ABSTRACT

Set theory is formulated using the notation of first order logic. It is assumed that the reader has a basic familiarity with the propositional and predicate calculi of the sort gained from a first course in symbolic logic. We use the following notation: https://www.niso.org/standards/z39-96/ns/oasis-exchange/table"> https://www.w3.org/1998/Math/MathML"> ( ∀ x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> universal quantifier https://www.w3.org/1998/Math/MathML"> ( ∃ x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> existential quantifier & conjunction v disjunction https://www.w3.org/1998/Math/MathML"> ⊃ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if... then https://www.w3.org/1998/Math/MathML"> ≡ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if and only if ~ negation = identity It is assumed that the student has a working knowledge of which formulas of logic follow from which. That may not be a reasonable expectation in the case of identity, because identity is often omitted in beginning courses in logic. By ‘identity' we mean ‘numerical identity', not ‘qualitative identity'. ‘x = y' means that x and y are literally the same thing. The basic principles regarding identity are simple and intuitive. All principles regarding identity in first order logic are consequences of the following four simple principles:

Reflexivity: V (Eveiything is identical to itself.)

2 Symmetry: https://www.w3.org/1998/Math/MathML"> ( ∀ x ) ( ∀ y ) ( x = y ⊃ y = x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (If one thing is identical to a second, then the second is identical to the first.)

Transitivity: https://www.w3.org/1998/Math/MathML"> ( ∀ x ) ( ∀ y ) ( ∀ z ) [ ( x = y & y = z ) ⊃ x = z ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (If one thing is identical to a second, and the second is identical to a third, then the first is identical to the third.)

Substitutivity: https://www.w3.org/1998/Math/MathML"> ( ∀ x ) ( ∀ y ) [ x = y ⊃ ( A x = A y ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429308321/75a126d7-3547-442c-b7fd-c110a4eb036e/content/inline-math7.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (If one thing is identical to a second, then anything true of the first is true of the second and vice versa.)