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Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion.

What is the cardinality of ${\frak P}$, and what's the largest cardinality that a chain in ${\frak P}$ can have?

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The prime ideals you ask about are the duals of the non-principal ultrafilters on $\omega$. They are all maximal, so the largest cardinality of a chain is $1$. The total number of such ultrafilters is $2^{2^{\aleph_0}}$, the same as the total number of subsets of the real line.

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