Exercise · ZW-AXIOM-OF-CHOICE-004 · ★☆☆ Basic
In what context is the axiom of choice formulated as an independent axiom?
Choose one option.
Show answer
- D. In ZF, as an additional axiom that yields ZFC.
Explanation
The axiom of choice is added to ZF as an independent axiom, and the resulting theory is ZFC. The other contexts are not where it is formulated.
Review the concept: Axiom of choice
Statement that every family of nonempty sets admits a choice function \(f\) that picks one element from each set in the family.
\[\forall X\,(\emptyset\notin X \rightarrow \exists f\,\forall A\in X\,(f(A)\in A))\]
More practice problems: axiom of choice
- ZW-AXIOM-OF-CHOICE-001 What does the axiom of choice state?
- ZW-AXIOM-OF-CHOICE-002 Given a family X of nonempty sets, what must a choice function f satisfy?
- ZW-AXIOM-OF-CHOICE-003 Which of these statements is equivalent to the axiom of choice over ZF?
- ZW-AXIOM-OF-CHOICE-005 What does the axiom of choice guarantee for a family X of nonempty sets?
- ZW-AXIOM-OF-CHOICE-006 Let X=\\1,2\,\3,4,5\\. Is the axiom of choice needed to find a choice function on X?
- ZW-AXIOM-OF-CHOICE-007 For an infinite family of pairs of indistinguishable socks, what is needed to choose one sock from…
- ZW-AXIOM-OF-CHOICE-008 Let f:A→ B be surjective. Using the axiom of choice, which function g:B→ A can be constructed?
- ZW-AXIOM-OF-CHOICE-009 A student claims that the family \(0,n): n∈ℕ, n≥ 1\ admits no choice function without the axiom. Wh…