10
I - Sets and Functions
x
c
fig.!.
/X~eIl}'{{eIl'{eIl}}}
/~l={eIl'{eIl}}
{ell} =B
' " A = el l
fig. 1 bis.
The significance of the signs { and } will be explained in the following n O •
2 - The set defined by a relation. Intersections and unions
In practice, a set is most often defined by a characteristic property of its
elements: "the set of integers between 2 and 25", ''the set of points distant
15 km from a given point 0", "the set of rational numbers < 11"", ''the set of
functions f defined on R with values in R" , etc. A seemingly obvious general
statement, due to Frege, is that for every proposition or relation P{ x} in
which there appears a variable x symbolising a totally undetermined object
one can speak of the set of those x such that P{ x} is true; this set will be
unique, by the axiom of extension. If you choose the relation x = x you
will thus obtain the set of all mathematical objects, a creature which justly
evoked great suspicion in Cantor; he spoke of it as a "class" of sets, a concept
which the logicians later used and developed.
In 1903, as Frege was on the point of publishing the second volume of his
Grundgesetze der A rithmetik, Bertrand RusseUl4 demolished Frege's whole
14 On Bertrand Russell, an exceptional personality and formidable source of ideas
of all sorts, see Ronald. W. Clark, The Life of Bertrand Russell (Penguin Books,
1975). A much more scholarly biography is in course of publication, but the 979
pages of Clark, of which 160 are notes, bibliography and index, already contain
much information; sold at £2.95 when it appeared, one cannot ask for much more
. .. Do not seek mathematical logic there: Clark is a "man of letters" , does not
I - Sets and Functions
x
c
fig.!.
/X~eIl}'{{eIl'{eIl}}}
/~l={eIl'{eIl}}
{ell} =B
' " A = el l
fig. 1 bis.
The significance of the signs { and } will be explained in the following n O •
2 - The set defined by a relation. Intersections and unions
In practice, a set is most often defined by a characteristic property of its
elements: "the set of integers between 2 and 25", ''the set of points distant
15 km from a given point 0", "the set of rational numbers < 11"", ''the set of
functions f defined on R with values in R" , etc. A seemingly obvious general
statement, due to Frege, is that for every proposition or relation P{ x} in
which there appears a variable x symbolising a totally undetermined object
one can speak of the set of those x such that P{ x} is true; this set will be
unique, by the axiom of extension. If you choose the relation x = x you
will thus obtain the set of all mathematical objects, a creature which justly
evoked great suspicion in Cantor; he spoke of it as a "class" of sets, a concept
which the logicians later used and developed.
In 1903, as Frege was on the point of publishing the second volume of his
Grundgesetze der A rithmetik, Bertrand RusseUl4 demolished Frege's whole
14 On Bertrand Russell, an exceptional personality and formidable source of ideas
of all sorts, see Ronald. W. Clark, The Life of Bertrand Russell (Penguin Books,
1975). A much more scholarly biography is in course of publication, but the 979
pages of Clark, of which 160 are notes, bibliography and index, already contain
much information; sold at £2.95 when it appeared, one cannot ask for much more
. .. Do not seek mathematical logic there: Clark is a "man of letters" , does not