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Set theory concept
-prewellordering. The prewellordering property is related to the stronger scale property; in practice, many pointclasses having the prewellordering property
Prewellordering
a map from A into the ordinal numbers. Each norm has an associated prewellordering, where one element of A precedes another element if the norm of the
Scale (descriptive set theory)
Scale_(descriptive_set_theory)
Topics referred to by the same term
group Norm map, a map from a pointset into the ordinals inducing a prewellordering Norm group, a group in class field theory that is the image of the
Norm
Class of mathematical orderings
generalization Ordinal number Well-founded set Well partial order Prewellordering Directed set Manolios P, Vroon D. Algorithms for Ordinal Arithmetic
Well-order
Order whose elements are all comparable
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Total_order
determinacy, the class of inductive sets has the scale property and thus the prewellordering property. The term can have a number of different meanings: Russell's
Inductive_set
Reflexive and transitive binary relation
is antisymmetric and total Directed set Category of preordered sets Prewellordering Well-quasi-ordering For "proset", see e.g. Eklund, Patrik; Gähler,
Preorder
Type of binary relation
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Transitive_relation
Binary relation that relates every element to itself
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Reflexive_relation
Set whose pairs have minima and maxima
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Lattice_(order)
Mathematical concept for comparing objects
{\displaystyle S} would constitute an infinite antichain. Better-quasi-ordering Prewellordering – Set theory concept Well-order – Class of mathematical orderings Here
Well-quasi-ordering
Mathematical ranking of a set
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Weak_ordering
Mathematical set with an ordering
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Partially_ordered_set
Partially ordered set equipped with a rank function
power set has the Sperner property Graded (mathematics) Prewellordering – a prewellordering with a norm is analogous to a graded poset, replacing a map
Graded_poset
Type of binary relation
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Antisymmetric_relation
Relationship between elements of two sets
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Binary_relation
Binary relation which never occurs in both directions
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Asymmetric_relation
Partial order with joins
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Semilattice
Overview of and topical guide to logic
equivalence relation Partial function Partially ordered set Preorder Prewellordering Propositional function Quasitransitive relation Reflexive relation
Outline_of_logic
Mathematical concept for comparing objects
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Equivalence_relation
theory) Tree (descriptive set theory) Suslin's problem Absorption law Prewellordering Stone duality Stone's representation theorem for Boolean algebras Specialization
List_of_order_theory_topics
Smallest transitive relation containing a given binary relation
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Transitive_closure
Type of binary relation
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Symmetric_relation
Concept in order theory
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Join_and_meet
Property of a relation on a set
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Connected_relation
Subfield of set theory
then Π12n+1 and Σ12n+2 have the prewellordering property (and that Σ12n+1 and Π12n+2 do not have the prewellordering property, but rather have the separation
Determinacy
Concept in mathematical logic
{\displaystyle \Theta } is also the supremum of the order types of all prewellorderings of the reals.[citation needed] It may not be obvious that it can be
Theta_(set_theory)
Type of binary relation
✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering
Well-founded_relation
Infinity-Borel set Lightface analytic game Perfect set property Polish space Prewellordering Projective set Property of Baire Uniformization (set theory) Universally
List_of_set_theory_topics
Topics referred to by the same term
to simplify the process of building a portfolio of financial assets Prewellordering in mathematics, a component of set theory Separation axiom in mathematics
Separation_property
Subfield of mathematical logic
as Kripke–Platek set theory and second-order arithmetic. Pointclass Prewellordering Scale property Kechris, Alexander S. (1994). Classical Descriptive
Descriptive_set_theory
Infinity-Borel set Lightface analytic game Perfect set property Polish space Prewellordering Projective set Property of Baire Uniformization (set theory) Universally
List of mathematical logic topics
List_of_mathematical_logic_topics
American mathematician
of all the projective ordinals (the suprema of the lengths of all prewellorderings of the real numbers at a particular level in the projective hierarchy)
Steve_Jackson_(mathematician)
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Boy/Male
Hindu, Indian, Kannada, Marathi, Oriya, Sanskrit, Sindhi, Telugu, Traditional
Eye; Eye of Star
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American, British, English, German
Wolf's Shield; Variants of Randolph; Surname
Female
English
 Old English name GILDA means "coated with gold; gilded." Compare with other forms of Gilda.
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Keen Eyed; Little; Small
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Of the Forest
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An Ancient Philosopher
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Zone
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Arthurian Legend
A pagan king.
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Arabic, Hindu, Indian, Marathi, Muslim, Parsi, Portuguese, Telugu
Brilliant; The Fair One; Brilliant and Lovely; Fair
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