Application of sets with higher cardinality

[Demystifier]

Sets with cardinality $2^{\aleph_0}$, that is, with cardinality of the set of real numbers, obviously have many applications in other branches of mathematics outside of pure set theory. For example, real any complex analysis is completely based on such sets.

How about higher cardinality? Is there a branch of mathematics (outside of pure set theory) which uses sets with cardinality larger than that of reals?

 

[PeroK]

Sets with cardinality $2^{\aleph_0}$, that is, with cardinality of the set of real numbers, obviously have many applications in other branches of mathematics outside of pure set theory. For example, real any complex analysis is completely based on such sets.

How about higher cardinality? Is there a branch of mathematics (outside of pure set theory) which uses sets with cardinality larger than that of reals?

The set of all real-valued functions has a higher cardinality.

 

[nomadreid]

For applications of Lebesgue integrals, they are handy: http://en.wikipedia.org/wiki/Solovay_model

In various algebraic fields, Kurepa's hyptothesis is purported to be useful (http://projecteuclid.org/euclid.rml/1081878063), but should be exercised with care, because its failure is equiconsistent with the existence of uncountable inaccessible cardinals.

 

[Demystifier]

The set of all real-valued functions has a higher cardinality.

But which branch of mathematics really deals with all such functions? Functional analysis?

 

[PeroK]

But which branch of mathematics really deals with all such functions? Functional analysis?

Yes, that's the basis of functional analysis. The study of sets of functions or operators on a topological space.

 

[lavinia]

Sets with cardinality $2^{\aleph_0}$, that is, with cardinality of the set of real numbers, obviously have many applications in other branches of mathematics outside of pure set theory. For example, real any complex analysis is completely based on such sets.

How about higher cardinality? Is there a branch of mathematics (outside of pure set theory) which uses sets with cardinality larger than that of reals?

In Category Theory, categories can be of any cardinality but generally they are too large to even be sets.
For instance, the category of vector spaces and linear maps is too large to be a set.

Different assumptions about the cardinality of the Reals, imply different results in Analysis. This is not exactly your question since whatever the assumption it is still about the cardinality of the Real numbers.

 

[Demystifier]

In Category Theory, categories can be of any cardinality but generally they are too large to even be sets.
For instance, the category of vector spaces and linear maps is too large to be a set.

Are you saying that it is a proper class?

 

[micromass]

Are you saying that it is a proper class?

Yes.

Now, I don't really know many situations where you are dealing with all subsets or functions on $\mathbb{R}$. But there are some important structures with cardinality bigger than $2^{\aleph_0}$. For example

1) The Stone-Cech compactification of $\mathbb{N}$ has cardinality $2^{2^{\aleph_0}}$. This is studied in topology.
2) The set of all Lebesgue-measurable subsets of $\mathbb{R}$ has cardinality $2^{2^{\aleph_0}}$. This shows up occasionally in analysis, although using the Borel sets is more popular (the Borel sets have cardinality $2^{\aleph_0}$).