}}\hfil\lx@intercol\\ \text{({$\mathsf{IR}$\ Set})}&\Gamma&\vdash&\mathsf{IsSet}({\mathsf{IR}({s}/{F})})\\ \text{({$\mathsf{IR}$\ Base Generation})}&\Gamma&\vdash&s\in\mathsf{IR}({s}/{F})\\ \text{({$\mathsf{IR}$\ Step Generation})}&\Gamma&\vdash&t\in\mathsf{IR}({s}/{F})\ \Rightarrow\ F(t)\ \in\mathsf{IR}({s}/{F})\\ \text{({$\mathsf{IR}$\ Induction})}&\Gamma&\vdash&(P(s)\,\wedge\,(\forall x\!\in\!\mathsf{IR}({s}/{F}).\,(P(x)\Rightarrow P(F(x))))\,\wedge\,t\in\mathsf{IR}({s}/{F}))\ \Rightarrow\ P(t)\\ \vspace{-2ex}\\ \lx@intercol\text{{Well-founded elementhood}}\hfil\lx@intercol\\ \text{({$\mathsf{W}_{\in}$\ Generation})}&\Gamma&\vdash&(r\in A\wedge\forall y\!\in\!A.\,R(y,x)\Rightarrow\mathsf{W}_{\in\!{A}}^{{R}}({y}))\ \Rightarrow\ \mathsf{W}_{\in\!{A}}^{{R}}({r})\\ \text{({$\mathsf{W}_{\in}$\ Induction})}&\Gamma&\vdash&((\forall x\!\in\!A.\,(\forall y\!\in\!A.\,(R(y,x)\Rightarrow P(y)))\Rightarrow P(x))\,\wedge\,\mathsf{W}_{\in\!{A}}^{{R}}({t}))\ \Rightarrow\ P(t)\\ \vspace{-2ex}\\ \lx@intercol\text{{Well-founded recursion}}\hfil\lx@intercol\\ \text{({$\mathsf{WR}_{\in}$\ $\mathsf{NC}$})}&\Gamma&\vdash&{\mathsf{W}_{\in\!{A}}^{{R}}({s})}\;\rightarrowtriangle_{\mathsf{NC}}\;{\mathsf{WR}_{\in\!{A}}^{{R}/{F}}({s})}\\ \text{({$\mathsf{WR}_{\in}$\ Specification})}&\Gamma&\vdash&\mathsf{W}_{\in\!{A}}^{{R}}({s})\ \Rightarrow\ \mathsf{WR}_{\in\!{A}}^{{R}/{F}}({s})\,=\,F(s,\{{\mathsf{WR}_{\in\!{A}}^{{R}/{F}}({x})\mid x\!\in\!A\mid R(x,s)}\})\\ \vspace{-2ex}\\ \lx@intercol\text{{Powerset}}\hfil\lx@intercol\\ \text{({$\mathcal{P}$\ Set})}&\Gamma&\vdash&\mathsf{IsSet}({\mathcal{P}A})\\ \text{({$\mathcal{P}$\ Element})}&\Gamma&\vdash&B\in\mathcal{P}A\ \Leftrightarrow\ (\mathsf{IsSet}({B})\wedge B\subseteq A)\\ \vspace{-2ex}\\ \lx@intercol\text{{Choice}}\hfil\lx@intercol\\ \text{({Choice})}&\Gamma&\vdash&((\forall x\!\in\!A.\,\forall w\!\in\!A.\forall y\!\in\!B(x)\cap B(w).\,x=w)\,\wedge\,(\forall x\!\in\!A.\,\exists y\!\in\!B(x).\,\mathsf{True}))\\ &&&\quad\Rightarrow\ \exists Y\!\in\!\mathcal{P}\bigcup_{x\in A}B(x).\,\forall x\!\in\!A.\,\exists!y\!\in\!B(x).\,y\in Y\end{array}$$

$${\mathsf{Decl}}_{\gamma }(\mathsf{nil})$$

$$tensy\text{vbox}spread0.0ptplus0.0001fil\text{penalty}1\psi \Rightarrow (x\in A\Rightarrow \varphi (x))plus0.0001fil\text{vrule}height=0.25002pt,depth=0.25002pt,width=86.95815ptx\text{ fresh for }\psi ,A,\varphi \psi \Rightarrow \forall x\in A.\varphi (x)$$

