Zorn's lemma
Chains yield maximals
Zorn's lemma is a principle in set theory stating that if every chain (totally ordered subset) in a partially ordered set has an upper bound, then the set contains at least one maximal element.
Though called a lemma, it functions as an axiom equivalent to the axiom of choice.
Mathematicians invoke it to prove existence theorems: By Zorn's lemma, every vector space has a basis.
Linked segments forming length
- upper boundMaximum permitted value
- well-orderingOrdering with guaranteed minimums
- antisymmetricNegated when transposed
- transitiveTaking a direct object
reflexive · comparability · predecessor · successor · lattice
Ordering with guaranteed minimums
- upper boundMaximum permitted value
- chainLinked segments forming length
- equivalenceState of being equal
- nonconstructiveNot contributing positively
postulate · theorem · corollary · proposition · proof · deduction · implication · principle
Mathematics of abstract collections
- subsetCollection within collection
- unionJoining into one
- intersectionPoint where paths meet
- elementBasic substance or factor
cardinality · nonempty · collection · family · inclusion · membership · ordinal · cardinal
Mathematics of vector spaces
- topologyStudy of spatial arrangement
- filterPorous separation device
- linear independenceNo vector redundant
chain
Linked segments forming length
- upper boundMaximum permitted value
- well-orderingOrdering with guaranteed minimums
- antisymmetricNegated when transposed
transitive · reflexive · comparability · predecessor · successor · lattice
well-ordering
Ordering with guaranteed minimums
- upper boundMaximum permitted value
- chainLinked segments forming length
- equivalenceState of being equal
nonconstructive · postulate · theorem · corollary · proposition · proof · deduction · implication · principle
set theory
Mathematics of abstract collections
- subsetCollection within collection
- unionJoining into one
- intersectionPoint where paths meet
element · cardinality · nonempty · collection · family · inclusion · membership · ordinal · cardinal · continuum
linear algebra
Mathematics of vector spaces
- topologyStudy of spatial arrangement
- filterPorous separation device
- linear independenceNo vector redundant