§ 1. Set Theory
7
§1. Set Theory
1 - Membership, equality, empty set
The concept of a set lO is a primitive concept in mathematics; one can no more
provide a definition than Euclid could define mathematically what a point is.
In my youth there were those who said that a set is "a collection of objects of
the same nature"; apart from the vicious circle (what indeed is a "collection"?
a set?), to talk of "nature" is empty and means nothing ll . Certain denigrators
of the introduction of "modern math" into elementary education have been
scandalised to see that in some textbooks they have had the temerity to form
the union of a set of apples with a set of pears; never mind that a normal
child will tell you that this gives a set of fruits, or even of things, and if asked
to count the number of elements of the union any moderately intelligent child
can explain to you that it does not matter that the first set consists of apples
rather than oranges and the second of pears rather than dessert spoons; the
fact that the Louvre Museum combines disparate collections - of pictures,
sculptures, ceramics, gold work, mummies, etc. - has never troubled anyone.
One calls this: to acquire the sense of abstraction.
The logicians have in any case long since invented a radical method of
eliminating questions concerning the "nature" of mathematical objects or
sets (the two terms are synonymous). One can describe this in a figurative
way by saying that a set is a "primary" box containing "secondary" boxes,
its elements, no two of which have identical contents, which in their turn
contain "tertiary" boxes themselves containing. .. The Louvre is a collection
of collections (of paintings, sculptures, etc.), the collection of paintings is
itself a collection of paintings stolen by Bonaparte, Monge and Berthollet in
Italy (we unfortunately had to return it in 1815), bequeathed by ... private
collectors, bought at sales, etc.
The whole of set theory rests on two sorts of relations. The membership
relation x E X which is read "x belongs to X" or "x is an element of X"; this
means that x is one of the secondary boxes contained in the box X, while
secondary means: not contained in any box other than X itself. The negation
of x E X is denoted x¢. X. To express the fact that an object x is an element
of a set which is itself an element of X, one might write x EE X; at the next
level one might write x EEE X. These last two notations have not found
common currency, but I will use them occasionally in this chapter. If one
considers the Louvre as a set whose elements are its collection of paintings,
its collection of sculpture etc., then the Mona Lisa EE Louvre.
On the other hand there is the equality relation x = y, whose intuitive
meaning is that the two sets are identical; its negation is written x =I- y. The
10 After several tentatives Cantor chose the word Menge (quantity, number,
amount, mass, multitude, crowd).
11 Cantor defined a set as "every assemblage as a whole (Zusammenfassung zu
einem Ganzen) M of defined and distinct objects m of our intuition or thought".
7
§1. Set Theory
1 - Membership, equality, empty set
The concept of a set lO is a primitive concept in mathematics; one can no more
provide a definition than Euclid could define mathematically what a point is.
In my youth there were those who said that a set is "a collection of objects of
the same nature"; apart from the vicious circle (what indeed is a "collection"?
a set?), to talk of "nature" is empty and means nothing ll . Certain denigrators
of the introduction of "modern math" into elementary education have been
scandalised to see that in some textbooks they have had the temerity to form
the union of a set of apples with a set of pears; never mind that a normal
child will tell you that this gives a set of fruits, or even of things, and if asked
to count the number of elements of the union any moderately intelligent child
can explain to you that it does not matter that the first set consists of apples
rather than oranges and the second of pears rather than dessert spoons; the
fact that the Louvre Museum combines disparate collections - of pictures,
sculptures, ceramics, gold work, mummies, etc. - has never troubled anyone.
One calls this: to acquire the sense of abstraction.
The logicians have in any case long since invented a radical method of
eliminating questions concerning the "nature" of mathematical objects or
sets (the two terms are synonymous). One can describe this in a figurative
way by saying that a set is a "primary" box containing "secondary" boxes,
its elements, no two of which have identical contents, which in their turn
contain "tertiary" boxes themselves containing. .. The Louvre is a collection
of collections (of paintings, sculptures, etc.), the collection of paintings is
itself a collection of paintings stolen by Bonaparte, Monge and Berthollet in
Italy (we unfortunately had to return it in 1815), bequeathed by ... private
collectors, bought at sales, etc.
The whole of set theory rests on two sorts of relations. The membership
relation x E X which is read "x belongs to X" or "x is an element of X"; this
means that x is one of the secondary boxes contained in the box X, while
secondary means: not contained in any box other than X itself. The negation
of x E X is denoted x¢. X. To express the fact that an object x is an element
of a set which is itself an element of X, one might write x EE X; at the next
level one might write x EEE X. These last two notations have not found
common currency, but I will use them occasionally in this chapter. If one
considers the Louvre as a set whose elements are its collection of paintings,
its collection of sculpture etc., then the Mona Lisa EE Louvre.
On the other hand there is the equality relation x = y, whose intuitive
meaning is that the two sets are identical; its negation is written x =I- y. The
10 After several tentatives Cantor chose the word Menge (quantity, number,
amount, mass, multitude, crowd).
11 Cantor defined a set as "every assemblage as a whole (Zusammenfassung zu
einem Ganzen) M of defined and distinct objects m of our intuition or thought".