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AlephNumbers

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**Lately I have been attempting (and failing miserably at) whatever sample Putnam questions I can find on the internet. Here is my latest endeavor. I found this question on the Kansas State University website, so I think I am allowed to post it. I must warn you that I know almost nothing about set theory.**

Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

I know that the set of all integers is "countably" infinite.

Let I be the set containing all integers. The set I contains an infinite number of subsets each containing any number of the infinite number of elements in the set I, so the number of subsets of I is uncountable. However, any two subsets of I can be shown to have a finite intersection.

For example G={1,2,3,4} , F={3,4,5}

G and F have 2 elements in common; the elements 3 and 4 belong to both sets

The answer is yes.

I know that the set of all integers is "countably" infinite.

Let I be the set containing all integers. The set I contains an infinite number of subsets each containing any number of the infinite number of elements in the set I, so the number of subsets of I is uncountable. However, any two subsets of I can be shown to have a finite intersection.

For example G={1,2,3,4} , F={3,4,5}

G and F have 2 elements in common; the elements 3 and 4 belong to both sets

The answer is yes.

I don't know if this would be a sufficient answer. It's awfully wordy and does not contain much mathematical rigor.