# Subset of $(P(\mathbb{N}),\subseteq)$

How one can find a subset of $(P(\mathbb{N}),\subseteq)$ without maximal element? ($P(\mathbb{N})$ is the power set of $\mathbb{N}$)

I think that I need point an antichain. Right?

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The empty set ? – Dan Brumleve Dec 27 '12 at 9:01

First note that every non-empty finite subset has a maximal element. Such subset has to be infinite.

If you require the subset to be a chain then $\Big\{\{j\in\mathbb N\mid j<k\}\mid k\in\mathbb N\Big\}$ is such subset. Otherwise just taking all the finite subsets would work.

You may also note that the collection of the co-infinite, i.e. $\{A\subseteq\mathbb N\mid\mathbb N\setminus A\text{ is infinite}\}$ is also without a maximal element.

Lastly, an antichain will not work. Every element is maximal within the antichain.

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To be more explicit: Every nonempty finite subset has a maximal element. – Hagen von Eitzen Dec 27 '12 at 9:29

Just consider the collection of finite subsets of $\mathbb{N}$. There cannot be a $\subseteq$-maximal element because given a finite subset $A$ of $\mathbb{N}$, you can find a finite subset $B$ of $\mathbb{N}$ that properly contains $A$.

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