All the numbers

A^w A^w B A A A ...     1/(w+w)
A^w B A A A ...         1/w
A A B A A A ...         1/4
A B A A A ...           1/2
A B B A A A ...         3/4
A B^w B A A A ...       1-1/w
B A A A ...             1
B B A A A ...           2
B^w A^w B A A A ...     w/2
B^w A B A A A ...       w-1 
B^w B A A A ...         w
B^w B B A A A ...       w+1
B^w B^w B A A A ...     w+w


Transfinite cardinals

An ordinal a is identified with the set { b : b < a } of all ordinals less
than a : 0 = {}, 1 = {0}, 2 = {0, 1} = {{}, {{}}}, 3 = {0, 1, 2}...
w = {0, 1, 2...}, w+1 = {0, 1, 2, ..., w}...

If a = lim(a_n) then b < a iff b is less than one of the a_n's. a is the least
ordinal greater than all the a_n's. Therefore, a = { b : b < a } 
= { b : exist n . b < a_n } = U { b : b < a_n } = U a_n.

Definitions
        sup A = first ordinal > any member of A.
        S is countable if 0, finite or card w.

Two principles of ordinal generation :
I: a -> a + 1 least ordinal > a
II: sequence a_n -> lim (a_n) = least ordinal > all the a_n
combined into
III: For every set A of ordinals, there is a least ordinal > every member of A 
called sup A.

Definitions
        First number class : w, finite ordinals
        Second number class : all additional ordinals that can be obtained 
                by repeatedly using I and II where II is applied only to
                countable sequences a_n. This is the collection of all
                countable infinite ordinals.
        aleph_1 = sup (second number class)

Aleph_1 lies beyond any countable sum of ordinals less than itself.
We can only reach aleph_1 by adding together aleph_1 ordinals.
An ordinal a that cannot be represented as a sum of less than a ordinals 
less than a is called regular. 0, 1, 2, w and aleph_1 are regular.

Cardinality

card S is the least ordinal a such that card a = card S.
aleph_0 = w
aleph_1 is the first uncountable cardinal.
aleph_2 is the first cardinal > aleph_1.
aleph_w = { 0, 1, ... aleph_0, ... , aleph_1, ... }
For every cardinal k, k < 2^k.
Generalized Continuum Hypothesis : for all k, 2^k = k+ the least cardinal
greater than k.
2^aleph_a = aleph_a+1

The continuum

Large cardinals

An ordinal is a generalized number that can be counted up to by a process 
of repeatedly taking the next step after all the preceding steps.
W (the collection of all ordinals) can be seen as an imaginary ordinal.
W is a regulal cardinal.
In ZF, regularity is introduced as the axiom of replacement :
{ f(y) : y in x } is a set.
This axiom is a weak form of the Reflection Principle :
For every conceivable property of ordinals P, if W has property P, then
there is at least one ordinal x < W that has also property P.
Not only is W regular, W does not have the form k+ for any cardinal k.
W is a limit cardinal, rather than successor. 
A cardinal that has this property is called inaccessible.
t is inaccessible if
- t is regular
- whenever k < t, k+ < t

Analogy :
        0       1       2       w       aleph_1
        0       w       aleph_1 theta   rho

There are W inaccessible cardinals less than W. W is hyperinaccessible.
0-hyperinaccessible = inaccessible
1-hyperinaccessible = hyperinaccessible
v is a+1-hyperinaccessible if v is a-hyperinaccessible preceded by v a-hyperinaccessibles
for limit l : v is l-hyperinaccessible if v is a-hyperinaccessible for every a<l

m is a-hyperinaccessible for every a < m : m is hyper-hyper-inaccessible or
hyper^2 inaccessible.
m is 1-hyper^2-inaccessible if m is inaccessible and there are m hyper^2-inaccessibles
less than m.
m is hyper^3-inaccessible if m is a-hyper^2-inaccessible for all a < m.
m is super-hyper-inaccessible if m is inaccessible and preceded by m
hyper^a-inaccessibles for every a < m.

r preceded by r inaccessibles, r hyperinaccessibles, r super-hyper-inaccessibles...
is called a Mahlo cardinal.
r is a Malho caridnal if r is inaccessible and whenever I is a set of ordinals 
such that there are r members of I less than r then there is an inaccessible 
cardinal k < r such that there are k members of I less than k.
r is Mahlo if it satisfies the Fixed-Point Reflexion Principle :
for every fixed-point property P, there is a k < r such that k enjoys P.
P is a fixed point property if it has the form : # is inaccessible and there
are # ordinals less than # that have property I.

The collection of all fixed-point properties is conceivable, and that is why the
Reflection Principle can be applied to get Mahlo cardinals. There are various other
standard conceivable collections of properties for which the same kind of argumentt
can be carried out. Large cardinals obtained in this way are called indescriptible
cardinals, and they are generally quite a bit bigger than the first Mahlo cardinal.

Strongly inaccessible cardinal t :
- t is regular
- whenever k < t, 2^k < t
GCH => strongly inaccessible = inaccessible

cardinals
        inaccessible
        Mahlo
        indescribable
        ineffable
        partition
        Ramsey
        measurable
        strongly compact
        supercompact
        extendible

        0       1       2       w       aleph_1
        0       w       aleph_1 theta   rho
        0       k               lambda

The first measurable cardinal is k.
k is measurable if there is a method of deciding which subsets of k are dense and
which are sparse. Dense subsets are called big or nodal.
The method for picking out the dense, big nodal subsets of k is simply
represented by taking the set N of all the dense subsets of k. k is 
measurable if we can find such an N that is k-complete, meaning that the
intersection of less than k of the members of N is also a member of k. 
k is measurable if there is a set M included in Parts(k) such that
a) no subsets of k with cardinality less than k are in M
b) the intersection of less than k members of M is also a member of M
c) For every A included in k, either A or the complement A' =
{ a in k : a not in A } of A is in M (but not both).
M is called a k-complete ultrafilter on k.


Reference

Infinity and the mind
The science and Philosophy of the Infinite
Rudy Rucker