Full text

1 Introduction

ZFC has long been established as a set-theoretic foundation of mathematics, but concerns about its meaningfulness have often been raised. Specifically, its use of unrestricted quantifiers seems to presuppose an absolute totality of sets. The purpose of this paper is to present a new set theory called TOPS that addresses this concern by speaking only about particular sets. Though weaker than ZFC, it is adequate for a large amount of mathematics.

To explain the need for a new theory, we begin in Section 2 by surveying some basic mathematical beliefs and conceptions. Section 3 presents the language of TOPS, and Section 4 the theory itself, using Natural Deduction. Section 5 adapts the theory to allow theorems with free variables.

Related work is discussed in Section 6. While TOPS is by no means the first set theory to use restricted quantifiers in order to avoid assuming a totality of sets, it is the first to do so using Natural Deduction, which turns out to be very helpful. This is apparent when TOPS is compared to a previously studied system [26] that proves essentially the same sentences but includes an inference rule that (we argue) cannot be considered truth-preserving. What saves TOPS from this pitfall is its use of Natural Deduction.

We sum up the paper’s argument in Section 7.

2 Motivating TOPS

2.1 Beliefs abut bivalence and definiteness

Before trying to determine whether a given sentence is true or false, one might wonder whether it has a truth value at all. A bivalent sentence is one that has an objective, determinate truth value (True or False)—regardless of whether anyone can ever know it.

Which mathematical sentences are bivalent? This is a contentious question, and there are various schools of thought. We shall consider in turn the languages of Peano arithmetic, second and third order arithmetic, and ZFC. In each of these, a sentence $\phi$ is built from atomic formulas using connectives and quantifiers. It is reasonable to say that $\phi$ is bivalent if the range of each quantifier is definite, i.e. clearly defined, with no ambiguity or haziness. So we examine each of our languages in the light of this principle.

• •

Arithmetical sentences, such as the Goldbach Conjecture, quantify over the set $\mathbb{N}=\{{0,1,\ldots}\}$ of natural numbers. The conception of $\mathbb{N}$ is “most restrictive”, allowing only what is generated by zero and successor. It is generally considered obvious (though finitists would disagree) that $\mathbb{N}$ is definite and so arithmetical sentences are bivalent. Let us accept this view and continue.

• •

Second order arithmetical sentences quantify over $\mathcal{P}\mathbb{N}$, or equivalently the set $2^{\mathbb{N}}$ of $\omega$-sequences of booleans. While some people (call them countabilists) believe in $\mathbb{N}$ but not in $2^{\mathbb{N}}$—see e.g. [46, 9]—the classical view is that, any definite set $X$ has a definite $\omega$-power, i.e. set $X^{\mathbb{N}}$ of $\omega$-sequences in $X$. Many adherents of this view believe also that $X^{\mathbb{N}}$ satisfies Dependent Choice, i.e. that for any $x\in X$ and entire111A relation $R$ from $X$ to $Y$ is entire when for every $x\!\in\!X$ there is $y\!\in\!Y$ such that $R(x,y)$. relation $R$ from $X$ to $X$, there is $s\!\in\!X^{\mathbb{N}}$ such that $s_{0}=x$ and $\forall n\!\in\!\mathbb{N}.\,R(s_{n},s_{n+1})$. These beliefs spring from a “most liberal” conception of $\omega$-sequences as consisting of arbitrary choices. Let us accept them and continue.

