User:Vel!/pu/List Of Infinities

$$\infin$$, $$\omega$$, $$\aleph_0$$

The smallest infinity. It has multiple interpretations.

This is the infinity when you talk about the limit as n goes to infinity. Take the series 1+1/2+1/4+1/8+1/16+.... It converges to 2 - it approaches 2 as the number of terms approaches infinity. 1+1/2+1/3+1/4+1/5+... will grow without bound - it diverges (but very slowly). The series 1-1+1-1+1-1+1-1+... will continually alternate between 1 and 0.

This is the infinity of the natural numbers. The set {0,1,2,3,4,5,...} is infinite in this infinity sort of way. So are the integers: {...,-3,-2,-1,0,1,2,3,...} - they can be matched up onto the natural numbers:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9,...} |  |  |  |  |  |  |  |  |  | {0, 1,-1, 2,-2, 3,-3, 4,-4, 5,...} The concept of one-to-one correspondence is at the core of set theory. Two sets are the same size (called cardinality) if one can be matched up so each member in one set gets a unique member in the other. The rational numbers also have this cardinality, although it may not seem like this at first: 0  1,   2,   3,   4,   5,   6,   7   1/2,   , 3/2,    , 5/2,    , 7/2,...  1/3, 2/3,    , 4/3, 5/3,    , 7/3,...   1/4,    , 3/4,    , 5/4,    , 7/4,...   1/5, 2/5, 3/5, 4/5,    , 6/5, 7/5,...   1/6,    ,    ,    , 5/6,    , 7/6,...   1/7, 2/7, 3/7, 4/7, 5/7, 6/7,    ,...    .    .    .    .    .    .   .    .    .    .    .    .    .     .   .    .    .    .    .    .       . You can go diagonal to get the following sequence:

0, 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5, 3/4, 4/3, 5/2, 6, 1/7, 3/5, 5/3, 7, 1/8,...

These are only the nonnegative rationals, but an alternation method can be done to show that the rationals have the same cardinality as the counting numbers.

When this infinity is used as a cardinality, it is called $$\aleph_0$$ (aleph-null).

When an infinite amount of things are in order, you use a type of infinity called ordinals. This ordinal is $$\omega$$ - the omega-th member in a set is the first one with a higher order than infinitely many other members. Ordinals will be discussed throughout.