Exercise · ZW-ZERMELO-FRAENKEL-AXIOMS-003 · ★☆☆ Basic
Which axiom guarantees the existence of a set containing the empty set that is closed under successor?
Choose one option.
Show answer
- C. Axiom of infinity
Explanation
The axiom of infinity ensures an inductive set, containing the empty set and closed under successor. The empty set axiom only ensures the empty set exists.
Review the concept: Zermelo–Fraenkel axioms
The Zermelo-Fraenkel axioms (ZF) are the standard axiom system of set theory, formulated in first-order logic. They allow the usual mathematics to be built from the primitive notions of set and membership (\( \in \)).
\[\forall x\,\forall y\,(\forall z\,(z\in x \leftrightarrow z\in y)\rightarrow x=y)\]
More practice problems: zermelo–fraenkel axioms
- ZW-ZERMELO-FRAENKEL-AXIOMS-001 Which of these statements is the axiom of extensionality?
- ZW-ZERMELO-FRAENKEL-AXIOMS-002 What is the name of the axiom that guarantees the existence of \a,b\ for any a and b ?
- ZW-ZERMELO-FRAENKEL-AXIOMS-004 Which axiom allows forming the set of all subsets of a given set?
- ZW-ZERMELO-FRAENKEL-AXIOMS-005 If A=\1,2\ and B=\3\ , how many elements does A× B have?
- ZW-ZERMELO-FRAENKEL-AXIOMS-006 What is the union of the family \\1\,\1,2\,\2,3\\ ?
- ZW-ZERMELO-FRAENKEL-AXIOMS-007 Which of these statements about regularity is correct?
- ZW-ZERMELO-FRAENKEL-AXIOMS-008 Which axiom allows defining an image set by applying a formula to a given set?
- ZW-ZERMELO-FRAENKEL-AXIOMS-009 Let A=\∅,\∅\\ . How many elements does P(A) have, and what is correct about A ∈ A under regularity?
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