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The first cardinal greater than inaccessible cardinals that do not reflect onto other cardinals, but are also extended with some special properties off of cardinals is the In-Compactness Cardinal T. (Inaccessible cardinals reflect downwards onto smaller inaccessibles.) A in-compactness cardinal T, would require a T amount of symbols to define. This is because an infinitely extending definition would not be able to reflect downwards due to it’s sheer amount of existing symbols after it and after that (infinitely looping). Also, a in-compactness cardinal T must be that largest cardinal definable with T symbols. Also, T+1 does reflect onto T, thus T+1 is a compact cardinal. |
The first cardinal greater than inaccessible cardinals that do not reflect onto other cardinals, but are also extended with some special properties off of cardinals is the In-Compactness Cardinal T. (Inaccessible cardinals reflect downwards onto smaller inaccessibles.) A in-compactness cardinal T, would require a T amount of symbols to define. This is because an infinitely extending definition would not be able to reflect downwards due to it’s sheer amount of existing symbols after it and after that (infinitely looping). Also, a in-compactness cardinal T must be that largest cardinal definable with T symbols. Also, T+1 does reflect onto T, thus T+1 is a compact cardinal. |
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− | The Compactness Cardinal T could also be called the Indescribable Cardinal T. |
+ | The In-Compactness Cardinal T could also be called the Indescribable Cardinal T. |

+ | [[Category:Numbers]] |
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+ | [[Category:Infinite]] |

## Latest revision as of 20:52, 25 April 2021

## Reflective-ness

I'll explain reflection using examples.

M cardinals are inaccessible with any and all inaccessible cardinals (besides itself) using any sort of sequence. M therefore has the sequence, a-inaccessible, hyper-inaccessible, richly-inaccessible,... but the sequence does not define M. Cof(M)=M. M cardinals reflect downward onto inaccessibles due to their previously defined inaccessibility.

However, N cardinals that are inaccessible with any and all M cardinals using any sort of sequence reflects onto M, which is trivial.

## Indescribable Cardinal T/In-Compactness Cardinal

The first cardinal greater than inaccessible cardinals that do not reflect onto other cardinals, but are also extended with some special properties off of cardinals is the In-Compactness Cardinal T. (Inaccessible cardinals reflect downwards onto smaller inaccessibles.) A in-compactness cardinal T, would require a T amount of symbols to define. This is because an infinitely extending definition would not be able to reflect downwards due to it’s sheer amount of existing symbols after it and after that (infinitely looping). Also, a in-compactness cardinal T must be that largest cardinal definable with T symbols. Also, T+1 does reflect onto T, thus T+1 is a compact cardinal.

The In-Compactness Cardinal T could also be called the Indescribable Cardinal T.