Combining Philosophers
Ideas for H.Putnam/P.Oppenheim, Thomas Jefferson and Herbert B. Enderton
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27 ideas
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
9703
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'dom R' indicates the 'domain' of objects having a relation [Enderton]
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9705
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'fld R' indicates the 'field' of all objects in the relation [Enderton]
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9704
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'ran R' indicates the 'range' of objects being related to [Enderton]
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9710
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We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton]
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9707
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'F(x)' is the unique value which F assumes for a value of x [Enderton]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
13206
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A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]
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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
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The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
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9699
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The 'powerset' of a set is all the subsets of a given set [Enderton]
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9700
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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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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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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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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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9714
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A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13200
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Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]
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13199
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The empty set may look pointless, but many sets can be constructed from it [Enderton]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
13203
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The singleton is defined using the pairing axiom (as {x,x}) [Enderton]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
9715
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An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton]
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9716
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We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13202
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Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13205
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We can only define functions if Choice tells us which items are involved [Enderton]
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