display all the ideas for this combination of texts
12 ideas
14427 | We can enumerate finite classes, but an intensional definition is needed for infinite classes [Russell] |
Full Idea: We know a great deal about a class without enumerating its members …so definition by extension is not necessary to knowledge about a class ..but enumeration of infinite classes is impossible for finite beings, so definition must be by intension. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) | |
A reaction: Presumably mathematical induction (which keeps apply the rule to extend the class) will count as an intension here. |
14428 | Members define a unique class, whereas defining characteristics are numerous [Russell] |
Full Idea: There is only one class having a given set of members, whereas there are always many different characteristics by which a given class may be defined. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], II) |
14440 | We may assume that there are infinite collections, as there is no logical reason against them [Russell] |
Full Idea: There is no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], VIII) |
14447 | Infinity says 'for any inductive cardinal, there is a class having that many terms' [Russell] |
Full Idea: The Axiom of Infinity may be enunciated as 'If n be any inductive cardinal number, there is at least one class of individuals having n terms'. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XIII) | |
A reaction: So for every possible there exists a set of terms for it. Notice that they are 'terms', not 'objects'. We must decide whether we are allowed terms which don't refer to real objects. |
14443 | The British parliament has one representative selected from each constituency [Russell] |
Full Idea: We have a class of representatives, who make up our Parliament, one being selected out of each constituency. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: You can rely on Russell for the clearest illustrations of these abstract ideas. He calls the Axiom of Choice the 'Multiplicative' Axiom. |
14444 | Choice is equivalent to the proposition that every class is well-ordered [Russell] |
Full Idea: Zermelo has shown that [the Axiom of Choice] is equivalent to the proposition that every class is well-ordered, i.e. can be arranged in a series in which every sub-class has a first term (except, of course, the null class). | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: Russell calls Choice the 'Multiplicative' Axiom. |
14445 | Choice shows that if any two cardinals are not equal, one must be the greater [Russell] |
Full Idea: The [Axiom of Choice] is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: It is illuminating for the uninitiated to learn that this result can't be taken for granted (with infinite cardinals). |
14446 | We can pick all the right or left boots, but socks need Choice to insure the representative class [Russell] |
Full Idea: Among boots we distinguish left and right, so we can choose all the right or left boots; with socks no such principle suggests itself, and we cannot be sure, without the [Axiom of Choice], that there is a class consisting of one sock from each pair. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XII) | |
A reaction: A deservedly famous illustration of a rather tricky part of set theory. |
14459 | Reducibility: a family of functions is equivalent to a single type of function [Russell] |
Full Idea: The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true? | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: I take this to say that in the theory of types, it is possible to reduce each level of type down to one type. |
14461 | Propositions about classes can be reduced to propositions about their defining functions [Russell] |
Full Idea: It is right (in its main lines) to say that there is a reduction of propositions nominally about classes to propositions about their defining functions. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII) | |
A reaction: The defining functions will involve the theory of types, in order to avoid the paradoxes of naïve set theory. This is Russell's strategy for rejecting the existence of sets. |
8469 | Russell's proposal was that only meaningful predicates have sets as their extensions [Russell, by Orenstein] |
Full Idea: Russell's solution (in the theory of types) consists of restricting the principle that every predicate has a set as its extension so that only meaningful predicates have sets as their extensions. | |
From: report of Bertrand Russell (Introduction to Mathematical Philosophy [1919]) by Alex Orenstein - W.V. Quine Ch.3 | |
A reaction: There might be a chicken-and-egg problem here. How do you decide the members of a set (apart from ostensively) without deciding the predicate(s) that combine them? |
8745 | Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell] |
Full Idea: The symbols for classes are mere conveniences, not representing objects called 'classes'. Classes are in fact logical fictions; they cannot be regarded as part of the ultimate furniture of the world. | |
From: Bertrand Russell (Introduction to Mathematical Philosophy [1919], Ch.18), quoted by Stewart Shapiro - Thinking About Mathematics 5.2 | |
A reaction: I agree. For 'logical fictions' read 'abstractions'. To equate abstractions with fictions is to underline the fact that they are a human creation. They are either that or platonic objects - there is no middle way. |