Naive Set Theory is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory.
[1] Originally published by Van Nostrand in 1960,[2] it was reprinted in the Springer-Verlag Undergraduate Texts in Mathematics series in 1974.
[3] While the title states that the set theory presented is 'naive', which is usually taken to mean without formal axioms, the book does introduce a system of axioms equivalent to that of ZFC set theory except the Axiom of foundation.
It also gives correct and rigorous definitions for many basic concepts.
[2][4] Where it differs from a "true" axiomatic set theory book is its character: there are no discussions of axiomatic minutiae, and there is next to nothing about advanced topics such as large cardinals or forcing.
[5] The statements of the axioms given below are as they appear in the book, with section references, and with explanatory commentary on each one.
The "principal primitive (undefined) concept of belonging" (that is, set membership) is the starting point, where "
are both sets, with the notational distinction of upper/lower case a purely stylistic choice.
The axioms govern the properties of this relation between sets.
Axiom of Extension (Section 1): two sets are equal if and only if they have the same elements.
This guarantees that the membership and (logical) equality relations interact appropriately.
Not specified as an named axiom, but instead stated to be "officially assumed".
This assumption is not necessary once the axiom of infinity is adopted later, which also specifies the existence of a set (with a certain property).
In Section 1 Halmos writes that "to avoid terminological monotony, we shall sometimes say collection instead of set."
From the axioms so far Halmos gives a construction of intersections of sets, and the usual Boolean operations on sets are described and their properties proved.
Again (noting that "collection" means "set") using the axiom (schema) of specification we can cut down to get the power set
The axioms so far are used to construct the cartesian product of sets.
Axiom of infinity (Section 11): There exists a set containing 0 and containing the successor of each of its elements.
This implies that there is a set containing all the elements of the first infinite von Neumann ordinal
And another application of the axiom (schema) of specification means
Note here that "family" is defined to be a function
), not as a certain type of set of ordered pairs, as in ZFC, for instance.
, and b) prove that every well-ordered set is order isomorphic to a unique von Neumann ordinal.
Additionally, axioms 1.-8. are nearly exactly those of Zermelo set theory ZC; the only difference being that the set existence assumption is replaced in ZC by the existence of the empty set, and the existence of singletons is stated outright for ZC, rather than proved, as above.
Additionally, the infinite set that is asserted to exist by the axiom of infinity is not the one that Zermelo originally postulated,[a] but Halmos' version is sometimes silently substituted for it in treatments of Zermelo set theory.
That the axiom (schema) of substitution is stated last and so late in the book is testament to how much elementary set theory—and indeed mathematics more generally—can be done without it.
As a very simple example of what is is needed for, the von Neumann ordinal
(that is, the second limit ordinal) cannot be proved to be a set using only axioms 1.-8., even though sets with well-orderings with this order type can be constructed from these axioms.
Working with von Neumann ordinals in place of generic well-orderings has technical advantages, not least the fact every well-ordering is order isomorphic to a unique von Neumann ordinal.
Halmos repeatedly dances around the issue of whether or not a set can contain itself.