Site of Denis Maksudov (Ufa, Russia)

Here I collected my following works:

1) extended arrows notation (December 2016)

2) the extended Wilfried Buchholz's functions (April 2017)

3) fundamental sequences for the functions collapsing -weakly inaccessible cardinals (August 2017).

Extended arrow notation

We define for non-zero natural numbers  and for ordinal number

1)  if

2)  if

3)  if

4)  if  is a countable limit ordinal

where  denotes the b-th element of the fundamental sequence assigned to the limit ordinal .

The fast-growing hierarchy

The fast-growing hierarchy is defined as follows:

1.  if  is a countable limit ordinal

where  are non-negative integers and  is the -th element of the fundamental sequence assigned to the limit ordinal .

The extended Wilfried Buchholz's functions

We rewrite Buchholz's definition as follows:

where

and  is the smallest infinite ordinal.

There is only one little detail difference with original Buchholz definition: ordinal  is not limited by , now ordinal  belongs to previous set . Limit of this notation must be omega fixed point

Normal form for the extended Wilfried Buchholz's functions

The normal form for 0 is 0. If  is a nonzero ordinal number  then the normal form for  is where  is a positive integer and  and each  are also written in normal form.

Fundamental sequences for the extended Wilfried Buchholz's functions

The fundamental sequence for an ordinal number  with cofinality  is a strictly increasing sequence  with length  and with limit , where  is the -th element of this sequence. If  is a successor ordinal then  and the fundamental sequence has only one element . If  is a limit ordinal then

For nonzero ordinals , written in normal form, fundamental sequences are defined as follows:

1. If  where  then  and
2. If , then  and
3. If , then  and
4. If  and , then  and
5. If  then  and  (and note: )
6. If  and  then  and
7. If  and  then  and  where

Limit of this notation . If  then  and  and

Fundamental sequences for the functions collapsing -weakly inaccessible cardinals

Definition of the functions collapsing -weakly inaccessible cardinals

An ordinal is -weakly inaccessible if it's an uncountable regular cardinal and it's a limit of -weakly inaccessible cardinals for all

Let  be the first -weakly inaccessible cardinal,  be the next -weakly inaccessible cardinal after , and  for limit ordinal

On this page the variables  are reserved for uncountable regular cardinals of the form  or

Then,

1.  for
2.  for

Standard form for ordinals

1. The standard form for 0 is 0
2. If  is of the form , then the standard form for  is  where  and  are expressed in standard form
3. If  is not additively principal and , then the standard form for  is , where the  are principal ordinals with , and the  are expressed in standard form
4. If  is an additively principal ordinal but not of the form , then  is expressible in the form . Then the standard form for  is  where and  are expressed in standard form

Fundamental sequences

The fundamental sequence for an ordinal number  with cofinality  is a strictly increasing sequence  with length  and with limit , where  is the -th element of this sequence.

Let  and  where  denotes the set of successor ordinals and  denotes the set of limit ordinals.

For non-zero ordinals  written in standard form fundamental sequences are defined as follows:

1. If  with  then  and
2. If  then  and
3. If  then  and
4. If  and  then  and
5. If  then  and  and
6. If  then  and  and
7. If  and  then  and  and
8. If  and  then  and
9. If  and  then  and
10. If  and  and  then  and
11. If  then  and
12. If  and  then  and
13. If  and  then  and
14. If  and  then  and  with  and

Limit of this notation . If  then  and  and

Let X be equal to  using the fast-growing hierarchy with fundamental sequences for the functions collapsing -weakly inaccessible cardinals. Let Rax be spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of X meters. Boundaries of this region are not  defined  by really existing  structures of the universe  or inhomogeneities in the distribution of matter because we nothing know about structures or inhomogeneities on so huge scales, but nevertheless this region can be defined at simple assumption that our universe is infinite.  The main motivation of this definition is wishing to apply somehow huge numbers, which can be obtained using the fast-growing hierarchy, to the real physical world.

Note 1: a sphere with center  and with radius  is the set of all points that are at distance  from .