
Re: Higherorder infinities question
Posted:
Jul 1, 2014 10:42 PM


On Tue, 1 Jul 2014, Port563 wrote:
> The set of all points on R x R has the cardinality of R, i.e. the > smallest uncountable infinity (if one assumes the Continuum hypothesis). > > The following question was put by me in the course of preparing my student > for his next mathematical leap. "Pseudoline" is his description. > > Consider the set S of all "pseudolines" in R x R. > > A "pseudoline" is an ordered collection of points on R x R.
What does that mean? Any subset of R^2 that's linearly ordered?
> There is no requirement for point adjacency (i.e. connectivity or what some > might term continuousness). > > To illustrate: Consider a subset T of R x R with n points in it. There > will exist exactly n! distinct "pseudolines" made up solely of all the > points in T. > > So, what is the cardinality of S? > (a) 2 ^ R, i.e. the cardinality of R', the set of all subsets of R > (b) 2 ^ (2 ^ R), i.e. the cardinality of R'', the set of all subsets of R' > (c) Something else.
Obviously (c) because the definition of pseudoline is too vague to actully be anything other than something else.
c = R, conventionsl definition; c = 2^(aleph_0), CH.
c_A = { <= : <= linear order for A }
For all A subset R, 1 <= c_A <= c^2
2^c <= sum{ c_A : A subset R } <= 2^c * c^2 = 2^c
Assuming CH, there are 2^c functions from R into R.

