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Comment by samoright | Hacker News Reader

> In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size

I thought there are only two types of infinity and Cantor already proved that they are different.

* Uncountable infinity which is the cardinality of the set of real numbers

* Countable infinity which is the cardinality of the set of integers

Cantor has already proved that uncountable infinity is larger than countable infinity.

Is this article claiming that mathematicians have proved that these two infinities are equal now? Doesn't that contradict Cantor's proof? What's going on here? Is the Cantor's proof flawed or have we introduced a contradiction to mathematics?

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If you take the power set of an infinite set, you /always/ get a cardinality bigger than the original set. So there are an infinite number of infinities.

This is a good point. But to be super pedantic, there isn't a size of infinity large enough to describe how many sizes of infinity there are.

Why not? The aleph numbers have a bijection with naturals.

See this answer on the mathematics stack exchange (it's really a book excerpt, but I don't know where else to find it.) https://math.stackexchange.com/a/5390/220797

This article is talking about the cardinality of two very specific sets. Before, many believed that t > p, but many believed that this was not provable in ZFC. Both sets contain only sets of integers, so their cardinality is bounded above by the cardinality of the real numbers (the continuum). Some people believed that this result was related to the Continuum hypothesis, and so was only provable one way or the other if you assume the CH or its negative.

As it turns out, both sets have the same cardinality (that of the real numbers) AND it is provable in ZFC.

The title is a little misleading, its really saying that two infinite sets are the same size, but prior to this their cardinality was unknown, and it wasn't even known if the cardinality was provable in ZFC.

There are infinite types of infinity not just two. The result is that they found that two types of infinity that were long assumed to be different turned out to be the same.

This is more interesting than it sounds because these infinities are in between (but possibly equal to one of) the size of the natural numbers and the size of the real numbers. There are hard limits on what can be known about such infinities in the usual mathematical framework (zfc): it's impossible to prove if an infinity which is strictly between the two exists. The hypothesis that no such infinity exists is called "the continuum hypothesis".

Writing this on mobile, but I hope I've made sense...

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.

My question exactly. I still don't get it.

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