The Philosophy of Set Theory by Mary Tiles

Author:Mary Tiles
Language: eng
Format: epub
ISBN: 9780486138558
Publisher: Dover Publications
Published: 2012-10-13T04:00:00+00:00

2 The ZF Axioms

In presenting the ZF axioms it is presumed that the domain of entities of which they are true is a universe of sets. All individual variables x, y,. . . are thus interpreted as ranging over the universe of sets. There is a single undefined relation ‘∈’ which is intended to stand for the membership relation, so that ‘a ∈ b’ means ‘a is an element of b’ or ‘a belongs to b’. The relation of set inclusion ‘⊆’ or the subset relation can be defined in terms of ‘∈’. If two sets b and c are such that every element of b is also an element of c, then b is said to be included in, or to be a subset of, c. (Formally b ⊆ c ≡df ∀x(x ∈ b ⇒ x ∈ c).)

1 Axiom of extensionality If two sets have the same elements then they are identical. ∀x∀y∀ z [(z ∈ x ⇔ z ∈ y) ⇒ x = y]

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