Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\ann{\operatorname{ann}}
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Appendix B Axioms for Set Theory
Axiom B.1 . Axiom of Extensionality.
Let \(x\) and \(y\) be sets. If \(x\) and \(y\) have the same elements, then \(x = y\text{.}\)
Axiom B.2 . Axiom of Empty Set.
There is a set \(z\) such that \(x \not\in z\) for all sets \(x\text{.}\)
Axiom B.3 . Axiom of Pairing.
Let \(x\) and \(y\) be sets. There is a set \(z\) such that \(w \in z\) if and only if \(w = x\) or \(w = y\text{.}\)
Axiom B.4 . Axiom of Union.
Let \(x\) be a set. There is a set \(z\) such that \(w \in z\) if and only if there is some \(y \in x\) such that \(w \in y\text{.}\)
Axiom B.5 . Axiom of Power Set.
Let \(x\) be a set. There is a set \(z\) such that \(w \in z\) if and only if \(w \sse x\text{.}\)
Axiom B.6 . Axiom of Regularity.
Let \(x\) be a set. Suppose that \(x \ne \es\text{.}\) Then there is some \(y \in x\) such that \(x \cap y = \es\text{.}\)
Axiom B.7 . Axiom of Selection.
Let \(P(t)\) be a logical property of sets with one free variable \(t\) that can be formulated in the context of the ZF axioms. Let \(x\) be a set. Then there is a set \(z\) such that \(y \in z\) if and only if y\(\in x\) and \(P(y)\) is true.
Axiom B.8 . Axiom of Infinity.
There is a set \(z\) such that \(\es \in z\text{,}\) and if \(x \in z\) then \(x \cup \{x\} \in z\text{.}\)
Axiom B.9 . Axiom of Replacement.
Let \(F(s,t)\) be a functional property of sets with two free variables \(s\) and \(t\) that can be formulated in the context of the ZF axioms. Let \(x\) be a set. Then there is a set \(z\) such that \(y \in z\) if and only if there is some \(w \in x\) such that \(F(w, y)\) is true.
Axiom B.10 . Axiom of Choice.
Let \(x\) be a set. Suppose that if \(y, w \in x\text{,}\) then \(y \ne \es\) and \(y \cap w = \es\text{.}\) Then there is a set \(z\) such that if \(y \in x\text{,}\) then \(y \cap z\) contains a single element.
