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 Randall M. Holmes.

Arguments supporting PST

There are at least two independent arguments in favor of a small set theory like PST. 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 (standard) natural numbers and the cardinality of the (standard) reals.

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

Some PST theorems

Once the above facts are settled, the following results can be proved: PST also verifies the: The well-foundedness of all sets is neither provable nor disprovable in PST.

Possible extensions