In mathematics, the axiom of dependent choice, denoted by , is a weak form of the axiom of choice () that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.[a] Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
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In mathematics, the axiomofdependentchoice, denoted by D C {\displaystyle {\mathsf {DC}}} , is a weak form of the axiomofchoice ( A C {\displaystyle...
the axiomofchoice, abbreviated AC or AoC, is an axiomof set theory equivalent to the statement that a Cartesian product of a collection of non-empty...
The axiomof countable choice or axiomof denumerable choice, denoted ACω, is an axiomof set theory that states that every countable collection of non-empty...
that ai+1 is an element of ai for all i. With the axiomofdependentchoice (which is a weakened form of the axiomofchoice), this result can be reversed:...
lemma Axiomof global choiceAxiomof countable choiceAxiomofdependentchoice Boolean prime ideal theorem Axiomof uniformization Axiomof real determinacy...
satisfy the axiomofdependentchoice but not the full axiomofchoice, the knowledge that a particular proof only requires dependentchoice can be useful...
realized this when introducing epsilon calculus. Axiomof countable choiceAxiomofdependentchoice Hausdorff paradox Hemicontinuity Zermelo, Ernst (1904)...
is equivalent over ZF to the axiom of dependentchoice, a weak form of the axiomofchoice. A restricted form of the Baire category theorem, in which the...
theories, the axiomof global choice is a stronger variant of the axiomofchoice that applies to proper classes of sets as well as sets of sets. Informally...
a_{2}\geq a_{3}\geq \cdots } of elements of P eventually stabilizes. Assuming the axiomofdependentchoice, the descending chain condition on (possibly...
an LCD technology dc (elliptic function), in complex analysis Axiomofdependentchoice, in set theory DC, 600 in Roman numerals DC, 220 in hexadecimal...
countable choice The product of a countable number of non-empty sets is non-empty Axiomofdependentchoice A weak form of the axiomofchoiceAxiomof determinacy...
Alphonse Pyramus de Candolle (1806–1893), French-Swiss botanist Axiomofdependentchoice This disambiguation page lists articles associated with the title...
elements, one is less than the other). Equivalently, assuming the axiomofdependentchoice, it is a totally ordered set without any infinite decreasing sequence — though...
existence of an inaccessible cardinal is consistent with ZFC. ZF stands for Zermelo–Fraenkel set theory, and DC for the axiomofdependentchoice. Solovay's...
the theory adopts An Axiomofdependentchoice, which is much weaker than the usual Axiomofchoice. Set theory in the flavor of Errett Bishop's constructivist...
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