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For each natural number $n\in\mathbb{N_0}$ list all the sets with at most $n$ elements and with maximal element as and including $n$, in numerical order. Now all the elements of $P(N)$ will be listed as n gets larger. Label each set of natural numbers with another natural number. Why is the above not a 1:1 correlation between N and P(N)? The sequence starts with the empty set, then $\{1\}, \{2\}, \{1,2\}, \{3\}, \{1,3\}, \{2,3\}, \{1,2,3\}, \{4\}, \{1,4\}, \{2,4\}, \{3,4\}, \{1,2,4\}, \{1,3,4\}, \{2,3,4\}, \{1,2,3,4\}$ …

Asaf Karagila
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Chris Ang
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  • The question has been answered I guess, but could you make more clear how your assignment looks like. Which number gets mapped to which concretely specified subset? Do you first map from ${\mathbb N}$ to ${\mathbb N}\times{\mathbb N}$ (is this injective?) and then to the power set? Of if not, have you computed which number gets replaced in the initial enumberations of sets of subsets? – Nikolaj-K Dec 11 '14 at 09:59

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Note that the all of the sets that you're mapping to are finite, so what you've listed is a one-to-one relationship between $\mathbb{N}$ and the finite subsets of $\mathcal{P}(\mathbb{N})$. That is completely possible. The real problem comes when you try to map $\mathbb{N}$ to the infinite subsets of $\mathcal{P}(\mathbb{N})$...

Michael Blakeman
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