Ordinals and cardinals form the backbone of mathematics. Ordinal numbers describe orderings. Cardinal numbers are the measure of size in mathematics.
They first ordinals are the integers.
, the set containing all integers,
is the ordinal
of the integers. The next ordinal is the successor of
. This set has one members 
just as the successor
of 0 has one set, the set containing the empty set. The successor
of 
is the set containing 
and the successor
of 
.
Ordinals can be thought of as general way of representing
induction or
iteration. 
corresponds to iteration that can be characterized
by a loop up to any integer or induction on all the integers.
contains 
and all finite successors to 
.
contains all finite successors to 
.
contains 
for all integers n.
corresponds to two nest loops each with a limit
of any integer. 
corresponds to loops on the integers nested
n deep. 
corresponds to loops on the integers nested
n deep where n is a parameter. All forms of iterations
and all induction can be characterized by some ordinal number.
By using infinite sets to describe iteration one masks over the
rich combinatorial structures that are required to define higher levels
of iteration. Recursive iteration is characterized by the recursive
ordinals but there is no recursive algorithm to describe the structure
of all recursive ordinal
s
although there is such an algorithm for any
recursive ordinal.
Each integer is a finite cardinal
number. Different integers have
different sizes. 
is the cardinal number of all integers.
No one knows what the next largest cardinal is. Cantor proved that
the Cardinal of the reals is larger than the cardinal of the integers
by showing that there could not exist a one to one map between the
integers and reals. Gödel
constructed a model for set theory
in which the first cardinal larger than the integers is the cardinal
of the real numbers. Gödel
proved this model was consistent of
ZF set theory was consistent[19,20]. Cohen proved that
if ZF set theory is consistent that it is consistent to assume there
exists cardinals greater than the cardinal of the integers but
less than the cardinal of the reals[11]. This question is known as
the Continuum Hypothesis
and is undecidable in ZF.
If there do not exist completed infinite totalities than there does not exist a single real number let alone the set of all numbers. In that case the set of all real numbers only make sense relative to a given axiom system. The continuum hypothesis is true, false or unprovable in a given formal system, but it has no absolute truth value.
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