• •

Third order arithmetical sentences, such as the Continuum Hypothesis, quantify over $\mathcal{P}(2^{\mathbb{N}})$. While some people (call them $\omega$-powerists) believe in all the above but not in $\mathcal{P}(2^{\mathbb{N}})$—see e.g. [36, Section 3.2.3]—the classical view is that any definite set has a definite powerset. This is equivalent to saying that, for any definite sets $X$ and $Y$, the function set $Y^{X}$ is definite.222To justify ($\Rightarrow$), take $Y^{X}\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}\{{r\!\in\!\mathcal{P}(X\times Y)\mid\forall x\!\in\!X.\,\exists!y\!\in\!Y.\langle{x,y}\rangle\in r}\}$. To justify ($\Leftarrow$), take $\mathcal{P}X\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}\{{\{{x\!\in\!X\mid f(x)=1}\}\mid f\!\in\!\{{0,1}\}^{X}}\}$. Many adherents of this view believe also that $Y^{X}$ satisfies the Axiom of Choice, i.e. that for any entire relation $R$ from $X$ to $Y$, there is $f\!\in\!Y^{X}$ such that $\forall x\!\in\!X.\,R(x,f(x))$. These beliefs (to paraphrase Bernays [3]) spring from a “most liberal” conception of functions or subsets as consisting of arbitrary choices. Let us accept them and continue.

• •

Recall next the definition of the cumulative hierarchy. It associates to each ordinal $\alpha$ a set $V_{\alpha}$ by transfinite recursion:

[TABLE]

For example, consider the ordinal $\omega_{1}$. (It can be implemented as the set of well-ordered subsets of $\mathbb{N}$, modulo isomorphism.) Is $V_{\omega_{1}}$ definite? The classical view—that, for any definite ordinal $\alpha$, the set $V_{\alpha}$ is definite—may be argued by induction on $\alpha$ as follows. For the successor case: any definite set has a definite powerset. For the limit case: given a definite set $I$, and for each $i\in I$ a definite set $A_{i}$, surely the union $\bigcup_{i\in I}A_{i}$ is definite. Let us accept this argument and continue.

• •

ZFC has sentences that quantify over the ordinals. To emphasize: over all the ordinals, however large. That is a “most liberal” conception. Is it definite? Arguably not, because of the Burali-Forti paradox: the totality of all ordinals is itself well-ordered, with an order-type too large to belong to it. This powerful argument led Parsons to describe the language of ZFC as “systematically ambiguous” [30], and was taken up by Dummett [7, pages 316–317]:

The Burali-Forti paradox ensures that no definite totality comprises everything intuitively recognisable as an ordinal number, where a definite totality is one over which quantification always yields a statement that is determinately true or false.

However, it was rejected by Boolos [4] and Cartwright [6], who held that quantifying over the ordinals does not imply that they form a totality.333Even for those who hold this view, it is hard to deny that quantifying over classes of ordinals, as in Morse-Kelley class theory, implies that the ordinals form a totality. And harder still for higher levels of class.

To summarize: we have accepted the classical view that (for example) the set $V_{\omega_{1}}$ is definite and, by the Axiom of Choice, well-orderable. But we have not accepted that quantification over the ordinals is definite. Since we believe in many particular sets, such as $V_{\omega_{1}}$, our view may be called particularism. Other names given to it, or to similar views, are “restricted platonism” [3, 28], “liberal intuitionism” [35] and “Zermelian potentialism” [1]—see the discussion in [13, Section 4.3].

2.2 ZFC and purity

Recall next that ZFC adopts the following assumptions.

• •

Everything is a set. This rules out, for example, urelements and primitive ordered pairs.

• •

Membership is well-founded. This rules out, for example, sets $a$ such that $a=\{{a}\}$, known as “Quine atoms”.

We call them purity assumptions. They can be combined into the statement Everything is a well-founded pure set, where “pure” means that every element, and every element of an element, etc., is a set. The significance of these assumptions depends on how we read quantifiers in ZFC.

