Naive Set Theory
The term "naïve set theory" may refer to very different things, like 1. informal presentational style of an axiomatic set theory 2.
It may refer to the early or later versions of Cantor's set theory or other informal systems.
It may refer to the decidedly inconsistent theories (axiomatic or not) such as Frege's set theory (which yielded Russell's paradox), Peano's set theory, Dedekind's set theory, etc.
A naive set theory is an informal approach to sets that was developed initially, in which the notion of a set is a primitive concept (so no proof required).
A primitive concepts such as set, number, space, and such, are the starting point of developing a theory. Usually, a primitive notion is a theory's primary subject matter, whose properties are then described by the axioms; alternatively, a primitive notion may also be stated as one of the axioms.
Primitive notions and axioms require no proof because in order to start, a theory must stop the infinite regression by picking a term that is the central notion of the theory. Then, the term is introduced, without proof, just by appealing to human intuition. Otherwise, we'd be off chasing definitions of definition terms across the spacetime, with no money or hair, wearing nothing but a non-Euclidean smile.
A naive set theory treats sets as collections without restricting what can constitute a set. It assumes the existence of sets, particularly the universal set, despite the fact that such a broad assumption is ridden with paradoxes. However, it is still regarded as a useful introductory tool for the general notions about sets.
Informal presentation of set theory
an informal presentation of an otherwise axiomatic set theory. For example, the book "Naïve Set Theory" by Paul Halmos
"Naive Set Theory", published in 1960 (reprint in 1974) is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory.
While the title states that it is "naive" (usually taken to mean without axioms) the book does introduce all the axioms of ZFC set theory (except the Axiom of Foundation) and gives correct and rigorous definitions for the basic objects.
Where it differs from a "true" axiomatic set theory book is its less formal style, lacking discussion about every possible detail and some of the advanced set-related topics like large cardinals. Instead, it targets people who have never before encountered set theory.
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