I think 0^0 has to be examined from the level of the Z* ( http://mathworld.wolfram.com/ZStar.html ) numbers.
By using the empty set (with the Von Neumann Hierarchy), we can construct the set of Z* numbers {0,1,2,3,4,...}:
Let us look at this question from a structural point of view, and we shall do it by using the base value expansion method where x is some Z* number.
For example we shall use number 26 represented by base 10 and base 3:
As we can see, no matter what base value > 1 is,
it is always reduced to 1 when power level = 0.
Now, let us check base value 1.
From the above we can learn that when base value = 1 it still can be represented by base value expansion method.
Now, what about base value 0 ?
Let us examine it from a structural point of view, for example base 10:
We can see that there exist two basic structural types:  AND _
 is what we call a singleton and it can be notated as {.} which is a singleton included in some set.
But what about _ ?
By standard Boolean logic every object which included in some set must have a single and unique value.
If we examine _ we shall find that the structural property of this element is out of the scope of standard Boolean logic.
And the reason is this: _ exists between any two different singletons, therefore its property is at least and simultaneously two different states, and this property (to be simultaneously in two different states)is not in the scope of Boolean logic.
But as we can see, without this element we can't develop our number system beyond 1.
The "job" of _ is to be the "platform of memory" that gives us the ability to move beyond 1.
The base value of this "platform" is 0 because there are no singletons in its scope.
But unlike nothingness this object exists, therefore its full notation is 0^0=1(set's contenet exists).
Now we have two basic objects: ^0=1 , _^0=1 where 1 stands for "there exist some object".
Those two different structures are indistinguishable by their quantitative property.
If we want to move beyond "there exist some object" (which are notated as 1^0=1 or 0^0=1) we have to associate between tham.
The result is the Natural numbers > 1, which are combinations of at least 2*1^0 singletons connected by at least 0^0 "platform of memory", for example:
From this point of view we have at least 3 structural types of set's contents:
{}, {.} AND {_} .
By quantitative point of view we have:
{} = 0 (set's content does not exist)
{_} = 0^0 = 1 (set's contenet exists)
{.} = 1^0 = 1 (set's contenet exists)
What do you think?
Organic
By using the empty set (with the Von Neumann Hierarchy), we can construct the set of Z* numbers {0,1,2,3,4,...}:
Code:
[b][i]0[/i][/b] = { } (notation = {})
[b][i]1[/i][/b] = {[b]{[/b] [b]}[/b]} (notation = {0})
[b][i]2[/i][/b] = {[b]{[/b] [b]}[/b],[b]{[/b]{ }[b]}[/b]} (notation = {0,1})
[b][i]3[/i][/b] = {[b]{[/b] [b]}[/b],[b]{[/b]{ }[b]}[/b],[b]{[/b]{ },{{ }}[b]}[/b]} (notation = {0,1,2})
[b][i]4[/i][/b] = {[b]{[/b] [b]}[/b],[b]{[/b]{ }[b]}[/b],[b]{[/b]{ },{{ }}[b]}[/b],[b]{[/b]{ },{{ }},{{ },{{ }}}[b]}[/b]} (notation = {0,1,2,3})
and so on.
For example we shall use number 26 represented by base 10 and base 3:
Code:
Number 26 represented by base 10:
^0 0123456789

_
__
___
Base 10 = ____
_____
______
_______
________
^1 
 1 0
( 2*10 + 6*10 )
^0 012345678901234567890123456

___
______
_________
____________
_______________
______________...
________________...
__________________...
^1  0  1  2
_________ 
___________________
_ ...

Number 26 represented by base 3:
^0 012
Base 3 = 
_
^1  3 2 1 0
( 0*3 + 2*3 + 2*3 + 2*3 )
^0 012012012012012012012012012

_________
^1 0 1 2 0 1 2 0 1 2
__  __  __ 
_____ _____ _____
^2  0  1  2
________ 
_________________
^3  0
_ ...

it is always reduced to 1 when power level = 0.
Now, let us check base value 1.
Code:
^0 0
Base 1 = 

^1 
Now, what about base value 0 ?
Let us examine it from a structural point of view, for example base 10:
Code:
^0 0123456789

_
__
___
Base 10 = ____
_____
______
_______
________
^1 

 is what we call a singleton and it can be notated as {.} which is a singleton included in some set.
But what about _ ?
By standard Boolean logic every object which included in some set must have a single and unique value.
If we examine _ we shall find that the structural property of this element is out of the scope of standard Boolean logic.
And the reason is this: _ exists between any two different singletons, therefore its property is at least and simultaneously two different states, and this property (to be simultaneously in two different states)is not in the scope of Boolean logic.
But as we can see, without this element we can't develop our number system beyond 1.
The "job" of _ is to be the "platform of memory" that gives us the ability to move beyond 1.
The base value of this "platform" is 0 because there are no singletons in its scope.
But unlike nothingness this object exists, therefore its full notation is 0^0=1(set's contenet exists).
Now we have two basic objects: ^0=1 , _^0=1 where 1 stands for "there exist some object".
Those two different structures are indistinguishable by their quantitative property.
If we want to move beyond "there exist some object" (which are notated as 1^0=1 or 0^0=1) we have to associate between tham.
The result is the Natural numbers > 1, which are combinations of at least 2*1^0 singletons connected by at least 0^0 "platform of memory", for example:
Code:
^0 0 1
. . = 2*1^0
Base 2 =  
___
^1  ^

0^0
{.} {.} = 2*1^0
Base 2 =  {} 
{__}
 ^

0^0
{}, {.} AND {_} .
By quantitative point of view we have:
{} = 0 (set's content does not exist)
{_} = 0^0 = 1 (set's contenet exists)
{.} = 1^0 = 1 (set's contenet exists)
What do you think?
Organic
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