1977 | Elements of Set Theory |
1.03 | p.3 | 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ |
1:02 | p.2 | 13199 | The empty set may look pointless, but many sets can be constructed from it |
1:04 | p.4 | 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other |
1:15 | p.18 | 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers |
2:19 | p.19 | 13203 | The singleton is defined using the pairing axiom (as {x,x}) |
3:36 | p.36 | 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} |
3:48 | p.48 | 13205 | We can only define functions if Choice tells us which items are involved |
3:62 | p.62 | 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy |
2001 | A Mathematical Introduction to Logic (2nd) |
1.1.3.. | p.1 | 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) |
1.1.7 | p.2 | 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity |
1.1.7 | p.2 | 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity |
1.10.1 | p.14 | 9724 | Until the 1960s the only semantics was truth-tables |
1.2 | p.23 | 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True |
1.3.4 | p.4 | 9721 | A logical truth or tautology is a logical consequence of the empty set |
1.6.4 | p.10 | 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B |
1.7 | p.62 | 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved |
1.7.3 | p.11 | 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' |
2.5 | p.142 | 9995 | Proof in finite subsets is sufficient for proof in an infinite set |
2.5 | p.142 | 9997 | For a reasonable language, the set of valid wff's can always be enumerated |
Ch.0 | p.2 | 9699 | The 'powerset' of a set is all the subsets of a given set |
Ch.0 | p.3 | 9700 | Two sets are 'disjoint' iff their intersection is empty |
Ch.0 | p.4 | 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation |
Ch.0 | p.4 | 9701 | A 'relation' is a set of ordered pairs |
Ch.0 | p.4 | 9703 | 'dom R' indicates the 'domain' of objects having a relation |
Ch.0 | p.4 | 9704 | 'ran R' indicates the 'range' of objects being related to |
Ch.0 | p.4 | 9705 | 'fld R' indicates the 'field' of all objects in the relation |
Ch.0 | p.5 | 9710 | 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 | 9707 | 'F(x)' is the unique value which F assumes for a value of x |
Ch.0 | p.5 | 9706 | A 'function' is a relation in which each object is related to just one other object |
Ch.0 | p.5 | 9708 | 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 | 9709 | 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 | 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions |
Ch.0 | p.5 | 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third |
Ch.0 | p.5 | 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself |
Ch.0 | p.6 | 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects |
Ch.0 | p.6 | 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects |
Ch.0 | p.6 | 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation |
Ch.0 | p.8 | 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second |