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Topic: Higher-order infinities question
Replies: 14   Last Post: Jul 3, 2014 6:24 PM

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William Elliot

Posts: 1,489
Registered: 1/8/12
Re: Higher-order infinities question
Posted: Jul 1, 2014 10:42 PM
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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. "Pseudo-line" is his description.
> Consider the set S of all "pseudo-lines" in |R x |R.
> A "pseudo-line" 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 "pseudo-lines" 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 pseudo-line 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.

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