• •

The full-blown interpretation of $\forall x$ is “for any thing $x$ whatsoever”. On this reading, ZFC denies the existence of urelements and Quine atoms, which makes it unsound if an urelement or Quine atom does in fact exist (whatever that means).

• •

The pure interpretation of $\forall x$ is “for any well-founded pure set $x$”. On this reading, ZFC neither denies nor affirms the existence of urelements and Quine atoms. It simply refrains from speaking about them.

Each interpretation raises a question.

• •

Is quantification over all things definite? That seems implausible. For example, it is hard to believe that the question “Are there precisely seven urelements?” has an objectively correct answer.

• •

Is quantification over the well-founded pure sets definite? The answer is yes if and only if quantification over the ordinals is definite.444To justify ($\Leftarrow$), note that the quantifier “for any well-founded pure set $x$” is equivalent to “for any ordinal $\alpha$ and any $x\!\in\!V_{\alpha}$”, since any well-founded pure set belongs to $V_{\alpha}$, where $\alpha$ is the successor of its rank. To justify ($\Rightarrow$), use von Neumann’s implementation of the ordinals as the transitive pure sets that are well-ordered by membership.

A full-blown totalist believes in quantification over all things. A pure totalist does not believe this, but does believe in quantification over the well-founded pure sets, or equivalently over the ordinals.

2.3 Set theory for particularists

We can now state the problem. ZFC is a set theory for pure totalists, in the sense that pure totalists (reading it purely) would view its sentences as bivalent and its theorems as true. What would be a set theory for particularists (i.e. us), in this sense? Such a theory would enable us to state and prove facts about particular sets, e.g. “The set $V_{\omega_{1}}$ is well-orderable”, but prevent us from forming sentences that quantify over all sets (“Every set is well-orderable”) or all ordinals (“Ordinal addition is associative”) or all things (“Everything is equal to itself”).

This paper presents a system meeting these requirements called TOPS—short for “Theory Of Particular Sets”. It differs from ZFC in two substantive ways: it avoids unrestricted quantification, and does not adopt the purity assumptions. The two points of difference are, in principle, orthogonal, but the purity assumptions are needed in ZFC to make the quantifiers intelligible to pure totalists. In a system where quantifiers are always restricted, they are not needed.

3 The language of TOPS

3.1 Informal account

We begin by informally introducing the TOPS notation. To this end, let $\mathfrak{T}$ be the totality of all things. It may contain urelements, Quine atoms, inaccessible cardinals, measurable cardinals, etc. Let $\mathfrak{S}$ (a subcollection of $\mathfrak{T}$) be the totality of all sets.

The TOPS notation is listed as follows; most of it is familiar.

Let $a$ be a thing. We write $\mathsf{IsSet}({a})$ to say that $a$ is a set. 2. 2.

Let $A$ be a set and $b$ a thing. We write $b\in A$ to say that $b$ is an element of $A$. 3. 3.

Let $A$ and $B$ be sets. We say that $A$ is included in $B$, written $A\subseteq B$, when every element of $A$ is in $B$. 4. 4.

We write $\emptyset$ for the empty set. 5. 5.

Let $A$ and $B$ be sets. We write $A\cup B$ for the union of $A$ and $B$. 6. 6.

Let $A$ be a set and $(B(a))_{a\in A}$ a family of sets. We write $\bigcup_{a\in A}B(a)$ for the union of the family. 7. 7.

Let $A$ be a set and $P$ a predicate on it. We write $\{{x\!\in\!A\mid P(x)}\}$ for the set of $x\in A$ satisfying $P$. 8. 8.

Let $a$ be a thing.

• •

The singleton $\{{a}\}$ is the set whose sole element is $a$.

• •

For an assertion $\phi$, the conditional singleton $\{{{a}\mid\mid{\phi}}\}$ is $\{{a}\}$ if $\phi$ is true and $\emptyset$ otherwise. 9. 9.

Let $A$ be a set.

• •

The range of a function $f\colon A\to\mathfrak{T}$ is written $\{{f(x)\mid x\!\in\!A}\}$.

