The meaning of "all" is not strictly defined. Of course, in most cases, this does not prevent us from correctly applying the concept of "set" and the nature of set to solve some problems. When applying the concept and theory of set, we require the so-called "certainty" of set:
That is, for any thing Y and any set B, "Y is a thing in set B" and "Y is not a thing in set B" must have one assertion and only one assertion is correct.
So, generally speaking, the definition of set is very clear. However, in some cases, the concept of set defined by the above descriptive definition will bring trouble. For example:
(1) barber paradox
The barber said; He shaves all those who don't shave.
At first glance, the barber's customers form a group B. However, there is a contradiction when discussing whether the barber himself belongs to B. If the barber doesn't shave himself, he should belong to B, that is, to be his customer, he should shave himself. In this way, he belongs to "the person who shaves himself", so he doesn't belong to B, but if he doesn't belong to B, that is, "shaving himself", he is not the object of service, so he shouldn't shave himself, thus creating contradictions.
There are many such paradoxes.
(2) semantic paradox
Because English has only a limited number of syllables, English expressions with less than 40 syllables in English can only be a limited number. In particular, only a limited number of positive integers can be expressed by such expressions. We use b to mean "the set of all positive integers that can be expressed by such an expression". Let x be the smallest positive integer that cannot be expressed by less than 40 syllables. But x can be expressed in the following English expressions:
this
minimum
active
integer
which
be
no
indicate
pass by
One; one
express
exist
this
English
language
include
fewer/ lesser
compare
forty
Syllables
The above expression only contains 37 syllables, so X belongs to B, which contradicts that X does not belong to B. ..
In view of the contradiction of the above types of examples, mathematicians have re-studied the basis of set theory and tried to avoid the paradox by various methods. They put forward the axiomatic system of set theory, whose function is to impose certain restrictions on the set as the object of mathematical research, thus eliminating the possibility of paradox. Under these restrictions, all the above "sets" are excluded from the object of mathematical research. Of course, these restrictions are also very loose, enough to keep all the valuable things of mathematical theory and meet the needs of mathematical development. In this axiomatic theory, the concept of set has not been defined, but its nature is embodied by the so-called "axiom of set". The research based on set theory has led to the rapid development of mathematical logic, an important branch of mathematics.