Pocket set theory¶
Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum).
Core Idea¶
Pocket set theory is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum). Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the.
Scope of Application¶
-
Arguments supporting 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.
-
Arguments supporting PST. 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.
-
Arguments supporting PST. Pocket set theory is a “minimalistic” set theory that allows for only two infinites: the cardinality \scriptstyle{\aleph0} of the (standard) natural numbers and the cardinality \scriptstyle{2^{\aleph0}} of the.
-
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.
-
Remarks on the axioms. Small case variables are used as mere abbreviations for various contexts; e.g.,.
Clarity¶
A clear use of Pocket set theory names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum).
Manages Complexity¶
Pocket set theory compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—a model for pocket set theory is given by taking the sets of pocket set theory to be the constructible elements of HC (the set of hereditarily countable sets), and the classes to be the constructible subsets of HC.—and the practical consequence—then by (A4), every proper class is a.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: Pocket set theory (PST) is an alternative set theory in which there are only two infinite cardinal numbers, ℵ 0 (aleph-naught, the cardinality of the set of all natural numbers) and c (the cardinality of the continuum).
- Check operation and conditions. Thus \scriptstyle{{\emptyset,i}\in R} holds by definition, so \scriptstyle{ \mathrm{R}} has at least two elements, \scriptstyle{ \emptyset } and \scriptstyle{{\emptyset,i}} . 4.
Knowledge Transfer¶
Within the home domain. Knowledge about Pocket set theory transfers literally when a new case preserves the same carrier type, relation, and recognition test. 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).
Relationships to Other Abstractions¶
Current abstraction Pocket set theory Domain-specific
Parents (1) — more general patterns this builds on
-
Pocket set theory is a kind of Theory Prime
Pocket set theory is a strict kind of Theory: its frozen identity entails the parent's defining structure while adding domain-specific restrictions.
Hierarchy paths (2) — routes to 2 parentless roots
- Pocket set theory → Theory → Formalization → Representation → Abstraction
- Pocket set theory → Theory → Formalization → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Pocket set theory sits in a sparse region of the domain-specific corpus (65th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Set-Theoretic & Order Structures (53 abstractions)
Nearest neighbors
- Finite set — 0.85
- Inaccessible cardinal — 0.85
- Maximal set (computability theory) — 0.84
- Whitehead problem — 0.84
- Admissible set — 0.84
Computed from structural-signature embeddings · 2026-10-08