Combining Philosophers

Ideas for Giordano Bruno, Herbert B. Enderton and Anon (Job)

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28 ideas

4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
9724 Until the 1960s the only semantics was truth-tables [Enderton]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
9703 'dom R' indicates the 'domain' of objects having a relation [Enderton]
9705 'fld R' indicates the 'field' of all objects in the relation [Enderton]
9704 'ran R' indicates the 'range' of objects being related to [Enderton]
9710 We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton]
9707 'F(x)' is the unique value which F assumes for a value of x [Enderton]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
9712 A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton]
9713 A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton]
13204 The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
13206 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]
9702 A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton]
9701 A 'relation' is a set of ordered pairs [Enderton]
9706 A 'function' is a relation in which each object is related to just one other object [Enderton]
9708 A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton]
9709 A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton]
9711 A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton]
9714 A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton]
9717 A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13200 Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]
13199 The empty set may look pointless, but many sets can be constructed from it [Enderton]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
13203 The singleton is defined using the pairing axiom (as {x,x}) [Enderton]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
9715 An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton]
9716 We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13202 Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13205 We can only define functions if Choice tells us which items are involved [Enderton]