1977 | Elements of Set Theory |
1.03 | p.3 | 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ |
Full Idea: Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ. A man with an empty container is better off than a man with nothing. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 1.03) |
1:02 | p.2 | 13199 | The empty set may look pointless, but many sets can be constructed from it |
Full Idea: It might be thought at first that the empty set would be a rather useless or even frivolous set to mention, but from the empty set by various set-theoretic operations a surprising array of sets will be constructed. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 1:02) | |||
A reaction: This nicely sums up the ontological commitments of mathematics - that we will accept absolutely anything, as long as we can have some fun with it. Sets are an abstraction from reality, and the empty set is the very idea of that abstraction. |
1:04 | p.4 | 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other |
Full Idea: To know if A ∈ B, we look at the set A as a single object, and check if it is among B's members. But if we want to know whether A ⊆ B then we must open up set A and check whether its various members are among the members of B. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 1:04) | |||
A reaction: This idea is one of the key ideas to grasp if you are going to get the hang of set theory. John ∈ USA ∈ UN, but John is not a member of the UN, because he isn't a country. See Idea 12337 for a special case. |
1:15 | p.18 | 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers |
Full Idea: It was observed by several people that for a satisfactory theory of ordinal numbers, Zermelo's axioms required strengthening. The Axiom of Replacement was proposed by Fraenkel and others, giving rise to the Zermelo-Fraenkel (ZF) axioms. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 1:15) |
2:19 | p.19 | 13203 | The singleton is defined using the pairing axiom (as {x,x}) |
Full Idea: Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 2:19) | |||
A reaction: An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter! |
3:36 | p.36 | 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} |
Full Idea: The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}; hence it can be proved that <u,v> = <x,y> iff u = x and v = y (given by Kuratowski in 1921). ...The definition is somewhat arbitrary, and others could be used. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 3:36) | |||
A reaction: This looks to me like one of those regular cases where the formal definitions capture all the logical behaviour of the concept that are required for inference, while failing to fully capture the concept for ordinary conversation. |
3:48 | p.48 | 13205 | We can only define functions if Choice tells us which items are involved |
Full Idea: For functions, we know that for any y there exists an appropriate x, but we can't yet form a function H, as we have no way of defining one particular choice of x. Hence we need the axiom of choice. | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 3:48) |
3:62 | p.62 | 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy |
Full Idea: A 'linear ordering' (or 'total ordering') on A is a binary relation R meeting two conditions: R is transitive (of xRy and yRz, the xRz), and R satisfies trichotomy (either xRy or x=y or yRx). | |||
From: Herbert B. Enderton (Elements of Set Theory [1977], 3:62) |
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) |
Full Idea: The point of logic is to give an account of the notion of validity,..in two standard ways: the semantic way says that a valid inference preserves truth (symbol |=), and the proof-theoretic way is defined in terms of purely formal procedures (symbol |-). | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.3..) | |||
A reaction: This division can be mirrored in mathematics, where it is either to do with counting or theorising about things in the physical world, or following sets of rules from axioms. Language can discuss reality, or play word-games. |
1.1.7 | p.2 | 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity |
Full Idea: If every proof-theoretically valid inference is semantically valid (so that |- entails |=), the proof theory is said to be 'sound'. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
1.1.7 | p.2 | 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity |
Full Idea: If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7) |
1.10.1 | p.14 | 9724 | Until the 1960s the only semantics was truth-tables |
Full Idea: Until the 1960s standard truth-table semantics were the only ones that there were. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.10.1) | |||
A reaction: The 1960s presumably marked the advent of possible worlds. |
1.2 | p.23 | 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True |
Full Idea: A truth assignment 'satisfies' a formula, or set of formulae, if it evaluates as True when all of its components have been assigned truth values. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.2) | |||
A reaction: [very roughly what Enderton says!] The concept becomes most significant when a large set of wff's is pronounced 'satisfied' after a truth assignment leads to them all being true. |
1.3.4 | p.4 | 9721 | A logical truth or tautology is a logical consequence of the empty set |
Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4) | |||
A reaction: So the final column of every line of the truth table will be T. |
1.6.4 | p.10 | 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B |
Full Idea: Not all sentences using 'if' are conditionals. Consider 'if you want a banana, there is one in the kitchen'. The rough test is that a conditional can be rewritten as 'that A implies that B'. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.6.4) |
1.7 | p.62 | 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved |
Full Idea: A set of expressions is 'decidable' iff there exists an effective procedure (qv) that, given some expression, will decide whether or not the expression is included in the set (i.e. doesn't contradict it). | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7) | |||
