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Ideas of Herbert B. Enderton, by Text

[American, fl. 1972, At the University of California, Los Angeles.]

1977 Elements of Set Theory
1.03 p.3 Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ
1:02 p.2 The empty set may look pointless, but many sets can be constructed from it
1:04 p.4 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other
1:15 p.18 Fraenkel added Replacement, to give a theory of ordinal numbers
2:19 p.19 The singleton is defined using the pairing axiom (as {x,x})
3:36 p.36 The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}
3:48 p.48 We can only define functions if Choice tells us which items are involved
3:62 p.62 A 'linear or total ordering' must be transitive and satisfy trichotomy
2001 A Mathematical Introduction to Logic (2nd)
1.1.3.. p.1 Validity is either semantic (what preserves truth), or proof-theoretic (following procedures)
1.1.7 p.2 A proof theory is 'sound' if its valid inferences entail semantic validity
1.1.7 p.2 A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
1.10.1 p.14 Until the 1960s the only semantics was truth-tables
1.2 p.23 A truth assignment to the components of a wff 'satisfy' it if the wff is then True
1.3.4 p.4 A logical truth or tautology is a logical consequence of the empty set
1.6.4 p.10 Sentences with 'if' are only conditionals if they can read as A-implies-B
1.7 p.62 Expressions are 'decidable' if inclusion in them (or not) can be proved
1.7.3 p.11 Inference not from content, but from the fact that it was said, is 'conversational implicature'
2.5 p.142 Proof in finite subsets is sufficient for proof in an infinite set
2.5 p.142 For a reasonable language, the set of valid wff's can always be enumerated
Ch.0 p.2 The 'powerset' of a set is all the subsets of a given set
Ch.0 p.3 Two sets are 'disjoint' iff their intersection is empty
Ch.0 p.4 A 'domain' of a relation is the set of members of ordered pairs in the relation
Ch.0 p.4 A 'relation' is a set of ordered pairs
Ch.0 p.4 'dom R' indicates the 'domain' of objects having a relation
Ch.0 p.4 'ran R' indicates the 'range' of objects being related to
Ch.0 p.4 'fld R' indicates the 'field' of all objects in the relation
Ch.0 p.5 We write F:A→B to indicate that A maps into B (the output of F on A is in B)
Ch.0 p.5 'F(x)' is the unique value which F assumes for a value of x
Ch.0 p.5 A 'function' is a relation in which each object is related to just one other object
Ch.0 p.5 A function 'maps A into B' if the relating things are set A, and the things related to are all in B
Ch.0 p.5 A function 'maps A onto B' if the relating things are set A, and the things related to are set B
Ch.0 p.5 A relation is 'reflexive' on a set if every member bears the relation to itself
Ch.0 p.5 A relation is 'symmetric' on a set if every ordered pair has the relation in both directions
Ch.0 p.5 A relation is 'transitive' if it can be carried over from two ordered pairs to a third
Ch.0 p.6 A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects
Ch.0 p.6 An 'equivalence relation' is a reflexive, symmetric and transitive binary relation
Ch.0 p.6 We 'partition' a set into distinct subsets, according to each relation on its objects
Ch.0 p.8 A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second