• •

For a predicate $P$ on $A$, the range of a function $f\colon\{{x\in A\mid P(x)}\}\to\mathfrak{T}$ is written $\{{f(x)\mid x\!\in\!A\mid P(x)}\}$. 10. 10.

Let $A$ be a singleton set. Its unique element is written $\mathsf{UE}({A})$. 11. 11.

Let $a$ be a thing and $F\colon\mathfrak{T}\to\mathfrak{T}$ a function, which may be written $x.F(x)$. The iterative reach of $F$ on $a$, written $\mathsf{IR}({a}/{F})$, is the set $\{{f^{n}(a)\mid n\in\mathbb{N}}\}$. 12. 12.

Let $A$ be a set and $R$ a relation on $A$, which may be written $x,y.R(x,y)$.

• •

We write $\mathsf{W}_{\in\!{A}}^{{R}}({a})$ when $a$ is an $R$-well-founded element of $A$. This property is generated inductively by the following rule: for $a\in A$, if every $b\in A$ such that $R(b,a)$ is $R$-well-founded, then so is $a$. By Dependent Choice, an element $a$ of $A$ is $R$-well-founded iff there is no sequence $(a_{n})_{n\in\mathbb{N}}$ of elements of $A$ such that $a_{0}=a$ and $\forall n\!\in\!\mathbb{N}.\,R(a_{n+1},a_{n})$.

• •

Let $F\colon A\times\mathfrak{S}\to\mathfrak{T}$ be a function, which may be written $z,Y.F(z,Y)$. For any $R$-well-founded element $a$ of $A$, we use well-founded recursion to define a thing written $\mathsf{WR}_{\in\!{A}}^{{R}/{F}}({a})$. Here is the recursive definition:

[TABLE] 13. 13.

Let $A$ be a set. Its powerset, written $\mathcal{P}A$, is the set of all subsets of $A$.

To ensure that our notation is defined in all cases, we adopt the following conventions.

Nonsense denotes the empty set. For a set $A$, we take

• •

$\mathsf{UE}({A})$ to be $\emptyset$ if $A$ is not singleton

• •

$\mathsf{WR}_{\in\!{A}}^{{R}/{F}}({a})$ to be $\emptyset$ if $a$ is not an $R$-well-founded element of $A$.

For an assertion $\phi$ and thing $a$, we say that $\phi$ is a Nonsense Convention prerequisite for $a$, written ${\phi}\;\rightarrowtriangle_{\mathsf{NC}}\;{a}$, when $\neg\phi$ implies $a=\emptyset$. This can also be stated in a positive way: either $\phi$ is true, or $a$ is a set such that, if it is inhabited, then $\phi$ is true. 2. 2.

Non-sets are treated like the empty set. Wherever the notation expects a set, any non-set provided is tacitly replaced by $\emptyset$. For example, if $a$ is not a set, we take

• •

the statement $b\in a$ to be false

• •

the statement $a\subseteq b$ to be true

• •

$\mathcal{P}a$ to be $\{{\emptyset}\}$.

Likewise, for a set $A$ and family of things $(b(a))_{a\in A}$, we take $\bigcup_{a\in A}b(a)$ to be $\bigcup_{a\in C}b(a)$, where $C$ is the set of $a\in A$ such that $b(a)$ is a set.

3.2 Formal syntax

We now present the syntax formally. Let $\mathsf{Vars}$ be an infinite set of variables, written $x,y,X,Y,\ldots$. The syntax of terms and that of propositional formulas are mutually inductively defined in Figure 1. We usually use a lowercase letter to suggest a thing, an uppercase one to suggest a set, and a calligraphic one (e.g. $\mathcal{A}$) to suggest a set of sets. But this is just for the sake of readability; the system does not distinguish these.