A reaction: This is obviously a highly desirable feature for a really reliable system of expressions to possess. All finite sets are decidable, but some infinite sets are not. |
1.7.3 | p.11 | 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' |
Full Idea: The process is dubbed 'conversational implicature' when the inference is not from the content of what has been said, but from the fact that it has been said. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.7.3) |
2.5 | p.142 | 9997 | For a reasonable language, the set of valid wff's can always be enumerated |
Full Idea: The Enumerability Theorem says that for a reasonable language, the set of valid wff's can be effectively enumerated. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |||
A reaction: There are criteria for what makes a 'reasonable' language (probably specified to ensure enumerability!). Predicates and functions must be decidable, and the language must be finite. |
2.5 | p.142 | 9995 | Proof in finite subsets is sufficient for proof in an infinite set |
Full Idea: If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5) | |||
A reaction: [Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that? |
Ch.0 | p.2 | 9699 | The 'powerset' of a set is all the subsets of a given set |
Full Idea: The 'powerset' of a set is all the subsets of a given set. Thus: PA = {x : x ⊆ A}. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.3 | 9700 | Two sets are 'disjoint' iff their intersection is empty |
Full Idea: Two sets are 'disjoint' iff their intersection is empty (i.e. they have no members in common). | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.4 | 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation |
Full Idea: The 'domain' of a relation is the set of all objects that are members of ordered pairs that are members of the relation. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.4 | 9701 | A 'relation' is a set of ordered pairs |
Full Idea: A 'relation' is a set of ordered pairs. The ordering relation on the numbers 0-3 is captured by - in fact it is - the set of ordered pairs {<0,1>,<0,2>,<0,3>,<1,2>,<1,3>,<2,3>}. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) | |||
A reaction: This can't quite be a definition of order among numbers, since it relies on the notion of a 'ordered' pair. |
Ch.0 | p.4 | 9703 | 'dom R' indicates the 'domain' of objects having a relation |
Full Idea: 'dom R' indicates the 'domain' of a relation, that is, the set of all objects that are members of ordered pairs and that have that relation. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.4 | 9704 | 'ran R' indicates the 'range' of objects being related to |
Full Idea: 'ran R' indicates the 'range' of a relation, that is, the set of all objects that are members of ordered pairs and that are related to by the first objects. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.4 | 9705 | 'fld R' indicates the 'field' of all objects in the relation |
Full Idea: 'fld R' indicates the 'field' of a relation, that is, the set of all objects that are members of ordered pairs on either side of the relation. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
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) |
Full Idea: We write F : A → B to indicate that A maps into B, that is, the domain of relating things is set A, and the things related to are all in B. If we add that F = B, then A maps 'onto' B. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.5 | 9707 | 'F(x)' is the unique value which F assumes for a value of x |
Full Idea:
F(x) is a 'function', which indicates the unique value which y takes in |
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From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.5 | 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third |
Full Idea: A relation is 'transitive' on a set if the relation can be carried over from two ordered pairs to a third. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.5 | 9706 | A 'function' is a relation in which each object is related to just one other object |
Full Idea: A 'function' is a relation which is single-valued. That is, for each object, there is only one object in the function set to which that object is related. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
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 |
Full Idea: A function 'maps A into B' if the domain of relating things is set A, and the things related to are all in B. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
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 |
Full Idea: A function 'maps A onto B' if the domain of relating things is set A, and the things related to are set B. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.5 | 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions |
Full Idea: A relation is 'symmetric' on a set if every ordered pair in the set has the relation in both directions. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.5 | 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself |
Full Idea: A relation is 'reflexive' on a set if every member of the set bears the relation to itself. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.6 | 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects |
Full Idea: Equivalence classes will 'partition' a set. That is, it will divide it into distinct subsets, according to each relation on the set. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.6 | 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation |
Full Idea: An 'equivalence relation' is a binary relation which is reflexive, and symmetric, and transitive. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
Ch.0 | p.6 | 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects |
Full Idea: A relation satisfies 'trichotomy' on a set if every ordered pair is related (in either direction), or the objects are identical. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |
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 |
Full Idea: A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second. | |||
From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], Ch.0) |