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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

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 prooftheoretic (following procedures)

1.1.7

p.2

9720

A proof theory is 'complete' if semantically valid inferences entail prooftheoretic 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 truthtables

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 AimpliesB

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

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.4

9701

A 'relation' is a set of ordered pairs

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

9712

A relation is 'symmetric' on a set if every ordered pair has the relation in both directions

Ch.0

p.5

9711

A relation is 'reflexive' on a set if every member bears the relation to itself

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

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

9706

A 'function' is a relation in which each object is related to just one other object

Ch.0

p.5

9707

'F(x)' is the unique value which F assumes for a value of x

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.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 onetoone function maps the first set into a subset of the second
