Is there an uncountable family of infinite parts of $\mathbb N$ with two to two intersection finite?
If the two to two intersections are empty, it is false; if not $\mathbb N$ would be uncountable.
What about this case?
Is there an uncountable family of infinite parts of $\mathbb N$ with two to two intersection finite?
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hmakholm left over Monica
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Yes. Look up "almost disjoint families". – bof Jan 21 '18 at 12:02
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Sets with finite intersection are called "almost disjoint".
View functions $\mathbb N\to\mathbb N$ in the usual ways as subsets of $\mathbb N\times\mathbb N$. This allows us to speak about whether two functions are almost disjoint.
If we have any countable family of functions, it is an easy diagonalization exercise to produce a new function that is almost disjoint from each of them.
On the other hand, Zorn's lemma proves that there is a maximal family of pairwise almost disjoint functions. By the previous paragraph, such a family cannot be countable.
This gives us an uncountable family of pairwise almost disjoint infinite subsets of $\mathbb N\times\mathbb N$. Since $\mathbb N\times\mathbb N$ is in bijective correspondence with $\mathbb N$, we can map this back to an uncountable family of pairwise almost disjoint subsets of $\mathbb N$.
hmakholm left over Monica
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The accepted answer to the duplicate question Asaf identified is better and simpler than this one; it does not depend (essentially) on choice. – hmakholm left over Monica Jan 21 '18 at 13:53