Space-Wide Cardinal

A Space-Wide Cardinal K is a very large (infinite) cardinal. Here is the definition (separated into parts).

"Define T as the first cardinal such that the set of all inaccessible cardinals (a-inaccessible, a+1-inaccessible, b-inaccessible, s is s-inaccessible, etc.) smaller than T are stationary (constant) relative to T. K is defined as the first cardinal greater than every value definable in a 1-argument function F such that each function F diagonalizes over T (as long as the extension F(n) for every n is not this extension, an extension of this extension, or an extension that creates cardinals greater than T.). "

Now we can define a sequence, n-Space-Wide Cardinals are defined below: "Define T as the first cardinal such that the set of all (n-1)-Space-Wide cardinals (a-inaccessible, a+1-inaccessible, b-inaccessible, s is s-inaccessible, etc.) smaller than T are stationary relative to T. K is defined as the first cardinal greater than every value definable in a 1-argument function F such that each function F n-diagonalizes (diagonalize n times) over T (as long as the extension F(n) for every n is not this extension, an extension of this extension, or an extension that creates cardinals greater than T.). "

The least Space-Wide Cardinal K is greater or equal to the least Mahlo Cardinal. However, the least 2-Space-Wide Cardinal K vastly exceeds an n-Mahlo Cardinal for all positive finite integers n. (I think) An w-Space-Wide-Cardinal K is defined here:

"T is defined as the least cardinal such that for every natural number N, the set of finite Space-Wide-Cardinals are stationary relative to T. K is defined as the first cardinal defined in a 1-argument function F such that each function F n-diagonalizes for any finite number n over T. (same rules to prevent contradictions)."

A Cardinal K is called nth Order-Space-Wide if it is w-Space-Wide for the n-1th order (a 0th order denotes a regular Space Wide Cardinal), and:

"T is defined as the least cardinal such that for every natural number N, the set of all definable Space-Wide-Cardinals are stationary relative to T. S is defined as the first cardinal defined in a 1-argument function F such that each function F n-diagonalizes (for any and all n) over (n-1)th order Space-Wide-Cardinals. (same rules to prevent contradictions)"

A cardinal K is Extremely Space-Wide if it is Kth Order-Space-Wide.