Zorn’s lemma

If $X$ is a partially ordered setsuch that every chain in $X$ has an upper bound,then $X$ has a maximal element.

Note that the empty chain in $X$ has an upper bound in $X$ if and only if $X$ is non-empty.Because this case is rather different from the case of non-empty chains,Zorn’s Lemma is often stated in the following form:If $X$ is a non-empty partially ordered setsuch that every non-empty chain in $X$ has an upper bound,then $X$ has a maximal element.(In other words: Any non-empty inductively ordered set has a maximal element.)

In ZF, Zorn’s Lemma is equivalent to the Axiom of Choice (http://planetmath.org/AxiomOfChoice).

单词	ZornsLemma
释义	Zorn’s lemma If $X$ is a partially ordered setsuch that every chain in $X$ has an upper bound,then $X$ has a maximal element. Note that the empty chain in $X$ has an upper bound in $X$ if and only if $X$ is non-empty.Because this case is rather different from the case of non-empty chains,Zorn’s Lemma is often stated in the following form:If $X$ is a non-empty partially ordered setsuch that every non-empty chain in $X$ has an upper bound,then $X$ has a maximal element.(In other words: Any non-empty inductively ordered set has a maximal element.) In ZF, Zorn’s Lemma is equivalent to the Axiom of Choice (http://planetmath.org/AxiomOfChoice).
随便看	InvolutoryRing involutory ring IrmgardFluggeLotz Irmgard Flügge-Lotz Irrational irrational IrrationalityMeasure irrationality measure IrrationalToAnIrrationalPowerCanBeRational irrational to an irrational power can be rational IrreducibilityOfBinomialsWithUnityCoefficients irreducibility of binomials with unity coefficients Irreducible irreducible irreducible Irreducible1 IrreducibleComponent irreducible component irreducible component IrreducibleComponent1 IrreducibleIdeal irreducible ideal IrreducibleNmanifold irreducible n-manifold IrreducibleOfAUFDIsPrime