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Single Idea 9698
[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
]
Full Idea
The 'induction clause' says that whenever one constructs more complex formulas out of formulas that have the property P, the resulting formulas will also have that property.
Gist of Idea
The 'induction clause' says complex formulas retain the properties of their basic formulas
Source
Graham Priest (Intro to Non-Classical Logic (1st ed) [2001], 0.2)
Book Ref
Priest,Graham: 'Introduction to Non-Classical Logic' [CUP 2001], p.-5
The
40 ideas
with the same theme
[main concepts which are distinctive of set theory]:
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13201
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∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
[Enderton]
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|
13204
|
The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
[Enderton]
|
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13206
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A 'linear or total ordering' must be transitive and satisfy trichotomy
[Enderton]
|
|
9699
|
The 'powerset' of a set is all the subsets of a given set
[Enderton]
|
|
9700
|
Two sets are 'disjoint' iff their intersection is empty
[Enderton]
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9702
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A 'domain' of a relation is the set of members of ordered pairs in the relation
[Enderton]
|
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9701
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A 'relation' is a set of ordered pairs
[Enderton]
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|
9706
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A 'function' is a relation in which each object is related to just one other object
[Enderton]
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9708
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A function 'maps A into B' if the relating things are set A, and the things related to are all in B
[Enderton]
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9709
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A function 'maps A onto B' if the relating things are set A, and the things related to are set B
[Enderton]
|
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9711
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A relation is 'reflexive' on a set if every member bears the relation to itself
[Enderton]
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9714
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A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
[Enderton]
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9717
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A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second
[Enderton]
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9712
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A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
[Enderton]
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9713
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A relation is 'transitive' if it can be carried over from two ordered pairs to a third
[Enderton]
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12337
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There is 'transivity' iff membership ∈ also means inclusion ⊆
[Badiou]
|
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15500
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Classes divide into subclasses in many ways, but into members in only one way
[Lewis]
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|
15499
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A subclass of a subclass is itself a subclass; a member of a member is not in general a member
[Lewis]
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18194
|
'Forcing' can produce new models of ZFC from old models
[Maddy]
|
|
9695
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An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order
[Priest,G]
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|
9696
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A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets
[Priest,G]
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9686
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A 'set' is a collection of objects
[Priest,G]
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9687
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A 'member' of a set is one of the objects in the set
[Priest,G]
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9688
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A 'singleton' is a set with only one member
[Priest,G]
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9689
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The 'empty set' or 'null set' has no members
[Priest,G]
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9690
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A set is a 'subset' of another set if all of its members are in that set
[Priest,G]
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9691
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A 'proper subset' is smaller than the containing set
[Priest,G]
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9694
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The 'relative complement' is things in the second set not in the first
[Priest,G]
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9693
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The 'intersection' of two sets is a set of the things that are in both sets
[Priest,G]
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9692
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The 'union' of two sets is a set containing all the things in either of the sets
[Priest,G]
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9698
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The 'induction clause' says complex formulas retain the properties of their basic formulas
[Priest,G]
|
|
10890
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A 'partial ordering' is reflexive, antisymmetric and transitive
[Zalabardo]
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|
10889
|
The 'Cartesian Product' of two sets relates them by pairing every element with every element
[Zalabardo]
|
|
10098
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The 'power set' of A is all the subsets of A
[George/Velleman]
|
|
10099
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The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}}
[George/Velleman]
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|
10101
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Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B
[George/Velleman]
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10859
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A set is 'well-ordered' if every subset has a first element
[Clegg]
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15914
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An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one
[Lavine]
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8665
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A 'proper subset' of A contains only members of A, but not all of them
[Friend]
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|
8672
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A 'powerset' is all the subsets of a set
[Friend]
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