Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum). The theory was first suggested by Rudy Rucker in his Infinity and the Mind. The details set out in this entry are due to the American mathematician M. Randall Holmes.
Arguments supporting PST There are at least two independent arguments in favor of a small set theory like PST.
One can get the impression from mathematical practice outside set theory that there are only two infinite cardinals which demonstrably are used "in classical mathematical practice outside set theory", (the cardinality of the natural numbers and the cardinality of the continuum), and therefore that "set theory produces far more superstructure than is needed to support classical mathematics". Although it may be an exaggeration (one can get into a situation in which one 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. 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 ℵ 0 {\displaystyle \scriptstyle {\aleph _{0}}} of the (standard) natural numbers and the cardinality 2 ℵ 0 {\displaystyle \scriptstyle {2^{\aleph _{0}}}} of the (standard) reals.
Theory PST uses standard first-order language with identity and the binary relation symbol ∈ {\displaystyle \scriptstyle {\in }} . Ordinary variables are upper case X, Y, etc. In the intended interpretation, the variables these stand for classes, and the atomic formula X ∈ Y {\displaystyle \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.
∀ z ( z ∈ X ↔ z ∈ Y ) → X = Y {\displaystyle \forall z\,(z\in X\leftrightarrow z\in Y)\rightarrow X=Y}
(A2) (class comprehension) — If ϕ ( x ) {\displaystyle \scriptstyle {\phi (x)}} is a formula, then there exists a class the elements of which are exactly those sets x that satisfy ϕ ( x ) {\displaystyle \scriptstyle {\phi (x)}} .
∃ Y ∀ x ( x ∈ Y ↔ ϕ ( x ) ) {\displaystyle \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.
∃ x ( i n f ( x ) ∧ ∀ y ( i n f ( y ) → x ≈ y ) ) {\displaystyle \exists x\,(\mathrm {inf} (x)\land \forall y\,(\mathrm {inf} (y)\rightarrow x\approx y))}
(inf(x) stands for “x is infinite”; x ≈ y {\displaystyle \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 is equinumerous with all proper classes.
∀ X ∀ Y ( ( p r ( X ) ∧ p r ( Y ) ) ↔ ( p r ( X ) ∧ X ≈ Y ) ) {\displaystyle \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 having the same extension. Small case variables are used as mere abbreviations for various contexts; e.g.,
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