A declaration context $\gamma$ is a finite subset of $\mathsf{Vars}$. We define $\mathsf{nil}\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}\emptyset$ (it may be written as nothing) and $\gamma,x\stackrel{{\scriptstyle\mbox{\rm{\tiny def}}}}{{=}}\gamma\cup\{{x}\}$. We write

• •

${\gamma}\vdash^{\mathsf{t}}{t}$ to say that $t$ is a term over $\gamma$, i.e. a term whose free variables are all in $\gamma$

• •

${\gamma}\vdash^{\mathsf{p}}{\phi}$ to say that $\phi$ is a propositional formula over $\gamma$.

These two judgements are defined by mutual induction in Figure 2.

As usual, we identify terms up to renaming of bound variables ($\alpha$-equivalence). We write $\phi[t/x]$ and $s[t/x]$ for the capture-avoiding substitution of $t$ for $x$ in $\phi$ and in $s$ respectively.

A sentence is a closed propositional formula, so ${}\vdash^{\mathsf{p}}{\phi}$ says that $\phi$ is a sentence.

4 Definition of TOPS

4.1 Sequents

We formulate TOPS using Natural Deduction [33], which is a convenient framework for proofs in general, but especially for proofs involving restricted quantification and nested dependencies, as dependent type theory has shown [22].

In the middle of a Natural Deduction proof, an assertion is made subject to some hypotheses and variable declarations. This information can be presented as a sequent. For example, given formulas and terms

[TABLE]

the following is a sequent:

[TABLE]

It is read: “Assuming $\phi_{0}$, and $x$ in $A$, and $\phi_{1}$, and $y$ in $B$, and $z$ in $C$, we assert $\psi$.” Thus it has the same meaning as the sentence

[TABLE]

The part of a sequent that appears to the left of the $\vdash$ symbol is called a logical context. Let us now make this precise.

Definition 1

We define the set of logical contexts, and to each of these we associate

• •

a declaration context $\mathsf{Decl}({\Gamma})$

• •

and a list $\mathsf{Hyp}({\Gamma})$ of formulas over $\mathsf{Decl}({\Gamma})$, called hypotheses,

inductively as follows.

• •

$\mathsf{nil}$* is a logical context (it may be written as nothing), with*

[TABLE]

• •

If $\Gamma$ is a logical context and $x\not\in\mathsf{Decl}({\Gamma})$ and ${\mathsf{Decl}({\Gamma})}\vdash^{\mathsf{t}}{A}$, then $\Gamma,\,x\!:\!A$ is a logical context, with

[TABLE]

• •

If $\Gamma$ is a logical context and ${\mathsf{Decl}({\Gamma})}\vdash^{\mathsf{p}}{\phi}$, then $\Gamma,\,\phi$ is a logical context, with

[TABLE]

Definition 2

A sequent, written $\Gamma\vdash\psi$, consists of

• •

a logical context $\Gamma$

• •

and a formula $\psi$ over $\mathsf{Decl}({\Gamma})$, called the conclusion.

Below (Definition 3) we represent a sentence $\phi$ as the sequent $\vdash\phi$.

4.2 Abbreviations

We shall introduce some abbreviations to aid readability.

For $n\in\mathbb{N}$, an $n$-ary abstracted term is a term $F$ that may use additional variables $z_{0},\ldots,z_{n-1}$ not in $\mathsf{Vars}$. It is said to be over $\gamma$ when all its free variables other than these are in $\gamma$. Given terms $\vec{t}=t_{0},\ldots,t_{n-1}$ we write

[TABLE]

If $F$ and $\vec{t}$ are over $\gamma$ then so is $F(\vec{t})$. We likewise define $P(\vec{t})$, where $P$ is an $n$-ary abstracted propositional formula. We often abbreviate $\vec{x}.\,F(\vec{x})$ as $F$ and $\vec{x}.\,P(\vec{x})$ as $P$.

