Pocket set theory


Pocket set theory is an alternative set theory in which there are only two infinite cardinal numbers, ℵ0 and c. 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 Randall M. Holmes.

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’,” therefore “set theory produces far more superstructure than is needed to support classical mathematics.” Although it may be an exaggeration, 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 of the natural numbers and the cardinality of the reals.

The theory

PST uses standard first-order language with identity and the binary relation symbol. Ordinary variables are upper case X, Y, etc. In the intended interpretation, the variables these stand for classes, and the atomic formula 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

Remarks on the axioms

;1. The Russell class is a proper class.
;2. The empty class is a set.
;3. The singleton class is a set.
;4. is infinite.
;5. Every finite class is a set.
Once the above facts are settled, the following results can be proved:
;6. The class V of sets consists of all hereditarily countable sets.
;7. Every proper class has the cardinality.
;8. The union class of a set is a set.
PST also verifies the:
The well-foundedness of all sets is neither provable nor disprovable in PST.

Possible extensions