- #1

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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?

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- Thread starter Demystifier
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- #1

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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?

- #2

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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.

- #3

nomadreid

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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.

- #4

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But which branch of mathematics really deals withThe set of all real-valued functions has a higher cardinality.

- #5

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But which branch of mathematics really deals withallsuch functions? Functional analysis?

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

- #6

lavinia

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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.

- #7

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Are you saying that it is a proper class?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.

- #8

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Are you saying that it is a proper class?

Yes.

Now, I don't really know many situations where you are dealing with

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}##).

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