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Types of numbers:
Natural numbers: The positive integers. The number of natural numbers is sometimes designated as n.
Cardinal Numbers: Cardinal numbers consider numbers as sets or collections of things that are equinumerous or can
be put in a one to one correspondence. The basic experience involved here is comparing a set to some paradigmatic
set to see if they are equivalent.
Ordinal Numbers: Ordinal numbers involve the concept of a well ordered set, or a set based upon some sequence or
ordering relation, such as counting, the successor relation. The basic experience involved here is going through a
collection by counting to determine its number.
For finite numbers, ordinals and cardinals are equivalent.
Rational numbers: Numbers that can be expressed as a ratio of integers. These can be represented by a repeating
infinite decimal expansion. E. g. 1/3= 0.33333. . . . These comprise a denumerable infinity.
Irrational numbers: Numbers that cannot be represented as a ratio of integers. These can be represented by a nonrepeating infinite decimal expansion.
Real numbers: All the rationals and the irrationals. Can be conceived of as all the points on the number line.
These comprise a non-denumerable infinity. The number of real numbers is sometimes designated as c, for the
number of points on the continuum.
Infinite numbers:
Infinite ordinals:
 - Infinity. The first infinite ordinal. The ordinal number corresponding to n, or the natural numbers. The limit of
the series {1, 2, 3, 4, . . .}. This can also be seen as 1+1+1+1 . . . . From this you can see that
1 + 
But + 1= +1
2 - Two infinities. The limit of { +1, +2, +3, +4, . . . }
 Infinity Infinities. The limit of { 1, 2, 3, 4, . . . } This can also be seen as 

So =

 - An Infinity of Infinities of Infinities. A really, really big number. (Normal Words fail us here.) The limit of
{ 1, 2, 3, 4, . . . }.
 - Infinity to the Infinitieth power. The limit of { , ,  . . .}. This can also be seen as 

S
=
But all of these, and any more arrived at in this way, are really the same cardinal number as .
Infinite Cardinal numbers:
The cardinal number corresponding to  is 0. (Aleph null) . All denumerable infinities are have this cardinal
number. it is the cardinal number of n, The natural numbers.
The cardinal number associated with the Real numbers ( by the Continuuum Hypothesis) or the first nondenumerable infinity is 1 (Aleph one).