The category of all categories
The category of all categories is an impossibility due to Russell's paradox.
Attempting to define a category that encompasses all possible categories inevitably leads to logical contradictions. This scenario mirrors the famous paradox of the "set of all sets that does not include itself".
As outlined by the renowned Russell's paradox, a hypothetical set of all sets $ U $ would theoretically contain every existing set.
This prompts a critical inquiry: Does the set of all sets $ U $ include itself as an element?
There are only two conceivable responses:
- Yes, the set $ U $ includes itself as an element. However, this scenario is untenable as no element can contain itself or list itself as an element.
- No, the set $ U $ does not include itself. Yet, this conclusion introduces another paradox because, as the set of all sets, it ought to include itself.
Either response results in a contradiction, underscoring the inherent logical flaws in the concept of a set of all sets.
Similarly, the notion of a "category of all categories" leads to comparable paradoxes within category theory.
Such a category would need to include itself, a requirement that is fundamentally inadmissible.
Therefore, defining the category of all categories ends up referring to itself in ways that introduce unavoidable contradictions.
This underscores that some theoretical constructs are too expansive to be seamlessly integrated within the structured confines of a coherent mathematical framework like category theory.
