What would be the examples of a Set of point sets in $\mathbb R^2$ having the cardinality greater than that of real numbers?

Any stipulations over these point sets themselves may please be specified.

EDIT: My mistake which i correct in bold face as an edit. Motivation here is purely for informative purposes.

This is also related to a previous question of mine asking for the cardinality of all planar curves.

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    $\begingroup$ There isn't any. All subsets of $\mathbb{R}^2$ have cardinality at most $\#\mathbb{R}^2=\#\mathbb{R}$. $\endgroup$ – Alex Youcis Nov 11 '13 at 6:50
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    $\begingroup$ None, obviously. You could ask for natural point classes of subsets of $\mathbb R^2$ with larger size, and then there would be some examples. $\endgroup$ – Andrés E. Caicedo Nov 11 '13 at 6:51
  • $\begingroup$ Why did you delete your previous question, by the way? There was something potentially interesting there. $\endgroup$ – Andrés E. Caicedo Nov 11 '13 at 6:51
  • $\begingroup$ @AndresCaicedo I am looking for a more fitting word than 'curve', I am rephrasing it. $\endgroup$ – ARi Nov 11 '13 at 6:53
  • $\begingroup$ @AlexYoucis of course a I edit the question to rectify my mistake. $\endgroup$ – ARi Nov 11 '13 at 6:57

A natural example is the collection of Lebesgue measurable sets. The reason is that there are sets of measure $0$ and size continuum, and any subset of a measure $0$ set is also of measure $0$. This example has size $2^{|\mathbb R|}$. Note that, on the other hand, there are only $|\mathbb R|$ many Borel sets, we really needed that the measure space is complete to obtain a larger collection.

For a different example: There are sets of reals of size $\aleph_1$, the first uncountable cardinal. For example, fix a ("natural") way of coding binary relations on $\mathbb N$, using reals (for example, a real $r$ can code a sequence $a_0,a_1,\dots$ of natural numbers, using the continued fraction of $r$, and we can identify each natural with a pair of naturals, so this gives us a binary relation; there are many other natural ways, of course). Now pick a representative from each equivalence class where two reals are equivalent iff either neither codes a well-ordering, or else both code a well-ordering with the same order type. The collection of all subsets of this set has size $2^{\aleph_1}$. Of course, if the continuum hypothesis holds, this example has the same size as the previous one, but it is consistent that the size is different: It is strictly larger than $|\mathbb R|$, and it may be strictly smaller than $2^{|\mathbb R|}$.

  • $\begingroup$ The set of all Lebesgue measurable sets, thanks. $\endgroup$ – ARi Nov 11 '13 at 7:05

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