Pocket set theory

Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinals. The theory is authored by American mathematician Randall M. Holmes, although the basic idea was suggested by Rudy Rucker in his Infinity and the Mind..

Arguments supporting PST

There are at least two independent arguments in favor of a small set theory like PST.

1. One can get the impression from mathematical practice outside set theory that there are “only two infinite cardinals which demonstrably ‘occur in nature’ (the cardinality of the natural numbers and the cardinality of the continuum),” therefore “set theory produces far more superstructure than is needed to support classical mathematics.” Although it may be an exagerration (one can get into a situation in which she has to talk about arbitrary sets of real numbers or real functions), with some technical tricks a considerable portion of mathematics can be reconstructed within PST; certainly enough for most of its practical applications.
2. A second argument arises from foundational considerations. Most of mathematics can be implemented in standard set theory or one of its large alternatives. Set theories, on the other hand, are introduced in terms of a logical system; in most cases it is first-order logic. The syntax and semantics of first-order logic, on the other hand, is built on set-theoretical grounds. Thus, there is a foundational circularity, which forces us to choose as weak a theory as possible for bootstrapping. This line of thought, again, leads to small set theories.

Thus, there are reasons to think that Cantor's infinite hierarchy of the infinites is superfluous. Pocket set theory is a “minimalistic” set theory that allows for only two infinites: the cardinality $\scriptstyle {\aleph _{0}}$ of the (standard) natural numbers and the cardinality $\scriptstyle {2^{\aleph _{0}}}$ of the (standard) reals.

The theory

PST uses standard first-order language with identity and the binary relation symbol $\scriptstyle {\in }$. Ordinary variables are upper case X, Y, etc. In the intended interpretation, the variables these stand for classes, and the atomic formula $\scriptstyle {X\in Y}$ means "class X is an element of class Y". A set is a class that is an element of a class. Small case variables x, y, etc. stand for sets. A proper class is a class that is not a set. Two classes are equinumerous iff a bijection exists between them. A class is infinite iff it is equinumerous with one of its proper subclasses. The axioms of PST are

(A1) (extensionality) — Classes that have the same elements are the same.

$\forall z\,(z\in X\leftrightarrow z\in Y)\rightarrow X=Y$

(A2) (class comprehension) — If $\scriptstyle {\phi (x)}$ is a formula, then there exists a class the elements of which are exactly those sets x that satisfy $\scriptstyle {\phi (x)}$.

$\exists Y\forall x\,(x\in Y\leftrightarrow \phi (x))$

(A3) (axiom of infinity) — There is an infinite set, and all infinite sets are equinumerous.

$\exists x\,(\mathrm {inf} (x)\land \forall y\,(\mathrm {inf} (y)\rightarrow x\approx y))$

(inf(x) stands for “x is infinite”; $\scriptstyle {x\approx y}$ abbreviates that x is equinumerous with y.)

(A4) (limitation of size) – A class is a proper class if and only if it s equinumerous with all proper classes.

$\forall X\forall Y\,((\mathrm {pr} (X)\land \mathrm {pr} (Y))\leftrightarrow (\mathrm {pr} (X)\land X\approx Y))$

(pr(X) stands for “X is a proper class”.)

Remarks on the axioms

• Although different kinds of variables are used for classes and sets, the language is not many-sorted; sets are identified with classes of the same extension. Small case variables are used as mere abbreviations for various contexts; e.g.,

$\forall x\,\phi (x)\Leftrightarrow _{\mathrm {def} }\forall X\,(\mathrm {set} (X)\rightarrow \phi (X))$

• Since quantification in $\scriptstyle {\phi (x)}$ is not set-bound, (A2) is the comprehension scheme of Morse–Kelley set theory, not that of Von Neumann–Bernays–Gödel set theory. This extra strength of (A2) is used, e.g., in the definition of ordinals (not presented here).
• Since there is no axiom of pairing, it requires further justification that the Kuratowski pairs of any two sets exist and are sets. For this reason the equinumerosity of two classes cannot be assumed on the basis that there is a formula expressing a one-to-one correspondence between them.

Some theorems of PST

1. The Russell class is a proper class. ($\scriptstyle {\mathrm {R} =_{\mathrm {def} }\{x\,|\,x\notin x\}}$)

Proof: This is actually Russell's paradox. ∎

2. The empty class is a set. ($\scriptstyle {\emptyset =_{\mathrm {def} }\{x\,|\,x\neq x\}}$)

Proof: Suppose (towards a contradiction) that the empty class is a proper class. According to (A4), it is equinumerous with R; that is, R is empty. Let i be an infinite set, and consider the class $\scriptstyle {\{i\}}$. It is not equinumerous with $\scriptstyle {\emptyset }$, thus it is a set. It is finite, but its single element is infinite, thus it cannot be an element of itself. Therefore, it is an element of R. This contradicts that R is empty. ∎