We write $A_{0}\cup\cdots\cup A_{n-1}$ and $\phi_{0}\vee\cdots\vee\phi_{n-1}$ and $\phi_{0}\wedge\cdots\wedge\phi_{n-1}$ in the usual way. For $n=0$ these are $\emptyset$ and $\mathsf{False}$ and $\mathsf{True}$ respectively, and for $n\geqslant 3$ we choose some arrangement of parentheses.

We make the following abbreviations.

[TABLE]

Here and throughout the paper, fresh variables are used for binding. For example, in the definition of $\exists!x\!\in\!A.\,P(x)$ above, $y$ must be fresh for $P$ and not be $x$.

4.3 Provability

TOPS consists of logical rules (Figure 3) and axiom schemes (Figure 4), which together define provability of sequents. Each rule and scheme refers to a logical context $\Gamma$. All (abstracted) terms and formulas mentioned are assumed to be over $\mathsf{Decl}({\Gamma})$. The logical rules are the usual introduction ($\mathcal{I}$) and elimination ($\mathcal{E}$) rules of Natural Deduction, adapted to restricted quantification. As there is no standard set of Natural Deduction rules for negation, the $\neg$Quodlibet and $\neg$Cases rules have been chosen for convenience and symmetry.

To complete the definition of Closed TOPS, we define provability of sentences.

Definition 3

A theorem is a sentence $\phi$ such that the sequent $\vdash\phi$ is provable.

4.4 Subsystems

Certain subsystems of TOPS may be of special interest. For example:

• •

the arithmetical fragment, which excludes $\mathsf{W}$, $\mathsf{WR}$, powerset and Choice

• •

the $W$-arithmetical fragment, which excludes powerset and Choice

• •

the intuitionistic fragment, which excludes negation and Choice

• •

TOPS without Choice.

Our formulation of TOPS has been arranged so as to keep these subsystems self-contained. However, this is a matter of taste; other presentations are possible and may have their own advantages.

5 Open TOPS

As we have seen, TOPS allows us to prove sentences. For a declaration context $\gamma$, we shall give a variant of TOPS called TOPS over $\gamma$ allowing us to prove formulas over $\gamma$. These systems are collectively called Open TOPS.

We first define logical context over $\gamma$ and sequent over $\gamma$ the same way as in Section 4.1, except that we replace $\mathsf{Decl}$ with $\mathsf{Decl}_{\gamma}$. The definition of the latter differs only in the following clause:

[TABLE]

Provability of sequents is defined as in Section 4.3. Finally, a theorem over $\gamma$ is a formula $\phi$ such that $\vdash\phi$ is provable. Some examples are shown in Figure 5.

To make sense of TOPS over $\gamma$ from a particularist viewpoint, let $\rho$ be a $\gamma$-valuation, which associates to each $x\in\gamma$ a thing $\rho(x)\in\mathfrak{T}$. Note that $\rho(x)$ may be an urelement, a Quine atom, an inaccessible cardinal, a measurable cardinal, etc. With respect to $\rho$, every formula or sequent over $\gamma$ is bivalent, every instance of an axiom scheme is true, and every instance of a logical rule preserves truth; so every provable sequent and every theorem over $\gamma$ is true.

Accordingly, although $\mathfrak{T}$ is not fixed, and we are free to admit to it anything we find credible and desirable, we would not admit things that violate a theorem of Open TOPS. For example, we would not admit a set that is not well-orderable, ordinals on which addition is not associative, or a thing that is not equal to itself. We deem such properties impossible, as they contradict the beliefs we have accepted.

6 Related work

6.1 General background

The following is just a selection, as the relevant literature is large.

• •

For a wide-ranging account of set-theoretic belief, see Maddy [20].

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For recent discussion of unrestricted quantification and the totality of sets, see e.g. the anthology [37] and the overviews [10, 44]. These issues have been considered in many contexts, such as reflection principles [47], categoricity theorems [5, 15, 21], modal logic [14] and categorical semantics [43, 2] .

