Appendices
273
to be a collection, e.g. the set of all dogs is a collection that contains all dogs
and nothing else.
Russell explored the question of whether a set can contain itself. In the
above example, this is clearly not the case; a set of dogs (i.e., a collection of
dogs) is not a dog. However, if we consider a set of sets, i.e., a collection of
collections, then it is possible for the set to contain itself. In fact, the set of
all sets must contain itself.
Suppose now we take this process one step further, and consider the set
of all sets that do not contain themselves. The paradox occurs when we ask
ourselves: “Does this set, so defined, contain itself, or not?” If we say, “no, it
does not contain itself ”, we see from its definition as the set of all sets that do
not contain themselves, that it does contain itself. Conversely, if we say that
it does contain itself, we see that this is wrong, because by definition it is the
set of all sets that do not contain themselves, and therefore must not contain
itself.
If you are having trouble getting your head around the paradox, spare a
thought for the innocent eight-year-olds of the sixties, who were expected to
master set theory on their way to learning that 2 + 2 = 4.
A.4.2 Gödel’s Incompleteness Theorem
In analogy with the Liar’s Paradox that we have already discussed, let us
consider the statement: This statement is unprovable. If the statement is provable, then the proof will prove something that is false, which does not bode
well for mathematics. The only alternative is that the statement is unprovable. In other words, although the statement is true, it cannot be proved, i.e.,
there are some statements that, although true, cannot be proved.
Of course, the above argument is only an outline of the logic that Gödel
employs in his proof. His genius lies in encoding This statement is unprovable into a natural number, and then examining the implications within the
rules of arithmetic. For a deeper insight into this intriguing topic, please see
Gödel’s Proof by Ernest Nagel and James R. Newman, referenced in Chap. 4.
A summary of Gödel’s approach is given by Natalie Wolchover [7].
A.5.1 Common Sense
The remark: common sense is not so common is often attributed to Voltaire: Le
sens commun n’est pas si commun in the Dictionnaire philosophique portatif ,
273
to be a collection, e.g. the set of all dogs is a collection that contains all dogs
and nothing else.
Russell explored the question of whether a set can contain itself. In the
above example, this is clearly not the case; a set of dogs (i.e., a collection of
dogs) is not a dog. However, if we consider a set of sets, i.e., a collection of
collections, then it is possible for the set to contain itself. In fact, the set of
all sets must contain itself.
Suppose now we take this process one step further, and consider the set
of all sets that do not contain themselves. The paradox occurs when we ask
ourselves: “Does this set, so defined, contain itself, or not?” If we say, “no, it
does not contain itself ”, we see from its definition as the set of all sets that do
not contain themselves, that it does contain itself. Conversely, if we say that
it does contain itself, we see that this is wrong, because by definition it is the
set of all sets that do not contain themselves, and therefore must not contain
itself.
If you are having trouble getting your head around the paradox, spare a
thought for the innocent eight-year-olds of the sixties, who were expected to
master set theory on their way to learning that 2 + 2 = 4.
A.4.2 Gödel’s Incompleteness Theorem
In analogy with the Liar’s Paradox that we have already discussed, let us
consider the statement: This statement is unprovable. If the statement is provable, then the proof will prove something that is false, which does not bode
well for mathematics. The only alternative is that the statement is unprovable. In other words, although the statement is true, it cannot be proved, i.e.,
there are some statements that, although true, cannot be proved.
Of course, the above argument is only an outline of the logic that Gödel
employs in his proof. His genius lies in encoding This statement is unprovable into a natural number, and then examining the implications within the
rules of arithmetic. For a deeper insight into this intriguing topic, please see
Gödel’s Proof by Ernest Nagel and James R. Newman, referenced in Chap. 4.
A summary of Gödel’s approach is given by Natalie Wolchover [7].
A.5.1 Common Sense
The remark: common sense is not so common is often attributed to Voltaire: Le
sens commun n’est pas si commun in the Dictionnaire philosophique portatif ,