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The power set of a set is the set of all distinct subsets of the set, with the smallest being {} and the largest the set itself. This means that if S is an infinite set, the power set pow(S) must contain S as its largest element. But if S is infinite then pow(S) must also be infinite, which means that pow(S) cannot contain a largest element. As the existence of pow(S) leads to a contradiction, pow(S) cannot exist.

What is the standard way of dealing with the contradiction inherent in the definition of an infinite power set?

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    *Largest* ...with respect to **what**? Set inclusion? Of course there can be a largest set then in $\;P(S)\;$ (in the sense that it isn't contained in any other element of the power set), not matter it is an infinite set. – 2017-02-08
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    "pow(S) must also be infinite, *which means that* pow(S) cannot contain a largest element": this is **false** (or, to be precise, it's a **non sequitur**). There are many infinite partial orders with a maximum. For instance, the interval $[0,1]$ with the usual order. – 2017-02-08
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    "largest" refers to the set within the power set that contains the most elements. I am assuming that the set S is ordered and has a least element, which means that it is possible to set up an ordering within the power set based on the ordering in S, such that for any n > 0 the first $2^n$ elements in pow(S) will be the power set of the first n elements of S, with the set containing the first n elements being placed last. – 2017-02-08

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There is no contradiction. It is entirely possible for an infinite set to have a largest element (with respect to some ordering); indeed, the example of the power set of an infinite set (with respect to the partial ordering $\subseteq$) simply proves that this is possible. For a perhaps more familiar example, consider the set $[0,1]$ of all real numbers between $0$ and $1$ (inclusive): this set is infinite, but it has a largest element (with respect to the usual ordering of real numbers), namely $1$. For another example, take the set $X=\mathbb{N}\cup\{\infty\}$, where $\infty$ is some symbol that we have declared to be greater than every element of $\mathbb{N}$. Then $X$ is infinite, and $\infty$ is its largest element.