§1. Set Theory
9
reside in Paris and park on the public roads know that the set of days of the
month of August on which this tax is obligatory is empty, in contrast to the
rest of the year.
The most remarkable property of the empty set is that everything one can
say about its elements (though not about itself) is simultaneously true and
false, and that, further, no logical catastrophe ensues. When I once informed
a friend that every man who has passed the age of five hundred years makes
love three times a day she replied "that's false, I'm sure you wouldn't be able
to"; to this kind of typically feminine ad hominem logic - it is well known that
women are incapable of reasoning impersonally, objectively and abstractly -
I obviously replied "certainly, with you"; she then exclaimed "false, I'd be
dead and I don't like old men" and, to finish, threw a fit of nerves when I
replied that that did not contradict the initial proposition one whit. To learn
to juggle with the innumerable properties of the empty set is an excellent
exercise for developing your powers of reasoning; you could in particular
set yourself to detect all statements, including those in the present treatise,
which, taken literally, are false because the author has forgotten to posit that
a certain set is not empty: "every continuous function on a compact set attains
its maximum at a point of this set", "every bounded set has a strict upper
bound", etc.; these statements are false if the set under consideration is empty
because they affirm the existence of an element (possessing certain properties)
of the empty set. The perpetrators of such gross errors will generally reply
that they have passed the age of priggishness and trust to the good sense of
the reader, who is asked, implicitly, to trust the competence of the author.
The empty set is a subset of every other set X: since 0 contains no elements
at all, all its elements are also elements of X. It is also true that the empty
set may be an element of another set X as we shall now see.
It is on the empty set that one leans to "lift oneself up from the void";
figure 1 below represents a primary box X containing three secondary boxes
A, B, C, which are the elements of X; A is the empty set and contains no
elements, B is a box containing an empty box, so has one element, namely
the empty box in question, while C is a box with two elements: an empty
box and a box containing an empty box.
The representation above might lead the reader to believe that there are
four distinct empty sets in the schema for X; now there is only one empty
set in Nature, but, like the Holy Ghost, it is everywhere simultaneously. One
can finesse this niggle by replacing the imagery of boxes by the schema of
the relation x E y; on writing x --+ y to facilitate the graphic representation,
figure 1 would be replaced by the following schema:
that parking is free between 7 in the evening and 9 in the morning as well as on
Saturday and Sunday, i.e. outside working hours. One should teach the abc of
formal logic (a few pages of Plato would suffice) to the bureaucrats who hope to
fight pollution by taxing the non-polluters. Let us add that in certain countries
the residents buy a permit at the beginning of each year, so freeing them from
the daily racket which one suffers in Paris, and for a modest sum.
9
reside in Paris and park on the public roads know that the set of days of the
month of August on which this tax is obligatory is empty, in contrast to the
rest of the year.
The most remarkable property of the empty set is that everything one can
say about its elements (though not about itself) is simultaneously true and
false, and that, further, no logical catastrophe ensues. When I once informed
a friend that every man who has passed the age of five hundred years makes
love three times a day she replied "that's false, I'm sure you wouldn't be able
to"; to this kind of typically feminine ad hominem logic - it is well known that
women are incapable of reasoning impersonally, objectively and abstractly -
I obviously replied "certainly, with you"; she then exclaimed "false, I'd be
dead and I don't like old men" and, to finish, threw a fit of nerves when I
replied that that did not contradict the initial proposition one whit. To learn
to juggle with the innumerable properties of the empty set is an excellent
exercise for developing your powers of reasoning; you could in particular
set yourself to detect all statements, including those in the present treatise,
which, taken literally, are false because the author has forgotten to posit that
a certain set is not empty: "every continuous function on a compact set attains
its maximum at a point of this set", "every bounded set has a strict upper
bound", etc.; these statements are false if the set under consideration is empty
because they affirm the existence of an element (possessing certain properties)
of the empty set. The perpetrators of such gross errors will generally reply
that they have passed the age of priggishness and trust to the good sense of
the reader, who is asked, implicitly, to trust the competence of the author.
The empty set is a subset of every other set X: since 0 contains no elements
at all, all its elements are also elements of X. It is also true that the empty
set may be an element of another set X as we shall now see.
It is on the empty set that one leans to "lift oneself up from the void";
figure 1 below represents a primary box X containing three secondary boxes
A, B, C, which are the elements of X; A is the empty set and contains no
elements, B is a box containing an empty box, so has one element, namely
the empty box in question, while C is a box with two elements: an empty
box and a box containing an empty box.
The representation above might lead the reader to believe that there are
four distinct empty sets in the schema for X; now there is only one empty
set in Nature, but, like the Holy Ghost, it is everywhere simultaneously. One
can finesse this niggle by replacing the imagery of boxes by the schema of
the relation x E y; on writing x --+ y to facilitate the graphic representation,
figure 1 would be replaced by the following schema:
that parking is free between 7 in the evening and 9 in the morning as well as on
Saturday and Sunday, i.e. outside working hours. One should teach the abc of
formal logic (a few pages of Plato would suffice) to the bureaucrats who hope to
fight pollution by taxing the non-polluters. Let us add that in certain countries
the residents buy a permit at the beginning of each year, so freeing them from
the daily racket which one suffers in Paris, and for a modest sum.