The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that requires forming the set of all partial chains first, hence the powerset axiom $\text{P}$, so that the choosing can occur. And it is the well ordering of the continuum that leads to most "paradoxes" usually blamed on $\text{AC}$.

People objected to completing a process of picking elements from 'too many' sets to form a set, why not to completing a process of creating too many sets to form a set?

Now concerning the consequences. There are many books detailing which forms of $\text{AC}$ are needed for which theorems of analysis and algebra, but I could not find any detailed accounts for $\text{P}$. There is a bold claim that "we can do practically all mathematics without the axiom of powerset" here, but the references are not particularly illuminating, and in some sense this can be said even of Peano arithmetic.

This is an interesting paper, but it deals more with logical and set-theoretic consequences of dropping $\text{P}$ than with its effect on 'ordinary' mathematics. It turns out that collection doesn't follow from replacement without $\text{P}$, and using it makes for a better theory denoted $\text{ZFC}^-$. Then there is an obvious issue of not being able to form real numbers, but there are similar 'catastrophes' if at least countable forms of $\text{AC}$ are not assumed. So in fairness let's assume that collection of at most countable subsets of a set is a set (such a set is formed by running only simple inductive processes "in parallel", like real numbers from decimal expansions).

How much of classical mathematics survives in $\text{ZFC}^-+\text{P}_\omega$ or similar theory? Is it so little that it's not worth considering?

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    $\begingroup$ Allowing countably many power sets from the integers, but no further, is a cumbersome axiom. All the "usual paradoxes" about the real numbers follow from this theory anyway. So you haven't gained too much by restricting the power set operations like that. Of course, one thing you do miss out is Borel determinacy which requires $\rm P_{\omega_1}$, as was shown by Harvey Friedman. $\endgroup$ – Asaf Karagila Aug 21 '14 at 3:33
  • $\begingroup$ @Asaf Karagila You are right, I didn't think of that. Vitali's construction doesn't need powerset. Equivalence classes form a set of continuum cardinality and AC produces non-measurable set by picking representatives. So there is no point restricting powerset from this perspective, it's AC's fault after all. $\endgroup$ – Conifold Aug 21 '14 at 17:45
  • $\begingroup$ As I wrote before that. It's not quite AC's fault. It's the fact that infinite objects don't obey our intuition because our intuition is laced with finite things. So if any axiom is to blame, it's the axiom of infinity really. $\endgroup$ – Asaf Karagila Aug 21 '14 at 17:48
  • $\begingroup$ @Asaf Karagila I thought of that after you said that things can be non-constructive even w/o AC. But the kinds of non-constructive objects in ZF (from excluded middle mostly), which already has infinity, are nowhere near as ghost like as non-measurable sets or non-linear additive functions. So there does seem to be a leap in 'paradoxality' from uncountable AC. But limiting the size of infinity doesn't help apparently, at least not without also limiting how it is manipulated. $\endgroup$ – Conifold Aug 21 '14 at 19:52

For the purpose of applying set theory as a foundational subject, one must be able to express typical mathematical reasoning, such as

  • Let $r$ be a real number...
  • Let $n$ be an integer...
  • Let $a$ be a sequence of real numbers...
  • Let $f$ be a real-valued function of the reals...
  • Let $S$ be a subset of the reals...

And in general, if you have a kind of thing $T$, mathematicians tend to become interested in sets of things, functions whose domain or range consists of things, predicates on things, and so forth.

The usual application of set theory requires there to be a set encompassing all of the choices. We need a set of real numbers, a set of integers, a set of sequences of real numbers, a set of real-valued functions of the reals, a set of subsets of the reals, a set of sets of things, a set of functions whose domain or range consists of things, a set of predicates on things, and so forth.

And thus, you need power sets.

Note that if you used function sets rather than power sets as your basic idea, you'd still get something that adequately serves the purpose of power sets in the form of the set of functions $S \to \{ 0, 1 \}$.

If you give up power sets, you have to find a whole new approach.

Incidentally, while rejecting the axiom of powersets kills typical mathematical constructions, rejecting the axiom of choice does not: even without the axiom of choice, you can still form arbitrary infinite products of nonempty sets

$$ \prod_{i \in I} S_i $$

it's just that you no longer have a guarantee that this product is nonempty. You only kill reasoning that depends on the construction giving nontrivial results.

  • $\begingroup$ I can't really think of a use for the set of all real-valued functions of the reals in classical analysis, indeed the whole theory of measurable functions is devoted to casting out such large sets. Cartesian products are a problem, existence of countable ones at least should be postulated separately. $\endgroup$ – Conifold Aug 21 '14 at 1:00
  • $\begingroup$ @Conifold: To single out the measurable functions from all functions requires first being able to speak of all functions. Also, note that if one believes only measurable functions exist, that is no reason to believe there is a not a set of all measurable functions. And if only measurable functions exist in some set-theoretic universe, then the set of all functions is the set of measurable functions. $\endgroup$ – Hurkyl Aug 21 '14 at 1:21
  • $\begingroup$ @ Hurkyl Measurable functions are limits of step functions with rational ends and values, so they can be proved to form a set much like real numbers. And after that all 'useful' $sets$ of functions in classical analysis are subsets of them. When there is a need for further sets of functions it's not for more general real valued functions, but distributions. We don't even have to believe that non-measurable functions don't exist, only that their totality isn't a set. $\endgroup$ – Conifold Aug 21 '14 at 17:51

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