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It's well known that $\mathbb{R}$ has the same cardinality as $\mathcal{P}(\mathbb{N})$; but I would fain know if there is a way to construct $(\mathbb{R}, +,\cdot, \leq )$ using only definitions that rely upon $\mathcal{P}(\mathbb{N})$'s elements and their respective properties.

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    Have you seen the [Dedekind cuts](http://math.stackexchange.com/q/95271/)?2012-01-25
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    @Srivatsan, but Dedekind cuts need the construction of rationals first. I want a direct construction.2012-01-25
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    The real numbers in interval $(0,1\rangle$ can be described using infinite binary expansions, which are in bijective correspondence with infinite subsets of $\mathbb N$.2012-01-25
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    Presumably, you want to do it so that you do not define an equivalence relationship on $\mathcal{P}(\mathbb{N})$, but essentially you want to define operations $\oplus$ and $\otimes$ on $\mathcal{P}(\mathbb{N})$ so that $(\mathcal{P}(\mathbb{N}),\oplus,\otimes)\cong (\mathbb R,+,\cdot)$?2012-01-25
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    @ThomasAndrews, exactly.2012-01-25
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    Have you read the following essay by Tim Gowers? http://www.dpmms.cam.ac.uk/~wtg10/decimals.html2012-01-25
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    I'm extending my comment of January 25: The following paper (in German) describes such a construction in detail: http://www.math.ethz.ch/~blatter/Dualbrueche_2.pdf2012-02-11
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    Not quite what you're asking for, but it's possible to construct $\mathbb R$ solely from $\mathbb Z$: [Wikipedia](http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_the_group_of_integers). The construction draws on the idea that drawing a (non-vertical) line through $0$ on an infinite computer display ($\mathbb Z \times \mathbb Z$) corresponds to choosing a slope - which can be any real number2012-02-11
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    For what it's worth, while interesting I don't see much point in such construction. The fact that we can prove that $\mathbb R$ has the same cardinality as $\mathcal P(\mathbb N)$ allows us to define it using the rationals and transfer the structure at will.2012-02-19
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    This may help: http://books.google.com/books?id=tXiVo8qA5PQC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false2012-02-20
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    [Another link](http://maths.mq.edu.au/~street/EffR.pdf)2012-02-20

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