Two card players 
and 
are given some cards such that each
gets an ace. Thus, both 
and 
know that the following is a
fact:
When asked whether they knew if the other one had an ace or not, they
both would answer ``no.'' If they are told that at least one of them
has an ace and asked the above question again, first they both would
answer ``no.'' But upon hearing 
answer ``no,'' 
would know
that 
has an ace. Because, if 
does not know 
has an
ace, having heard that at least one of them does, it can only be
because 
has an ace. Obviously, 
would reason the same way,
too. So, they would conclude that each has an ace. Therefore, being
told that at least one of them has an ace must have added some
information to the situation. How can being told a fact that each of
them already knew increase their information? This is known as
Conway's Paradox. The solution relies on the fact that initially
was known by each of them, but it was not common
knowledge. Only after it became common knowledge, it gave more
information.
Hence, common knowledge can be viewed as iterated knowledge of
of the following form: 
knows 
, 
knows
, 
knows 
knows 
, 
knows 
knows
, and so on. This iteration can be represented by an infinite
sequence of facts (where K is the relation ``knows'' and s is the
situation in which the above game takes place, hence 
):
, 
, 
, 
, 
However, considering the system
of equations
the Solution Lemma asserts the existence of the unique sets 
and
satisfying these equations, respectively, where
Then, the fact that s is common knowledge can more effectively be
represented by 
which contains just two infons and is circular.