"But, really, mathematics is not religion; it cannot be founded on faith. And what was most important, the methods yielding such remarkable results in the hands of the great masters began to lead to errors and paradoxes when employed by their less talented students. The masters were kept from error by their perfect mathematical intuition, that subconscious feeling that often leads to the right answer more quickly than lengthy logical reasoning. But the students did not possess this intuition […]"(Naum Ya. Vilenkin, "Stories about Sets", 1968)
"In order to define a concept we have to indicate first of
all that it is a special case of a more general concept. This is impossible for
the concept of set, since this concept is already as broad as possible and is
thus not a special case of any other concept."
"In practice, let us note, the determination of sets by means
of characterizing criteria runs into difficulty because of the ambiguity of our
language. The task of separating the objects belonging to a set from those that
do not is often made difficult by the large number of objects of intermediate type."
"Infinite sets possess remarkable properties. In studying these
properties mathematicians were led to continually perfect their reasoning and
to further develop mathematical logic."
"It would not be an exaggeration to say that all of mathematics derives from the concept of infinity. In mathematics, as a rule, we are not interested in individual objects (numbers, geometric figures), but in whole classes of such objects: all natural numbers, all triangles, and so on. But such a collection consists of an infinite number of individual objects." (Naum Ya. Vilenkin, "Stories about Sets", 1968)
"Many examples occur in the theory of sets in which the definition of the set is self-contradictory. The study of the question of the conditions under which this takes place leads to deep questions of logic. Consideration of these questions has completely changed the face of the subject." (Naum Ya. Vilenkin, "Stories about Sets", 1968)
"[…] mathematicians and philosophers have always been
interested in the concept of infinity. This interest arose at the very moment
when it became clear that each natural number has a successor, i.e., that the number
sequence is infinite. However, even the first attempts to cope with infinity
lead to numerous paradoxes."
"The very name 'set' leads us to think that any set must contain many elements (at least two). But this is not the case. In mathematics it is sometimes necessary to examine sets having only one element and sometimes even a set having no elements at all." (Naum Ya. Vilenkin, "Stories about Sets", 1968)
"Two kinds of sets turn up in geometry. First of all, in geometry
we ordinarily talk about the properties of some set of geometric figures. For
example, the theorem stating that the diagonals of a parallelogram bisect each
other relates to the set of all parallelograms. Secondly, the geometric figures
are themselves sets composed of the points occurring within them. We can
therefore speak of the set of all points contained within a given circle, of
the set of all points within a given cone, etc."
"Unfortunately, we are not in a position to give a rigorous definition of the fundamental concept of the theory : the concept of set. Of course, we could say that a set is a collection, a union, an ensemble, a family, a system, a class, etc. But this would not be a mathematical definition, but rather a misuse of the multitude of words available in the English language." (Naum Ya. Vilenkin, "Stories about Sets", 1968)
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