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BTW: Why then does wikipedia say that the cardinality of the set of all real numbers (denoted c and called cardinality of the continuum) is strictly greater th

BTW: Why then does wikipedia say that the cardinality of the set of all real numbers (denoted c and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ 0 'aleph-naught')? UPD: I get it, they are actually talking about a third set in the article in a way that's not immediately apparent.

schoen 9y ago
Three things: First, this research didn't show that all infinities are equal, it just showed that two particular ones are equal. Indeed, in Cantor's formulation, the powerset of any set is larger than the original set, which immediately shows that there is no largest infinite set. Second, this research wasn't about ℵ₀ and c at all, but about another pair. Third, the issue about ℵ₀ and c isn't about whether they're equal, but whether there is or isn't a third infinity that's larger than ℵ₀ and smaller than c. It's already known that both answers (yes and no) are consistent with standard mathematical axioms. Edit: this result does have some connection with the continuum hypothesis, but I don't know exactly what that connection is. However, these researchers definitely didn't claim either to prove or disprove the continuum hypothesis.
gwf 9y ago
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.
- samoright 9y ago
  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?
  - ColinWright 9y ago
    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.
- jasomill 9y ago
  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.

schoen 9y ago

Three things: First, this research didn't show that all infinities are equal, it just showed that two particular ones are equal. Indeed, in Cantor's formulation, the powerset of any set is larger than the original set, which immediately shows that there is no largest infinite set. Second, this research wasn't about ℵ₀ and c at all, but about another pair. Third, the issue about ℵ₀ and c isn't about whether they're equal, but whether there is or isn't a third infinity that's larger than ℵ₀ and smaller than c. It's already known that both answers (yes and no) are consistent with standard mathematical axioms. Edit: this result does have some connection with the continuum hypothesis, but I don't know exactly what that connection is. However, these researchers definitely didn't claim either to prove or disprove the continuum hypothesis.

gwf 9y ago

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.

samoright 9y ago
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?
- ColinWright 9y ago
  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.
jasomill 9y ago
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.