Infinite Sets and Countability
* Infinite Sets
* Russell's Paradox
* The Beginnings of Theoretical Computer Science
set, we can always designate one element as the first member, another as the second
If there are k elements in the set, then these can be listed in the order we have
Therefore, a finite set is one which can be enumerated by the positive integers from 1 up to some
More precisely, A is finite if there is a positive integer k such that there is a one-to-one
correspondence between A and the set of all natural numbers less than k.
We define one-to-one
correspondence between the elements of a set P and the elements of a set Q if it is possible to pair
off the elements of of P and Q such that every element of P is paired off with a distinct element of Q.
We are defining a function that maps the elements of P to the elements of Q; this function must be
both one-to-one and onto (or bijective for you function enthusiasts out there).
If a set is infinite, we may still be able to select a first element s
, and a second element s
, but we
have no limit k.
So the list of chosen elements may look like this:
Such an infinite set is called
Both finite and denumerable sets are
because we can count, or enumerate the elements in the set.
Being countable, however, does not
always mean that we can give a value for the total number of elements in the set; it just means we
can say "Here is a first one, here is a second one.
In other words, a denumerable set is one
where we can define a one-to-one correspondence between the elements in the set and
Thus, the set
is, in a sense, the most basic of all infinite sets.
The reason for this is that the one-
to-one correspondence can be used to "count" the elements of an infinite set.
If f is a one-to-one and
onto function from
to some infinite set A, then f(1) can be designated as the first element of A,
f(2), the second, and so forth.
Because f is one-to-one, no element is ever counted twice; and
because f is onto, every element of A is counted eventually.