Elementarv
of Logic
In this chapter, we begin to apply the principles of logic developed in
Chapters 2 and 3. In Article 4.1 for the first time we look at mathematical
proofs, limiting our consideration at this stage to proofs of elementary prop-
erties of sets. In Article 4.2 we introduce the notion of an infinite collection
of sets and analyze basic properties of such collections. In Article 4.3 we
examine in detail the epsilon-delta definition of limit.4.1 Applications of Logic to
Set Theory-Some Proofs/-- We now take a somewhat more formal approach to the theory of sets,
introduced informally in Chapter 1. In particular, we begin here to apply
the principles oflogic developed in Chapters 2 and 3 to the problem of
constructing proofs of theorems in set theory.
Recall our informal definitions of set equality and set inclusion from
Article 1.1 (Remarks 2 and 3). Two sets are equal if and only if they have
precisely the same elements. Set A is a subset of set B if and only if every
element of A is also an element of B. Using connectives and quantifiers,
we can now restate these definitions in a more formal way. The precision
we gain from this added formality will enable us to deal with some questions
that were not fully resolved in the informal context of Chapter 1 (e.g.,
Example 1).