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The potency axiom asserts that the class of all subsets of any set is itself represented by a set, usually called the first set's "power set".

The potency axiom is pretty much universally acceepted in formal systems of set theory; it is not controversial like the axiom of choice or the continuum hypothesis.

# Potency Axiom -- InnocentAxiom, or the Villain behind the Continuum Hypothesis?

As Gorgonzola points out, the potency axiom is universally accepted, and "obviously" true. However, together with the axiom of infinity (which guarantees us the existence of sets we may call "infinite"), it forces us into the bogs of the continuum hypothesis and the generalized continuum hypothesis. It can also be accused of being indirectly responsible for the whole axiom of choice mess.

This seemingly-peaceful pair of axioms is, in fact, the root of all evil in set theory. Most evil, anyway. For one thing, without it we could easily believe in a countable model of mathematics. Of course, giving it up would mean doing mathematics without the real numbers, which is probably not a good idea. I am not seriously suggesting doing away with potency -- just pointing out that with its great power (apologies for the pun) should have come a bit more responsibility!

The axiom of infinity gives us an infinite set. Say we get some countable set N. The potency axiom allows us to form PN or 2N (take your notational pick), consisting of all the sets of N. The thing is, we've no idea what these sets all are!

Indeed, our first intuition -- "give a method to list all subsets of N" -- fails. NOT (just) because we have no method -- but because there can be no list! (See Cantor's theorem, or the proof that for every set there is a larger set, aka Cantor diagonalization and a few other names). We literally have no idea what a "generic subset" of N would look like.

The result is complete and utter havoc:

The continuum hypothesis
We've no idea just how many elements there are of PN.
Normal real numbers
We have great difficulty with things like normal numbers. Almost every real number is normal. But of the numbers we know which we didn't expressly design to be normal, we don't know of any that are indeed normal. Indeed, we don't know if there exists an algebraic number (i.e. a number which we really can specify exactly using only integers) which is normal. So we'd have the problem no matter what -- but without potency, we wouldn't be so embarassed about not being able to find one algebraic number which belongs to such a huge set...
Transcendental numbers
Almost every number is transcendental. We don't know of a good many numbers whether they are transcendental or not. Indeed, in the deathmatch: e vs pi we see that we don't even know if e+π is rational or not. And these are 2 numbers which we know "explicitly"! They'll be on anybody's attempted "list of all the real numbers"! The potency axiom generates numbers which will be on nobody's list -- we know nothing of (almost) all of these numbers!
Turing machines with oracles
A Turing machine with an "oracle" is a Turing machine which is allowed to call a "magical subroutine" fA such that fA(x)=1 iff x∈A, for some subset A∈PN. Every Oracle for a nonrecursive set A helps us compute membership in a great many other uncomputable sets (e.g. it is trivial to compute membership in 1+A, given the Oracle fA). Is there a "universal Oracle", a set A which will make all sets in PN computable? NO -- there are only countably many Turing machines using any given Oracle, so not only is there no universal Oracle, but there are also very many "Turing degrees". And since almost all of them are completely generic, we can know nothing about them.
The list goes on and on.

The paradox is that even before the axiom of choice, the axiom of potency is necessary to do mathematics, yet of necessity also introduces the unknowable "monsters" that rocked mathematics in the first half of the 20th century.

The potency axiom creates the problematic sets; "all" the axiom of choice does is drag them out into the light of day.

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