How to build linear order on $\mathcal P({\bf N})$

Question

How to build linear order on $\mathcal P({\bf N})$?

Had idea about inclusion relation, but it does not satisfy linearity.

Answer 1

Hint:

Given two sets of natural numbers, $A,B$ we can say that $A\prec B$ if the minimal element of $A\triangle B$ is a member of $A$.

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Another alternative is to find an injective function from $\mathcal P(\Bbb N)$ into $[0,1]$ and use that to define the order.

A third alternative is to think about it as infinite binary sequences, and order them lexicographically.