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Comment by gwf | Hacker News Reader

The cardinality of the reals has been known to be strictly greater than the cardinality of the naturals since Cantor. What the Continuum Hypothesis considers is the cardinality of the reals and the cardinality of the power set of the naturals. The power set of another set is the set of unique subsets of the first set. If the first set has cardinality of N, then the power set has cardinality 2^N. Thus, the reals can be considered to be exponentially more dense than the naturals.

Some intuition: while there may be N unique stocks on the NYSE, there could in theory be 2^N unique mutual funds that are based on the different combinations of stocks.

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To be a bit more specific: the cardinality of the reals is equal to the cardinality of the power set of the naturals. This has also been known since the time of Cantor (read: late 19th century, at the dawn of set theory). CH states that no cardinal numbers exist between the cardinalities of the reals and the naturals[1].

Cantor believed CH to be true, but couldn’t prove it, and it was eventually (ca. 1960) proven to be independent of the usual (ZFC) axioms of set theory.

[1] Though CH could of course be stated in infinitely many equivalent ways, by replacing “reals” and “naturals” in this statement with any other pair of sets whose cardinalities are equal to those of the reals and naturals, respectively.

For (uncountably) infinitely many stupid examples, take any non-empty open subset of any connected, finite-dimensional (Hausdorff, second-countable) manifold with at least two points, and any infinite subset of the integers.

Can you simplify for us what exactly have the mathematicians in this article proven?

I get that they have proven some infinity A = some infinity B but can you tell us what these A and B are.

Also, isn't the cardinality of reals equinumerous with that of the cardinality of the power set of naturals?

Quoting from the article:

> Briefly, p is the minimum size of a collection of infinite sets of the natural numbers that have a “strong finite intersection property” and no “pseudointersection,” which means the subsets overlap each other in a particular way; t is called the “tower number” and is the minimum size of a collection of subsets of the natural numbers that is ordered in a way called “reverse almost inclusion” and has no pseudointersection.

In short, there are two infinities, p and t, that are implicitly defined by some characteristics. It was previously known that there was a relationship between them, but it was not suspected that there were, in fact, equal.

Considerable work is required to understand what these are, and what the result means.

And to answer your explicit question, yes, the reals can be put in 1-1 correspondence with 2^N, the power set of the naturals.

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