Shiraito983
He/Him- 1,237
- 404
and their leyers
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explain Inaccessible cardinal
- Thread starter Shiraito983
- Start date
- 373
- 111
In set theory, an inaccessible cardinal is a cardinal number that possesses a particular set-theoretic largeness and structural robustness, rendering it impervious to standard methods of constructability within the established hierarchies of cardinality. Formally denoted as κ, an inaccessible cardinal manifests as an unattainable point along the progression of the cumulative hierarchy, resisting the grasp of ordinal operations, and exhibiting an insusceptibility to iterative set formation.
The foundational significance of an inaccessible cardinal resides in its role as an obstruction to the potential recursive exhaustion of the universe of sets, thereby establishing a threshold beyond which conventional set-theoretic constructions falter. Specifically, an inaccessible cardinal κ is characterized by two primary attributes: regularity and inaccessibility. Regularity connotes that the cardinality of sets below κ is stabilized under cardinal exponentiation, conferring a degree of stability and coherence upon the cumulative hierarchy.
Moreover, the property of inaccessibility endows κ with a certain unreachability from below through the process of ordinal exponentiation. That is to say, for any ordinal α, the ordinal power α^+ (the successor of α) is strictly less than κ. This imparts a hierarchical inaccessibility to κ, precluding its representation as the result of iterative processes applied to smaller cardinals.
To elaborate further, the conceptual stratification of inaccessible cardinals into a hierarchy is accomplished through the iterative introduction of larger and larger inaccessible cardinals, yielding a cumulative progression of increasingly impervious strata. This stratified arrangement engenders a multi-layered structure, where each layer, defined by an inaccessible cardinal, serves as a robust foundation resistant to the intrusions of conventional set-theoretic operations.
The foundational significance of an inaccessible cardinal resides in its role as an obstruction to the potential recursive exhaustion of the universe of sets, thereby establishing a threshold beyond which conventional set-theoretic constructions falter. Specifically, an inaccessible cardinal κ is characterized by two primary attributes: regularity and inaccessibility. Regularity connotes that the cardinality of sets below κ is stabilized under cardinal exponentiation, conferring a degree of stability and coherence upon the cumulative hierarchy.
Moreover, the property of inaccessibility endows κ with a certain unreachability from below through the process of ordinal exponentiation. That is to say, for any ordinal α, the ordinal power α^+ (the successor of α) is strictly less than κ. This imparts a hierarchical inaccessibility to κ, precluding its representation as the result of iterative processes applied to smaller cardinals.
To elaborate further, the conceptual stratification of inaccessible cardinals into a hierarchy is accomplished through the iterative introduction of larger and larger inaccessible cardinals, yielding a cumulative progression of increasingly impervious strata. This stratified arrangement engenders a multi-layered structure, where each layer, defined by an inaccessible cardinal, serves as a robust foundation resistant to the intrusions of conventional set-theoretic operations.
Scottycj256
He/Him- 1,770
- 924
complete bullshit that will mean nothing once ultima's revision goes through
(which will be after he finishes his argument with DT, which will take forever)for now though what taiwanese said
Shiraito983
He/Him- 1,237
- 404
- Thread starter
- #4
It
Its still going?complete bullshit that will mean nothing once ultima's revision goes through
for now though what taiwanese said
Scottycj256
He/Him- 1,770
- 924
On the Many, Many Incoherences of the Tiering System
Or, "Why did I make this system? Am I stupid?" This thread has certainly been a long time coming. And though its placement in the general discussion board makes this clear already, it is probably best that I say it explicitly: I don't actually have any interest in bringing any changes to the...
ye, when this passes, cardinality/set theory is being bumped down to High 1-B+ and R>F transcendences are becoming 1-A, provided they don't contradict our standards for an R>F transcendence
