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Prove that $\mathbb{N}$ contains uncountably many infinite subsets $(N_{\alpha})_{\alpha \in \mathbb{R}}$ such that $N_{\alpha}\cap N_{\beta}$ is finite if $\alpha \neq \beta$.

I think we have to assume that $\mathbb{N}$ has only countably many subsets of that kind and then arrive at a contradiction due to the finiteness of the intersection of any two distinct subsets between them. But I am unable to find the correct argument. What will be the possible approach here?

Asaf Karagila
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  • It looks like you want to show that there is an uncountable collection of infinite subsets of natural numbers such that any two distinct subsets from this collection have finite intersection . Is this your problem ? Your indexing is confusing to me . – user439545 Jul 01 '17 at 06:04
  • Also https://math.stackexchange.com/questions/linked/162387 has several other duplicates. – Asaf Karagila Jul 01 '17 at 08:04

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