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Following [17], it is usual to classify sentences by their number of alternations of unrestricted quantifiers. Mathias [23] extensively analyzes subsystems of ZFC, such as the theories of Mac Lane [19] and Kripke-Platek, that use this classification to limit (in various ways) the permissible use of Separation and/or Replacement. These are classical first-order theories, but other authors restrict also the use of excluded middle [45, 35, 34, 48, 11, 12, 8, 36, 41]. The legitimacy of classical vs intuitionstic reasoning for the totality of sets is further discussed in [16, 31, 18, 39].

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The formation of sequents in TOPS is adapted from dependent type theory. Indeed, extensional dependent type theory [22], without universes, can be seen as a subsystem of TOPS.

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Set theories in which terms and propositional formulas are simultaneously defined can be seen e.g. in [42, 29, 34].

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The important role of well-founded recursion in set theory has been argued e.g. in [49, 23, 38].

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The convention that “nonsense denotes the empty set” is followed by the proof assistants Isabelle/ZF [32, Section 7.2] and Metamath [40].

6.2 Comparison with Mayberry’s Local ZF

After writing the first version of this paper, the author learned of the system “Local ZF” studied by Mayberry [26, 25], with variants appearing in [27, 24]. It clearly proves the same sentences (modulo notational differences) as TOPS. Therefore the following valuable results about Local ZF given in [26] apply also to TOPS.

• •

It has a conservative extension that provides global choice function.

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It has a further conservative extension that allows formulas with unrestricted quantification, governed by intuitionistic logic without induction and various set-theoretic axioms.

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It can be embedded in a weak fragment of ZFC, whose consistency is provable in ZFC.

Despite the close similarity, Local ZF differs from TOPS in several ways. It adopts the purity assumptions, unlike TOPS, and lacks the modular arrangement mentioned in Section 4.4. More importantly, it is not quite a theory of particular sets, as it uses open formulas rather than sequents.

To see the problem this causes, recall first the conventional understanding of open formula systems. A formula’s denotation is given with respect to a valuation, i.e. map $\mathsf{Vars}\to\mathfrak{T}$. A universally true formula is one that is true with respect to every valuation. A theory is deemed acceptable when each axiom is universally true and each inference rule preserves universal truth.

For example, consider the following Hilbert-style counterpart of our $\forall_{\in}\mathcal{I}$ rule:

[TABLE]

(Rule III(3) in [26] is similar.) For a totalist, it clearly preserves universal truth. But for a particularist, who doubts the notion of universal truth, it is hard to see what kind of truth this rule can be said to preserve.

By contrast, the logical rules for TOPS preserve not universal truth but simply truth—that is what makes them immediately convincing. Sequents are bivalent just as sentences are, because every variable is declared. Specifically, the premise of the $\forall_{\in}\mathcal{I}$ rule in Figure 3 would not be a sequent if $x\!:\!A$ were replaced by $x\!\in\!A$.

What about Open TOPS? The reason TOPS over $\gamma$ provides an acceptable way of reasoning about a particular $\gamma$-valuation $\rho$ is that each axiom is true with respect to $\rho$, and each rule preserves truth with respect to $\rho$. No other valuation need be considered.

Thus, whereas one can present first-order logic either in Hilbert style or via Natural Deduction, this is not the case for TOPS. The latter must be presented via Natural Deduction, distinguishing between a declaration $x\!:\!A$ and a mere hypothesis of the form $x\!\in\!A$, in order to have a notion of truth that every rule preserves.

7 Conclusion

Despite the popularity of ZFC as a set-theoretic foundation, the Burali-Forti paradox (for example) raises serious concerns about its meaningfulness. As an alternative, we propose TOPS, a Natural Deduction system that avoids the problem by speaking only about particular sets. This allows us to maintain, as particularists, that its sentences are bivalent and its theorems objectively true. An example theorem is the well-orderability of $V_{\omega_{1}}$. Theorems of Open TOPS have a different character; they lack objective meaning but are true with respect to any valuation.

Acknowledgements

I thank Ehud Hrushovski, Adrian Mathias and Norman Megill for helpful comments.