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Not all uncountable infinities are the same cardinality.

So does this article prove that two uncountable infinities are equal? Can you tell us precisely what two entities have been proven equal in this article?

I understand aleph-0, aleph-1 and 2^aleph-0 and I also understand that if continuum hypothesis is true, then aleph-1 = 2^aleph-0. Is this what these mathematicians have proven, or have they proven something else?

The paper proves that the smallest cardinality of a set of integers (such that every finite subset has infinite intersections and has no almost-intersections) is equal to the smallest cardinality of a tower.

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