3. The singleton class of the empty set is a set.

Proof: Suppose that $\scriptstyle {\{\emptyset \}}$ is a proper class. Then, according to (A4), every proper class is a singleton. Let i be an infinite set and consider the class $\scriptstyle {\{\emptyset ,i\}}$. It is neither a proper class (because it is not singleton) nor an element of itself (because it is neither empty nor infinite). Thus by definition $\scriptstyle {\{\emptyset ,i\}\in R}$, so R has at least two elements, $\scriptstyle {\emptyset }$ and $\scriptstyle {\{\emptyset ,i\}}$. This contradicts our assumption that proper classes are singletons. ∎

4. The Russell class is infinite.

Proof: Let $\scriptstyle {\mathrm {R} ^{-}=_{\mathrm {def} }\mathrm {R} \setminus \{\emptyset \}}$. Suppose that this class is a set. Then either $\scriptstyle {\mathrm {R} ^{-}\in \mathrm {R} ^{-}}$ or $\scriptstyle {\mathrm {R} ^{-}\notin \mathrm {R} ^{-}}$. In the first case by definition $\scriptstyle {\mathrm {R} ^{-}\in \mathrm {R} ^{-}}$, which is a contradiction. In the second case either $\scriptstyle {\mathrm {R} ^{-}\notin \mathrm {R} ^{-}}$, which is a contradiction, or $\scriptstyle {\mathrm {R} ^{-}=\emptyset }$. But $\scriptstyle {\mathrm {R} ^{-}}$ is not empty, because $\scriptstyle {\{\emptyset \}}$ is an element of it. ∎

5. Every finite class is a set.

Proof: Let X be a proper class. According to (A4), there exists an $\scriptstyle {F:X\longrightarrow \mathrm {R} }$ such that F is a bijection. This contains a pair $\scriptstyle {\langle x_{0},\emptyset \rangle }$, and for each member r of $\scriptstyle {\mathrm {R} ^{-}}$ , a pair $\scriptstyle {\langle x,r\rangle }$. Let $\scriptstyle {X^{-}=X\setminus \lbrace x_{0}\rbrace }$ and $\scriptstyle {F^{-}=F\setminus \lbrace \langle x_{0},\emptyset \rangle \rbrace }$. According to (A4), both of these classes exist. Now, $\scriptstyle {F^{-}:X^{-}\longrightarrow \mathrm {R} ^{-}}$ is a bijection. Thus, according to (A4), $\scriptstyle {X^{-}}$ is a proper class, too. Clearly, $\scriptstyle {X^{-}\subseteq X}$ and $\scriptstyle {X^{-}\neq X}$. Now, another application of (A4) shows that there exists a bijection $\scriptstyle {G:X\longrightarrow X^{-}}$. This proves that X is infinite.

Once the above facts are settled, the following results can be proved:

5. The class V of sets ($\scriptstyle {\mathrm {V} =_{\mathrm {def} }\{x\,|\mathrm {set} (x)\}}$) consists of all hereditarily countable sets.

6. Every proper class has the cardinality ${\mathfrak {c}}$.

Proof: Let i be an infinite set. The class $\scriptstyle {{\mathcal {P}}(i)}$ has the cardinality $\scriptstyle {2^{\aleph _{0}}}$. According to (A4), all proper classes have the cardinality $\scriptstyle {2^{\aleph _{0}}}$. ∎

7. The union class of a set is a set.

PST also verifies

• the continuum hypothesis

See (5) and (6) above.

• the axiom of replacement

This is actually part of the axiom of limitation of size.

• the axiom of choice

Both the class V of sets and the class Ord of ordinals are proper classes (because of Cantor's paradox and the Burali-Forti paradox, respectively, therefore there exists a bijection between them, which well-orders the universe.

The theory neither approves nor rejects well-foundedness.

Possible extensions

• Ha a PST-t bővítjük az ún. axiom of free construction, akkor bármely, az extenzionalitást kielégítő konzisztens halmazelméleti axiómarendszernek lesz belső modellje az elméletben.

• It is an unfriendly feature of PST that it cannot handle classes of sets of real numbers or classes of sets of real functions. However, it is not a necessary one. (A3) can be modified various ways to allow for various portions of the usual hierarchy of infinites, with or without supporting the continuum hypothesis. One example is

$\exists x\exists y\,(\mathrm {inf} (x)\land \mathrm {inf} (y)\land |{\mathcal {P}}(x)|\neq |{\mathcal {P}}(y)|\land \forall z(\mathrm {inf} (z)\rightarrow (|z|=|x|\lor |z|=|y|)))$

In this version, the cardinality of an infinite set is either $\aleph _{0}$ or $2^{\aleph _{0}}$, and the cardinality of a proper class is $2^{2^{\aleph _{0}}}$ (which means that the generalized continuum hypothesis holds.

External links

• Randall Holmes: Alternative Set Theories
• Randall Holmes: Notes on "Pocket Set Theory"