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Here I am collecting some comments I made on this question into a CW answer, which should be easier to read and respond to.

Is there a better argument for showing that the powerset of a transfinite set is vastly larger than the set? I'm, in particular, looking for an argument that I can explain to talented high school math students.

As others have touched on in their answers, it's somewhat debatable what the first sentence even means: there is no agreed upon meaning for "vastly larger" among infinite sets. One could even argue (as Nate Eldredge did above, rather cogently in my opinion) that "having larger cardinality" is a reasonable way to construe "vastly larger": thus the set of rational numbers is larger than that of the natural numbers, but the set of real numbers has larger cardinality so is "vastly larger".

To say much more than this goes into subtleties of set theory that the average working mathematician need not be conversant with. You might want to mention the Generalized Continuum Hypothesis -- which argues against any stronger sense of "vastly larger" -- and you might try expressing that in some precise sense GCH has been shown to be equiconsistent with the usual axioms of set theory. But to understand this one needs training in mathematical logic and exposure to the specific axioms of ZFC set theory, which again plenty of research mathematicians do not have. So I don't think it will make much sense to high school students, however bright. In particular I would advise against the sort of thing Asaf Karagila mentions in his comment:

[E]xplain vaguely that the power set can be almost anything larger, so it must be vastly larger.

The first part of this comment alludes to rather sophisticated ideas in logic and set theory -- i.e., for which ordinals $\alpha$ is the equation $2^{\aleph_0} = \aleph_{\alpha}$ equiconsistent with ZFC? (And to answer one must, presumably, mention things like cofinality...) On the other hand, the last part -- "so it must be vastly larger" -- is, as far as I understand it, a heuristic / intuition rather than a precise mathematical statement. One should be careful not to conflate these kinds of things.

At least, these are my opinions. If I were a professional set theorist I might have more confidence in my insight. But that's a key point too -- the OP sounds like he is not fully conversant with the post-Cohen age of axiomatic set theory. (That's not meant to be a stinging insult: it applies equally well to me.) I would tell anyone working with high school students -- especially bright ones -- to stick to things they know and understand very well.

3 deleted 8 characters in body

Here I am collecting some comments I made on this question into a CW answer, which should be easier to read and respond to. I said:

Is there a better argument for showing that the powerset of a transfinite set is vastly larger than the set? I'm, in particular, looking for an argument that I can explain to talented high school math students.

As others have touched on in their answers, it's somewhat debatable what the first sentence even means: there is no agreed upon meaning for "vastly larger" among infinite sets. One could even argue (as Nate Eldredge did above, rather cogently in my opinion) that "having larger cardinality" is a reasonable way to construe "vastly larger": thus the set of rational numbers is larger than that of the natural numbers, but the set of real numbers has larger cardinality so is "vastly larger".

To say much more than this goes into subtleties of set theory that the average working mathematician need not be conversant with. You might want to mention the Generalized Continuum Hypothesis -- which argues against any stronger sense of "vastly larger" -- and you might try expressing that in some precise sense GCH has been shown to be equiconsistent with the usual axioms of set theory. But to understand this one needs training in mathematical logic and exposure to the specific axioms of ZFC set theory, which again plenty of research mathematicians do not have. So I don't think it will make much sense to high school students, however bright. In particular I would advise against the sort of thing Asaf Karagila mentions in his comment:

[E]xplain vaguely that the power set can be almost anything larger, so it must be vastly larger.

The first part of this comment alludes to rather sophisticated ideas in logic and set theory -- i.e., for which ordinals $\alpha$ is the equation $2^{\aleph_0} = \aleph_{\alpha}$ equiconsistent with ZFC? (And to answer one must, presumably, mention things like cofinality...) On the other hand, the last part -- "so it must be vastly larger" -- is, as far as I understand it, a heuristic / intuition rather than a precise mathematical statement. One should be careful not to conflate these kinds of things.

At least, these are my opinions. If I were a professional set theorist I might have more confidence in my insight. But that's a key point too -- the OP sounds like he is not fully conversant with the post-Cohen age of axiomatic set theory. I would tell anyone working with high school students -- especially bright ones -- to stick to things they know and understand very well.

2 added 162 characters in body

Here I am collecting some comments I made on this question into a CW answer, which should be easier to read and respond to. I said:

"

Is there a better argument for showing that the powerset of a transfinite set is vastly larger than the set? I'm, in particular, looking for an argument that I can explain to talented high school math students." students.

As others have touched on in their answers, it's somewhat debatable what the first sentence even means: there is no agreed upon meaning for "vastly larger" among infinite sets. One could even argue (as Nate Eldredge did above, rather cogently in my opinion) that "having larger cardinality" is a reasonable way to construe "vastly larger": thus the set of rational numbers is larger than that of the natural numbers, but the set of real numbers has larger cardinality so is "vastly larger".

To say much more than this goes into subtleties of set theory that the average working mathematician need not be conversant with. You might want to mention the Generalized Continuum Hypothesis -- which argues against any stronger sense of "vastly larger" -- and you might try expressing that in some precise sense GCH has been shown to be equiconsistent with the usual axioms of set theory. But to understand this one needs training in mathematical logic and exposure to the specific axioms of ZFC set theory, which again plenty of research mathematicians do not have. So I don't think it will make much sense to high school students, however bright. In particular I would advise against the sort of thing Asaf Karagila mentions in his comment: "explain

[E]xplain vaguely that the power set can be almost anything larger, so it must be vastly larger"larger.

The first part of this comment alludes to rather sophisticated ideas in logic and set theory -- i.e., for which ordinals $\alpha$ is the equation $2^{\aleph_0} = \aleph_{\alpha}$ equiconsistent with ZFC? (And to answer one must, presumably, mention things like cofinality...) On the other hand, the last part -- "so it must be vastly larger" -- is, as far as I understand it, a heuristic / intuition rather than a precise mathematical statement. One should be careful not to conflate these kinds of things.

At least, these are my opinions. If I were a professional set theorist I might have more confidence in my insight. But that's a key point too -- the OP sounds like he is not fully conversant with the post-Cohen age of axiomatic set theory. I would tell anyone working with high school students -- especially bright ones -- to stick to things they know and understand very well.