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The difference between 'naive' and axiomatic approaches to set theory

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College students who study only basic mathematics are unlikely to experience anything more than 'naive' set theory. There is nothing wrong with this, of course. Not everyone is a mathematician, and not everyone requires a knowledge of advanced mathematics. But what is a naive approach to set theory, and with what is it to be contrasted? Naive set theory simply refers to an approach of explaining set theory that consists of an exposition of its basic concepts (such as union, intersection, set-builder notation, etc.) as opposed to an axiomatic approach, which demonstrates the actual logical foundation of the concepts. For example, take a well-known and very basic concept of set theory like that of union. It simply refers to a combination of all elements that are members of at least one set in a collection of sets. We might speak of the "union" of A and B. This union would include all elements from both A and B.

However, there is a specific logical axiom which makes such a union possible. Students who take a finite mathematics course are not going to encounter this formulation. Indeed, students who take an undergraduate course in formal logic are unlikely to encounter it, because such logic will likely be truth-functional propositional logic of the "zeroth" order, rather than first-order logic that possesses predicates and quantifiers. As an example, the logical axiom undergirding the concept of union is the Axiom of Unions, which is articulated as follows:

∀x∃y∀z[z∈y ≡ ∃w(w∈x & z∈w)]

An "axiomatic" approach to set theory is a considerably more formal approach which demonstrates how each of the concepts of set theory presuppose certain more fundamental axioms which make them possible. The student of very basic formal logic will recognize some of the notation in the above formula. However, ∀ (a universal quantifier) and ∃ (an existential quantifier, referring to "at least one" of a variable rather than predicating a quality of all such a variable) are introduced only in first-order logic known as "predicate calculus." Students of set theory who wish to proceed further are encouraged to research this form of logic, as it is essential for even introductory, relatively user-friendly 'naive' approaches to axiomatic set theory, such as Paul Halmos' "Naive Set Theory." His book is not so much a straightforward "naive" set theory as the title suggests, but is, as he himself notes in the beginning of the book, a kind of 'naive' approach to axiomatic set theory, which articulates the relevant axioms in informal logic rather than depending on the highly technical and formal approach we find in ordinary formal approaches, which articulate the material in terms of predicate calculus.

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