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Well-order: Quiz

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Question 1: If a set is ________, then the following are equivalent:
Total orderPartially ordered setOrder theoryBinary relation

Question 2: With the ________ of this set, 1 is a limit point of the set.
Topological spaceLong line (topology)Compact spaceOrder topology

Question 3: Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the ________ is not a well-ordering, since, for example, the set of negative integers does not contain a least element.
Ring (mathematics)Field (mathematics)IntegerRational number

Question 4: From the ________ axioms of set theory (including the axiom of choice) one can show that there is a well-order of the reals; it is also possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well-order of the reals.
Von Neumann–Bernays–Gödel set theoryZermelo–Fraenkel set theoryMorse–Kelley set theoryFirst-order logic

Question 5: The size (number of elements, ________) of a finite set is equal to the order type.
Natural numberCardinal numberGeorg CantorOrdinal number

Question 6: For an infinite set the order type determines the ________, but not conversely: well-ordered sets of a particular cardinality can have many different order types.
Georg CantorCardinal numberCardinalitySet theory

Question 7: The well-ordering theorem, which is equivalent to the ________, states that every set can be well-ordered.
Set theoryAxiom of choiceMathematical logicZermelo–Fraenkel set theory

Question 8: In the case of a finite set, the basic operation of ________, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects.
Cardinal numberNatural numberCountingCountable set

Question 9: In ________, a well-order relation (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
Set theoryMathematical logicGeometryMathematics

Question 10: Every well-ordered set is uniquely order isomorphic to a unique ________, called the order type of the well-ordered set.
Ordinal numberNatural numberSurreal numberCardinal number







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