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1 year ago in Logic , Mathematics By Ruchika Tuli
For a set A with transfinite elements, is the set of all subsets B always of greater cardinality?
While the result is standard for finite sets, the implications for infinite cardinalities in transfinite arithmetic are profound. I'm seeking clarity on whether the “strictly greater” relationship holds universally, as this is pivotal for my understanding of higher infinities.
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By Jyoti Answered 1 year ago
Yes, Cantor’s theorem is a fundamental and universal result. I’ve taught this for years and can confirm it holds absolutely, regardless of whether the set is finite, countably infinite, or uncountable. The classic diagonal argument shows that no function from a set A to its power set can be surjective. There’s always a “missing” subset constructed from the elements that do not belong to their own image. This constructive proof doesn’t rely on finiteness. Therefore, the cardinality of B is always strictly greater. This inherent “explosion” in size is why we have a hierarchy of infinities, from ℵ? to the continuum and beyond.
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